YES Problem 1: (VAR v_NonEmpty:S IL:S L:S M:S N:S V:S V1:S V2:S X:S X1:S X2:S X3:S X4:S) (RULES a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) a__U62(tt) -> tt a__U62(X:S) -> U62(X:S) a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) a__U72(tt,L:S) -> s(a__length(mark(L:S))) a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) a__U81(tt) -> nil a__U81(X:S) -> U81(X:S) a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) a__isNatList(nil) -> tt a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) a__take(X1:S,X2:S) -> take(X1:S,X2:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) mark(U62(X:S)) -> a__U62(mark(X:S)) mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) mark(tt) -> tt mark(zeros) -> a__zeros ) (STRATEGY INNERMOST) Problem 1: Dependency Pairs Processor: -> Pairs: A__U41(tt,V2:S) -> A__U42(a__isNatIList(V2:S)) A__U41(tt,V2:S) -> A__ISNATILIST(V2:S) A__U51(tt,V2:S) -> A__U52(a__isNatList(V2:S)) A__U51(tt,V2:S) -> A__ISNATLIST(V2:S) A__U61(tt,V2:S) -> A__U62(a__isNatIList(V2:S)) A__U61(tt,V2:S) -> A__ISNATILIST(V2:S) A__U71(tt,L:S,N:S) -> A__U72(a__isNat(N:S),L:S) A__U71(tt,L:S,N:S) -> A__ISNAT(N:S) A__U72(tt,L:S) -> A__LENGTH(mark(L:S)) A__U72(tt,L:S) -> MARK(L:S) A__U91(tt,IL:S,M:S,N:S) -> A__U92(a__isNat(M:S),IL:S,M:S,N:S) A__U91(tt,IL:S,M:S,N:S) -> A__ISNAT(M:S) A__U92(tt,IL:S,M:S,N:S) -> A__U93(a__isNat(N:S),IL:S,M:S,N:S) A__U92(tt,IL:S,M:S,N:S) -> A__ISNAT(N:S) A__U93(tt,IL:S,M:S,N:S) -> MARK(N:S) A__ISNAT(length(V1:S)) -> A__U11(a__isNatList(V1:S)) A__ISNAT(length(V1:S)) -> A__ISNATLIST(V1:S) A__ISNAT(s(V1:S)) -> A__U21(a__isNat(V1:S)) A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) A__ISNATILIST(cons(V1:S,V2:S)) -> A__U41(a__isNat(V1:S),V2:S) A__ISNATILIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) A__ISNATILIST(V:S) -> A__U31(a__isNatList(V:S)) A__ISNATILIST(V:S) -> A__ISNATLIST(V:S) A__ISNATLIST(cons(V1:S,V2:S)) -> A__U51(a__isNat(V1:S),V2:S) A__ISNATLIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) A__ISNATLIST(take(V1:S,V2:S)) -> A__U61(a__isNat(V1:S),V2:S) A__ISNATLIST(take(V1:S,V2:S)) -> A__ISNAT(V1:S) A__LENGTH(cons(N:S,L:S)) -> A__U71(a__isNatList(L:S),L:S,N:S) A__LENGTH(cons(N:S,L:S)) -> A__ISNATLIST(L:S) A__TAKE(0,IL:S) -> A__U81(a__isNatIList(IL:S)) A__TAKE(0,IL:S) -> A__ISNATILIST(IL:S) A__TAKE(s(M:S),cons(N:S,IL:S)) -> A__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) A__TAKE(s(M:S),cons(N:S,IL:S)) -> A__ISNATILIST(IL:S) MARK(U11(X:S)) -> A__U11(mark(X:S)) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X:S)) -> A__U21(mark(X:S)) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> A__U31(mark(X:S)) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> A__U41(mark(X1:S),X2:S) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> A__U42(mark(X:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> A__U51(mark(X1:S),X2:S) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> A__U52(mark(X:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S)) -> A__U61(mark(X1:S),X2:S) MARK(U61(X1:S,X2:S)) -> MARK(X1:S) MARK(U62(X:S)) -> A__U62(mark(X:S)) MARK(U62(X:S)) -> MARK(X:S) MARK(U71(X1:S,X2:S,X3:S)) -> A__U71(mark(X1:S),X2:S,X3:S) MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U72(X1:S,X2:S)) -> A__U72(mark(X1:S),X2:S) MARK(U72(X1:S,X2:S)) -> MARK(X1:S) MARK(U81(X:S)) -> A__U81(mark(X:S)) MARK(U81(X:S)) -> MARK(X:S) MARK(U91(X1:S,X2:S,X3:S,X4:S)) -> A__U91(mark(X1:S),X2:S,X3:S,X4:S) MARK(U91(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> A__U92(mark(X1:S),X2:S,X3:S,X4:S) MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> A__U93(mark(X1:S),X2:S,X3:S,X4:S) MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> A__ISNAT(X:S) MARK(isNatIList(X:S)) -> A__ISNATILIST(X:S) MARK(isNatList(X:S)) -> A__ISNATLIST(X:S) MARK(length(X:S)) -> A__LENGTH(mark(X:S)) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> A__TAKE(mark(X1:S),mark(X2:S)) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(X2:S) MARK(zeros) -> A__ZEROS -> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) a__U62(tt) -> tt a__U62(X:S) -> U62(X:S) a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) a__U72(tt,L:S) -> s(a__length(mark(L:S))) a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) a__U81(tt) -> nil a__U81(X:S) -> U81(X:S) a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) a__isNatList(nil) -> tt a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) a__take(X1:S,X2:S) -> take(X1:S,X2:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) mark(U62(X:S)) -> a__U62(mark(X:S)) mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) mark(tt) -> tt mark(zeros) -> a__zeros Problem 1: SCC Processor: -> Pairs: A__U41(tt,V2:S) -> A__U42(a__isNatIList(V2:S)) A__U41(tt,V2:S) -> A__ISNATILIST(V2:S) A__U51(tt,V2:S) -> A__U52(a__isNatList(V2:S)) A__U51(tt,V2:S) -> A__ISNATLIST(V2:S) A__U61(tt,V2:S) -> A__U62(a__isNatIList(V2:S)) A__U61(tt,V2:S) -> A__ISNATILIST(V2:S) A__U71(tt,L:S,N:S) -> A__U72(a__isNat(N:S),L:S) A__U71(tt,L:S,N:S) -> A__ISNAT(N:S) A__U72(tt,L:S) -> A__LENGTH(mark(L:S)) A__U72(tt,L:S) -> MARK(L:S) A__U91(tt,IL:S,M:S,N:S) -> A__U92(a__isNat(M:S),IL:S,M:S,N:S) A__U91(tt,IL:S,M:S,N:S) -> A__ISNAT(M:S) A__U92(tt,IL:S,M:S,N:S) -> A__U93(a__isNat(N:S),IL:S,M:S,N:S) A__U92(tt,IL:S,M:S,N:S) -> A__ISNAT(N:S) A__U93(tt,IL:S,M:S,N:S) -> MARK(N:S) A__ISNAT(length(V1:S)) -> A__U11(a__isNatList(V1:S)) A__ISNAT(length(V1:S)) -> A__ISNATLIST(V1:S) A__ISNAT(s(V1:S)) -> A__U21(a__isNat(V1:S)) A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) A__ISNATILIST(cons(V1:S,V2:S)) -> A__U41(a__isNat(V1:S),V2:S) A__ISNATILIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) A__ISNATILIST(V:S) -> A__U31(a__isNatList(V:S)) A__ISNATILIST(V:S) -> A__ISNATLIST(V:S) A__ISNATLIST(cons(V1:S,V2:S)) -> A__U51(a__isNat(V1:S),V2:S) A__ISNATLIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) A__ISNATLIST(take(V1:S,V2:S)) -> A__U61(a__isNat(V1:S),V2:S) A__ISNATLIST(take(V1:S,V2:S)) -> A__ISNAT(V1:S) A__LENGTH(cons(N:S,L:S)) -> A__U71(a__isNatList(L:S),L:S,N:S) A__LENGTH(cons(N:S,L:S)) -> A__ISNATLIST(L:S) A__TAKE(0,IL:S) -> A__U81(a__isNatIList(IL:S)) A__TAKE(0,IL:S) -> A__ISNATILIST(IL:S) A__TAKE(s(M:S),cons(N:S,IL:S)) -> A__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) A__TAKE(s(M:S),cons(N:S,IL:S)) -> A__ISNATILIST(IL:S) MARK(U11(X:S)) -> A__U11(mark(X:S)) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X:S)) -> A__U21(mark(X:S)) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> A__U31(mark(X:S)) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> A__U41(mark(X1:S),X2:S) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> A__U42(mark(X:S)) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> A__U51(mark(X1:S),X2:S) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> A__U52(mark(X:S)) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S)) -> A__U61(mark(X1:S),X2:S) MARK(U61(X1:S,X2:S)) -> MARK(X1:S) MARK(U62(X:S)) -> A__U62(mark(X:S)) MARK(U62(X:S)) -> MARK(X:S) MARK(U71(X1:S,X2:S,X3:S)) -> A__U71(mark(X1:S),X2:S,X3:S) MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U72(X1:S,X2:S)) -> A__U72(mark(X1:S),X2:S) MARK(U72(X1:S,X2:S)) -> MARK(X1:S) MARK(U81(X:S)) -> A__U81(mark(X:S)) MARK(U81(X:S)) -> MARK(X:S) MARK(U91(X1:S,X2:S,X3:S,X4:S)) -> A__U91(mark(X1:S),X2:S,X3:S,X4:S) MARK(U91(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> A__U92(mark(X1:S),X2:S,X3:S,X4:S) MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> A__U93(mark(X1:S),X2:S,X3:S,X4:S) MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> A__ISNAT(X:S) MARK(isNatIList(X:S)) -> A__ISNATILIST(X:S) MARK(isNatList(X:S)) -> A__ISNATLIST(X:S) MARK(length(X:S)) -> A__LENGTH(mark(X:S)) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> A__TAKE(mark(X1:S),mark(X2:S)) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(X2:S) MARK(zeros) -> A__ZEROS -> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) a__U62(tt) -> tt a__U62(X:S) -> U62(X:S) a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) a__U72(tt,L:S) -> s(a__length(mark(L:S))) a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) a__U81(tt) -> nil a__U81(X:S) -> U81(X:S) a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) a__isNatList(nil) -> tt a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) a__take(X1:S,X2:S) -> take(X1:S,X2:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) mark(U62(X:S)) -> a__U62(mark(X:S)) mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) mark(tt) -> tt mark(zeros) -> a__zeros ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__U41(tt,V2:S) -> A__ISNATILIST(V2:S) A__U51(tt,V2:S) -> A__ISNATLIST(V2:S) A__U61(tt,V2:S) -> A__ISNATILIST(V2:S) A__ISNAT(length(V1:S)) -> A__ISNATLIST(V1:S) A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) A__ISNATILIST(cons(V1:S,V2:S)) -> A__U41(a__isNat(V1:S),V2:S) A__ISNATILIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) A__ISNATILIST(V:S) -> A__ISNATLIST(V:S) A__ISNATLIST(cons(V1:S,V2:S)) -> A__U51(a__isNat(V1:S),V2:S) A__ISNATLIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) A__ISNATLIST(take(V1:S,V2:S)) -> A__U61(a__isNat(V1:S),V2:S) A__ISNATLIST(take(V1:S,V2:S)) -> A__ISNAT(V1:S) ->->-> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) a__U62(tt) -> tt a__U62(X:S) -> U62(X:S) a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) a__U72(tt,L:S) -> s(a__length(mark(L:S))) a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) a__U81(tt) -> nil a__U81(X:S) -> U81(X:S) a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) a__isNatList(nil) -> tt a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) a__take(X1:S,X2:S) -> take(X1:S,X2:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) mark(U62(X:S)) -> a__U62(mark(X:S)) mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) mark(tt) -> tt mark(zeros) -> a__zeros ->->Cycle: ->->-> Pairs: A__U71(tt,L:S,N:S) -> A__U72(a__isNat(N:S),L:S) A__U72(tt,L:S) -> A__LENGTH(mark(L:S)) A__U72(tt,L:S) -> MARK(L:S) A__U91(tt,IL:S,M:S,N:S) -> A__U92(a__isNat(M:S),IL:S,M:S,N:S) A__U92(tt,IL:S,M:S,N:S) -> A__U93(a__isNat(N:S),IL:S,M:S,N:S) A__U93(tt,IL:S,M:S,N:S) -> MARK(N:S) A__LENGTH(cons(N:S,L:S)) -> A__U71(a__isNatList(L:S),L:S,N:S) A__TAKE(s(M:S),cons(N:S,IL:S)) -> A__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S)) -> MARK(X1:S) MARK(U62(X:S)) -> MARK(X:S) MARK(U71(X1:S,X2:S,X3:S)) -> A__U71(mark(X1:S),X2:S,X3:S) MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U72(X1:S,X2:S)) -> A__U72(mark(X1:S),X2:S) MARK(U72(X1:S,X2:S)) -> MARK(X1:S) MARK(U81(X:S)) -> MARK(X:S) MARK(U91(X1:S,X2:S,X3:S,X4:S)) -> A__U91(mark(X1:S),X2:S,X3:S,X4:S) MARK(U91(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> A__U92(mark(X1:S),X2:S,X3:S,X4:S) MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> A__U93(mark(X1:S),X2:S,X3:S,X4:S) MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(length(X:S)) -> A__LENGTH(mark(X:S)) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> A__TAKE(mark(X1:S),mark(X2:S)) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(X2:S) ->->-> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) a__U62(tt) -> tt a__U62(X:S) -> U62(X:S) a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) a__U72(tt,L:S) -> s(a__length(mark(L:S))) a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) a__U81(tt) -> nil a__U81(X:S) -> U81(X:S) a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) a__isNatList(nil) -> tt a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) a__take(X1:S,X2:S) -> take(X1:S,X2:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) mark(U62(X:S)) -> a__U62(mark(X:S)) mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) mark(tt) -> tt mark(zeros) -> a__zeros The problem is decomposed in 2 subproblems. Problem 1.1: Subterm Processor: -> Pairs: A__U41(tt,V2:S) -> A__ISNATILIST(V2:S) A__U51(tt,V2:S) -> A__ISNATLIST(V2:S) A__U61(tt,V2:S) -> A__ISNATILIST(V2:S) A__ISNAT(length(V1:S)) -> A__ISNATLIST(V1:S) A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) A__ISNATILIST(cons(V1:S,V2:S)) -> A__U41(a__isNat(V1:S),V2:S) A__ISNATILIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) A__ISNATILIST(V:S) -> A__ISNATLIST(V:S) A__ISNATLIST(cons(V1:S,V2:S)) -> A__U51(a__isNat(V1:S),V2:S) A__ISNATLIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) A__ISNATLIST(take(V1:S,V2:S)) -> A__U61(a__isNat(V1:S),V2:S) A__ISNATLIST(take(V1:S,V2:S)) -> A__ISNAT(V1:S) -> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) a__U62(tt) -> tt a__U62(X:S) -> U62(X:S) a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) a__U72(tt,L:S) -> s(a__length(mark(L:S))) a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) a__U81(tt) -> nil a__U81(X:S) -> U81(X:S) a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) a__isNatList(nil) -> tt a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) a__take(X1:S,X2:S) -> take(X1:S,X2:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) mark(U62(X:S)) -> a__U62(mark(X:S)) mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) mark(tt) -> tt mark(zeros) -> a__zeros ->Projection: pi(A__U41) = 2 pi(A__U51) = 2 pi(A__U61) = 2 pi(A__ISNAT) = 1 pi(A__ISNATILIST) = 1 pi(A__ISNATLIST) = 1 Problem 1.1: SCC Processor: -> Pairs: A__U41(tt,V2:S) -> A__ISNATILIST(V2:S) A__U51(tt,V2:S) -> A__ISNATLIST(V2:S) A__U61(tt,V2:S) -> A__ISNATILIST(V2:S) A__ISNATILIST(V:S) -> A__ISNATLIST(V:S) -> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) a__U62(tt) -> tt a__U62(X:S) -> U62(X:S) a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) a__U72(tt,L:S) -> s(a__length(mark(L:S))) a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) a__U81(tt) -> nil a__U81(X:S) -> U81(X:S) a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) a__isNatList(nil) -> tt a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) a__take(X1:S,X2:S) -> take(X1:S,X2:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) mark(U62(X:S)) -> a__U62(mark(X:S)) mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) mark(tt) -> tt mark(zeros) -> a__zeros ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Reduction Pairs Processor: -> Pairs: A__U71(tt,L:S,N:S) -> A__U72(a__isNat(N:S),L:S) A__U72(tt,L:S) -> A__LENGTH(mark(L:S)) A__U72(tt,L:S) -> MARK(L:S) A__U91(tt,IL:S,M:S,N:S) -> A__U92(a__isNat(M:S),IL:S,M:S,N:S) A__U92(tt,IL:S,M:S,N:S) -> A__U93(a__isNat(N:S),IL:S,M:S,N:S) A__U93(tt,IL:S,M:S,N:S) -> MARK(N:S) A__LENGTH(cons(N:S,L:S)) -> A__U71(a__isNatList(L:S),L:S,N:S) A__TAKE(s(M:S),cons(N:S,IL:S)) -> A__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S)) -> MARK(X1:S) MARK(U62(X:S)) -> MARK(X:S) MARK(U71(X1:S,X2:S,X3:S)) -> A__U71(mark(X1:S),X2:S,X3:S) MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U72(X1:S,X2:S)) -> A__U72(mark(X1:S),X2:S) MARK(U72(X1:S,X2:S)) -> MARK(X1:S) MARK(U81(X:S)) -> MARK(X:S) MARK(U91(X1:S,X2:S,X3:S,X4:S)) -> A__U91(mark(X1:S),X2:S,X3:S,X4:S) MARK(U91(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> A__U92(mark(X1:S),X2:S,X3:S,X4:S) MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> A__U93(mark(X1:S),X2:S,X3:S,X4:S) MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(length(X:S)) -> A__LENGTH(mark(X:S)) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> A__TAKE(mark(X1:S),mark(X2:S)) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) a__U62(tt) -> tt a__U62(X:S) -> U62(X:S) a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) a__U72(tt,L:S) -> s(a__length(mark(L:S))) a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) a__U81(tt) -> nil a__U81(X:S) -> U81(X:S) a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) a__isNatList(nil) -> tt a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) a__take(X1:S,X2:S) -> take(X1:S,X2:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) mark(U62(X:S)) -> a__U62(mark(X:S)) mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) mark(tt) -> tt mark(zeros) -> a__zeros -> Usable rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) a__U62(tt) -> tt a__U62(X:S) -> U62(X:S) a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) a__U72(tt,L:S) -> s(a__length(mark(L:S))) a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) a__U81(tt) -> nil a__U81(X:S) -> U81(X:S) a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) a__isNatList(nil) -> tt a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) a__take(X1:S,X2:S) -> take(X1:S,X2:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) mark(U62(X:S)) -> a__U62(mark(X:S)) mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) mark(tt) -> tt mark(zeros) -> a__zeros ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__U11](X) = X [a__U21](X) = X [a__U31](X) = X [a__U41](X1,X2) = X1 [a__U42](X) = X [a__U51](X1,X2) = X1 [a__U52](X) = X [a__U61](X1,X2) = X1 [a__U62](X) = X [a__U71](X1,X2,X3) = X1 + 2.X2 + X3 [a__U72](X1,X2) = X1 + 2.X2 [a__U81](X) = X [a__U91](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 [a__U92](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 [a__U93](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 [a__isNat](X) = 2 [a__isNatIList](X) = 2 [a__isNatList](X) = 2 [a__length](X) = 2.X + 2 [a__take](X1,X2) = 2.X1 + 2.X2 + 2 [a__zeros] = 2 [mark](X) = X [0] = 0 [U11](X) = X [U21](X) = X [U31](X) = X [U41](X1,X2) = X1 [U42](X) = X [U51](X1,X2) = X1 [U52](X) = X [U61](X1,X2) = X1 [U62](X) = X [U71](X1,X2,X3) = X1 + 2.X2 + X3 [U72](X1,X2) = X1 + 2.X2 [U81](X) = X [U91](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 [U92](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 [U93](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 [cons](X1,X2) = 2.X1 + X2 [fSNonEmpty] = 0 [isNat](X) = 2 [isNatIList](X) = 2 [isNatList](X) = 2 [length](X) = 2.X + 2 [nil] = 2 [s](X) = X [take](X1,X2) = 2.X1 + 2.X2 + 2 [tt] = 2 [zeros] = 2 [A__U11](X) = 0 [A__U21](X) = 0 [A__U31](X) = 0 [A__U41](X1,X2) = 0 [A__U42](X) = 0 [A__U51](X1,X2) = 0 [A__U52](X) = 0 [A__U61](X1,X2) = 0 [A__U62](X) = 0 [A__U71](X1,X2,X3) = X1 + 2.X2 + 2.X3 [A__U72](X1,X2) = X1 + 2.X2 [A__U81](X) = 0 [A__U91](X1,X2,X3,X4) = 2.X2 + 2.X4 [A__U92](X1,X2,X3,X4) = 2.X2 + 2.X4 [A__U93](X1,X2,X3,X4) = 2.X2 + 2.X4 [A__ISNAT](X) = 0 [A__ISNATILIST](X) = 0 [A__ISNATLIST](X) = 0 [A__LENGTH](X) = 2.X + 2 [A__TAKE](X1,X2) = 2.X2 [A__ZEROS] = 0 [MARK](X) = 2.X Problem 1.2: SCC Processor: -> Pairs: A__U71(tt,L:S,N:S) -> A__U72(a__isNat(N:S),L:S) A__U72(tt,L:S) -> A__LENGTH(mark(L:S)) A__U91(tt,IL:S,M:S,N:S) -> A__U92(a__isNat(M:S),IL:S,M:S,N:S) A__U92(tt,IL:S,M:S,N:S) -> A__U93(a__isNat(N:S),IL:S,M:S,N:S) A__U93(tt,IL:S,M:S,N:S) -> MARK(N:S) A__LENGTH(cons(N:S,L:S)) -> A__U71(a__isNatList(L:S),L:S,N:S) A__TAKE(s(M:S),cons(N:S,IL:S)) -> A__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S)) -> MARK(X1:S) MARK(U62(X:S)) -> MARK(X:S) MARK(U71(X1:S,X2:S,X3:S)) -> A__U71(mark(X1:S),X2:S,X3:S) MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U72(X1:S,X2:S)) -> A__U72(mark(X1:S),X2:S) MARK(U72(X1:S,X2:S)) -> MARK(X1:S) MARK(U81(X:S)) -> MARK(X:S) MARK(U91(X1:S,X2:S,X3:S,X4:S)) -> A__U91(mark(X1:S),X2:S,X3:S,X4:S) MARK(U91(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> A__U92(mark(X1:S),X2:S,X3:S,X4:S) MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> A__U93(mark(X1:S),X2:S,X3:S,X4:S) MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(length(X:S)) -> A__LENGTH(mark(X:S)) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> A__TAKE(mark(X1:S),mark(X2:S)) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) a__U62(tt) -> tt a__U62(X:S) -> U62(X:S) a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) a__U72(tt,L:S) -> s(a__length(mark(L:S))) a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) a__U81(tt) -> nil a__U81(X:S) -> U81(X:S) a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) a__isNatList(nil) -> tt a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) a__take(X1:S,X2:S) -> take(X1:S,X2:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) mark(U62(X:S)) -> a__U62(mark(X:S)) mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) mark(tt) -> tt mark(zeros) -> a__zeros ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__U71(tt,L:S,N:S) -> A__U72(a__isNat(N:S),L:S) A__U72(tt,L:S) -> A__LENGTH(mark(L:S)) A__LENGTH(cons(N:S,L:S)) -> A__U71(a__isNatList(L:S),L:S,N:S) ->->-> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) a__U62(tt) -> tt a__U62(X:S) -> U62(X:S) a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) a__U72(tt,L:S) -> s(a__length(mark(L:S))) a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) a__U81(tt) -> nil a__U81(X:S) -> U81(X:S) a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) a__isNatList(nil) -> tt a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) a__take(X1:S,X2:S) -> take(X1:S,X2:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) mark(U62(X:S)) -> a__U62(mark(X:S)) mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) mark(tt) -> tt mark(zeros) -> a__zeros ->->Cycle: ->->-> Pairs: A__U91(tt,IL:S,M:S,N:S) -> A__U92(a__isNat(M:S),IL:S,M:S,N:S) A__U92(tt,IL:S,M:S,N:S) -> A__U93(a__isNat(N:S),IL:S,M:S,N:S) A__U93(tt,IL:S,M:S,N:S) -> MARK(N:S) A__TAKE(s(M:S),cons(N:S,IL:S)) -> A__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S)) -> MARK(X1:S) MARK(U62(X:S)) -> MARK(X:S) MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U72(X1:S,X2:S)) -> MARK(X1:S) MARK(U81(X:S)) -> MARK(X:S) MARK(U91(X1:S,X2:S,X3:S,X4:S)) -> A__U91(mark(X1:S),X2:S,X3:S,X4:S) MARK(U91(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> A__U92(mark(X1:S),X2:S,X3:S,X4:S) MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> A__U93(mark(X1:S),X2:S,X3:S,X4:S) MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> A__TAKE(mark(X1:S),mark(X2:S)) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(X2:S) ->->-> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) a__U62(tt) -> tt a__U62(X:S) -> U62(X:S) a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) a__U72(tt,L:S) -> s(a__length(mark(L:S))) a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) a__U81(tt) -> nil a__U81(X:S) -> U81(X:S) a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) a__isNatList(nil) -> tt a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) a__take(X1:S,X2:S) -> take(X1:S,X2:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) mark(U62(X:S)) -> a__U62(mark(X:S)) mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) mark(tt) -> tt mark(zeros) -> a__zeros The problem is decomposed in 2 subproblems. Problem 1.2.1: Reduction Pairs Processor: -> Pairs: A__U71(tt,L:S,N:S) -> A__U72(a__isNat(N:S),L:S) A__U72(tt,L:S) -> A__LENGTH(mark(L:S)) A__LENGTH(cons(N:S,L:S)) -> A__U71(a__isNatList(L:S),L:S,N:S) -> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) a__U62(tt) -> tt a__U62(X:S) -> U62(X:S) a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) a__U72(tt,L:S) -> s(a__length(mark(L:S))) a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) a__U81(tt) -> nil a__U81(X:S) -> U81(X:S) a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) a__isNatList(nil) -> tt a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) a__take(X1:S,X2:S) -> take(X1:S,X2:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) mark(U62(X:S)) -> a__U62(mark(X:S)) mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) mark(tt) -> tt mark(zeros) -> a__zeros -> Usable rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) a__U62(tt) -> tt a__U62(X:S) -> U62(X:S) a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) a__U72(tt,L:S) -> s(a__length(mark(L:S))) a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) a__U81(tt) -> nil a__U81(X:S) -> U81(X:S) a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) a__isNatList(nil) -> tt a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) a__take(X1:S,X2:S) -> take(X1:S,X2:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) mark(U62(X:S)) -> a__U62(mark(X:S)) mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) mark(tt) -> tt mark(zeros) -> a__zeros ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__U11](X) = X [a__U21](X) = X [a__U31](X) = 2 [a__U41](X1,X2) = 2 [a__U42](X) = X [a__U51](X1,X2) = 2.X2 [a__U52](X) = 2.X [a__U61](X1,X2) = X1 + X2 [a__U62](X) = X [a__U71](X1,X2,X3) = 2.X2 [a__U72](X1,X2) = 2.X2 [a__U81](X) = X [a__U91](X1,X2,X3,X4) = 2.X2 + 2.X3 [a__U92](X1,X2,X3,X4) = 2.X2 + 2.X3 [a__U93](X1,X2,X3,X4) = 2.X2 + 2.X3 [a__isNat](X) = X [a__isNatIList](X) = 2 [a__isNatList](X) = X [a__length](X) = X [a__take](X1,X2) = X1 + X2 [a__zeros] = 0 [mark](X) = X [0] = 2 [U11](X) = X [U21](X) = X [U31](X) = 2 [U41](X1,X2) = 2 [U42](X) = X [U51](X1,X2) = 2.X2 [U52](X) = 2.X [U61](X1,X2) = X1 + X2 [U62](X) = X [U71](X1,X2,X3) = 2.X2 [U72](X1,X2) = 2.X2 [U81](X) = X [U91](X1,X2,X3,X4) = 2.X2 + 2.X3 [U92](X1,X2,X3,X4) = 2.X2 + 2.X3 [U93](X1,X2,X3,X4) = 2.X2 + 2.X3 [cons](X1,X2) = 2.X2 [fSNonEmpty] = 0 [isNat](X) = X [isNatIList](X) = 2 [isNatList](X) = X [length](X) = X [nil] = 2 [s](X) = 2.X [take](X1,X2) = X1 + X2 [tt] = 2 [zeros] = 0 [A__U11](X) = 0 [A__U21](X) = 0 [A__U31](X) = 0 [A__U41](X1,X2) = 0 [A__U42](X) = 0 [A__U51](X1,X2) = 0 [A__U52](X) = 0 [A__U61](X1,X2) = 0 [A__U62](X) = 0 [A__U71](X1,X2,X3) = 2.X1 + 2.X2 [A__U72](X1,X2) = 2.X2 + 2 [A__U81](X) = 0 [A__U91](X1,X2,X3,X4) = 0 [A__U92](X1,X2,X3,X4) = 0 [A__U93](X1,X2,X3,X4) = 0 [A__ISNAT](X) = 0 [A__ISNATILIST](X) = 0 [A__ISNATLIST](X) = 0 [A__LENGTH](X) = 2.X [A__TAKE](X1,X2) = 0 [A__ZEROS] = 0 [MARK](X) = 0 Problem 1.2.1: SCC Processor: -> Pairs: A__U72(tt,L:S) -> A__LENGTH(mark(L:S)) A__LENGTH(cons(N:S,L:S)) -> A__U71(a__isNatList(L:S),L:S,N:S) -> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) a__U62(tt) -> tt a__U62(X:S) -> U62(X:S) a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) a__U72(tt,L:S) -> s(a__length(mark(L:S))) a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) a__U81(tt) -> nil a__U81(X:S) -> U81(X:S) a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) a__isNatList(nil) -> tt a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) a__take(X1:S,X2:S) -> take(X1:S,X2:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) mark(U62(X:S)) -> a__U62(mark(X:S)) mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) mark(tt) -> tt mark(zeros) -> a__zeros ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2.2: Reduction Pairs Processor: -> Pairs: A__U91(tt,IL:S,M:S,N:S) -> A__U92(a__isNat(M:S),IL:S,M:S,N:S) A__U92(tt,IL:S,M:S,N:S) -> A__U93(a__isNat(N:S),IL:S,M:S,N:S) A__U93(tt,IL:S,M:S,N:S) -> MARK(N:S) A__TAKE(s(M:S),cons(N:S,IL:S)) -> A__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S)) -> MARK(X1:S) MARK(U62(X:S)) -> MARK(X:S) MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U72(X1:S,X2:S)) -> MARK(X1:S) MARK(U81(X:S)) -> MARK(X:S) MARK(U91(X1:S,X2:S,X3:S,X4:S)) -> A__U91(mark(X1:S),X2:S,X3:S,X4:S) MARK(U91(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> A__U92(mark(X1:S),X2:S,X3:S,X4:S) MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> A__U93(mark(X1:S),X2:S,X3:S,X4:S) MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> A__TAKE(mark(X1:S),mark(X2:S)) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) a__U62(tt) -> tt a__U62(X:S) -> U62(X:S) a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) a__U72(tt,L:S) -> s(a__length(mark(L:S))) a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) a__U81(tt) -> nil a__U81(X:S) -> U81(X:S) a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) a__isNatList(nil) -> tt a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) a__take(X1:S,X2:S) -> take(X1:S,X2:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) mark(U62(X:S)) -> a__U62(mark(X:S)) mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) mark(tt) -> tt mark(zeros) -> a__zeros -> Usable rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) a__U62(tt) -> tt a__U62(X:S) -> U62(X:S) a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) a__U72(tt,L:S) -> s(a__length(mark(L:S))) a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) a__U81(tt) -> nil a__U81(X:S) -> U81(X:S) a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) a__isNatList(nil) -> tt a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) a__take(X1:S,X2:S) -> take(X1:S,X2:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) mark(U62(X:S)) -> a__U62(mark(X:S)) mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) mark(tt) -> tt mark(zeros) -> a__zeros ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__U11](X) = X [a__U21](X) = X [a__U31](X) = X [a__U41](X1,X2) = X1 [a__U42](X) = X [a__U51](X1,X2) = X1 [a__U52](X) = X [a__U61](X1,X2) = X1 [a__U62](X) = X [a__U71](X1,X2,X3) = X1 + 2.X2 + 2.X3 [a__U72](X1,X2) = X1 + 2.X2 [a__U81](X) = X [a__U91](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 [a__U92](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 [a__U93](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 [a__isNat](X) = 2 [a__isNatIList](X) = 2 [a__isNatList](X) = 2 [a__length](X) = 2.X + 2 [a__take](X1,X2) = 2.X1 + 2.X2 + 2 [a__zeros] = 2 [mark](X) = X [0] = 0 [U11](X) = X [U21](X) = X [U31](X) = X [U41](X1,X2) = X1 [U42](X) = X [U51](X1,X2) = X1 [U52](X) = X [U61](X1,X2) = X1 [U62](X) = X [U71](X1,X2,X3) = X1 + 2.X2 + 2.X3 [U72](X1,X2) = X1 + 2.X2 [U81](X) = X [U91](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 [U92](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 [U93](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 [cons](X1,X2) = X1 + X2 [fSNonEmpty] = 0 [isNat](X) = 2 [isNatIList](X) = 2 [isNatList](X) = 2 [length](X) = 2.X + 2 [nil] = 2 [s](X) = X [take](X1,X2) = 2.X1 + 2.X2 + 2 [tt] = 2 [zeros] = 2 [A__U11](X) = 0 [A__U21](X) = 0 [A__U31](X) = 0 [A__U41](X1,X2) = 0 [A__U42](X) = 0 [A__U51](X1,X2) = 0 [A__U52](X) = 0 [A__U61](X1,X2) = 0 [A__U62](X) = 0 [A__U71](X1,X2,X3) = 0 [A__U72](X1,X2) = 0 [A__U81](X) = 0 [A__U91](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 [A__U92](X1,X2,X3,X4) = 2.X2 + 2.X3 + 2.X4 + 1 [A__U93](X1,X2,X3,X4) = 2.X2 + X3 + 2.X4 + 1 [A__ISNAT](X) = 0 [A__ISNATILIST](X) = 0 [A__ISNATLIST](X) = 0 [A__LENGTH](X) = 0 [A__TAKE](X1,X2) = 2.X1 + 2.X2 + 2 [A__ZEROS] = 0 [MARK](X) = 2.X + 1 Problem 1.2.2: SCC Processor: -> Pairs: A__U92(tt,IL:S,M:S,N:S) -> A__U93(a__isNat(N:S),IL:S,M:S,N:S) A__U93(tt,IL:S,M:S,N:S) -> MARK(N:S) A__TAKE(s(M:S),cons(N:S,IL:S)) -> A__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S)) -> MARK(X1:S) MARK(U62(X:S)) -> MARK(X:S) MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U72(X1:S,X2:S)) -> MARK(X1:S) MARK(U81(X:S)) -> MARK(X:S) MARK(U91(X1:S,X2:S,X3:S,X4:S)) -> A__U91(mark(X1:S),X2:S,X3:S,X4:S) MARK(U91(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> A__U92(mark(X1:S),X2:S,X3:S,X4:S) MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> A__U93(mark(X1:S),X2:S,X3:S,X4:S) MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> A__TAKE(mark(X1:S),mark(X2:S)) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) a__U62(tt) -> tt a__U62(X:S) -> U62(X:S) a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) a__U72(tt,L:S) -> s(a__length(mark(L:S))) a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) a__U81(tt) -> nil a__U81(X:S) -> U81(X:S) a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) a__isNatList(nil) -> tt a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) a__take(X1:S,X2:S) -> take(X1:S,X2:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) mark(U62(X:S)) -> a__U62(mark(X:S)) mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) mark(tt) -> tt mark(zeros) -> a__zeros ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__U92(tt,IL:S,M:S,N:S) -> A__U93(a__isNat(N:S),IL:S,M:S,N:S) A__U93(tt,IL:S,M:S,N:S) -> MARK(N:S) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S)) -> MARK(X1:S) MARK(U62(X:S)) -> MARK(X:S) MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U72(X1:S,X2:S)) -> MARK(X1:S) MARK(U81(X:S)) -> MARK(X:S) MARK(U91(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> A__U92(mark(X1:S),X2:S,X3:S,X4:S) MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> A__U93(mark(X1:S),X2:S,X3:S,X4:S) MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(X2:S) ->->-> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) a__U62(tt) -> tt a__U62(X:S) -> U62(X:S) a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) a__U72(tt,L:S) -> s(a__length(mark(L:S))) a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) a__U81(tt) -> nil a__U81(X:S) -> U81(X:S) a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) a__isNatList(nil) -> tt a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) a__take(X1:S,X2:S) -> take(X1:S,X2:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) mark(U62(X:S)) -> a__U62(mark(X:S)) mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) mark(tt) -> tt mark(zeros) -> a__zeros Problem 1.2.2: Subterm Processor: -> Pairs: A__U92(tt,IL:S,M:S,N:S) -> A__U93(a__isNat(N:S),IL:S,M:S,N:S) A__U93(tt,IL:S,M:S,N:S) -> MARK(N:S) MARK(U11(X:S)) -> MARK(X:S) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X:S)) -> MARK(X:S) MARK(U41(X1:S,X2:S)) -> MARK(X1:S) MARK(U42(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S)) -> MARK(X1:S) MARK(U62(X:S)) -> MARK(X:S) MARK(U71(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U72(X1:S,X2:S)) -> MARK(X1:S) MARK(U81(X:S)) -> MARK(X:S) MARK(U91(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> A__U92(mark(X1:S),X2:S,X3:S,X4:S) MARK(U92(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> A__U93(mark(X1:S),X2:S,X3:S,X4:S) MARK(U93(X1:S,X2:S,X3:S,X4:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(X2:S) -> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) a__U62(tt) -> tt a__U62(X:S) -> U62(X:S) a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) a__U72(tt,L:S) -> s(a__length(mark(L:S))) a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) a__U81(tt) -> nil a__U81(X:S) -> U81(X:S) a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) a__isNatList(nil) -> tt a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) a__take(X1:S,X2:S) -> take(X1:S,X2:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) mark(U62(X:S)) -> a__U62(mark(X:S)) mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) mark(tt) -> tt mark(zeros) -> a__zeros ->Projection: pi(A__U92) = 4 pi(A__U93) = 4 pi(MARK) = 1 Problem 1.2.2: SCC Processor: -> Pairs: A__U92(tt,IL:S,M:S,N:S) -> A__U93(a__isNat(N:S),IL:S,M:S,N:S) A__U93(tt,IL:S,M:S,N:S) -> MARK(N:S) -> Rules: a__U11(tt) -> tt a__U11(X:S) -> U11(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt) -> tt a__U31(X:S) -> U31(X:S) a__U41(tt,V2:S) -> a__U42(a__isNatIList(V2:S)) a__U41(X1:S,X2:S) -> U41(X1:S,X2:S) a__U42(tt) -> tt a__U42(X:S) -> U42(X:S) a__U51(tt,V2:S) -> a__U52(a__isNatList(V2:S)) a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) a__U52(tt) -> tt a__U52(X:S) -> U52(X:S) a__U61(tt,V2:S) -> a__U62(a__isNatIList(V2:S)) a__U61(X1:S,X2:S) -> U61(X1:S,X2:S) a__U62(tt) -> tt a__U62(X:S) -> U62(X:S) a__U71(tt,L:S,N:S) -> a__U72(a__isNat(N:S),L:S) a__U71(X1:S,X2:S,X3:S) -> U71(X1:S,X2:S,X3:S) a__U72(tt,L:S) -> s(a__length(mark(L:S))) a__U72(X1:S,X2:S) -> U72(X1:S,X2:S) a__U81(tt) -> nil a__U81(X:S) -> U81(X:S) a__U91(tt,IL:S,M:S,N:S) -> a__U92(a__isNat(M:S),IL:S,M:S,N:S) a__U91(X1:S,X2:S,X3:S,X4:S) -> U91(X1:S,X2:S,X3:S,X4:S) a__U92(tt,IL:S,M:S,N:S) -> a__U93(a__isNat(N:S),IL:S,M:S,N:S) a__U92(X1:S,X2:S,X3:S,X4:S) -> U92(X1:S,X2:S,X3:S,X4:S) a__U93(tt,IL:S,M:S,N:S) -> cons(mark(N:S),take(M:S,IL:S)) a__U93(X1:S,X2:S,X3:S,X4:S) -> U93(X1:S,X2:S,X3:S,X4:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__U11(a__isNatList(V1:S)) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__U41(a__isNat(V1:S),V2:S) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__U31(a__isNatList(V:S)) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__U51(a__isNat(V1:S),V2:S) a__isNatList(nil) -> tt a__isNatList(take(V1:S,V2:S)) -> a__U61(a__isNat(V1:S),V2:S) a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U71(a__isNatList(L:S),L:S,N:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__take(0,IL:S) -> a__U81(a__isNatIList(IL:S)) a__take(s(M:S),cons(N:S,IL:S)) -> a__U91(a__isNatIList(IL:S),IL:S,M:S,N:S) a__take(X1:S,X2:S) -> take(X1:S,X2:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X:S)) -> a__U11(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X:S)) -> a__U31(mark(X:S)) mark(U41(X1:S,X2:S)) -> a__U41(mark(X1:S),X2:S) mark(U42(X:S)) -> a__U42(mark(X:S)) mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) mark(U52(X:S)) -> a__U52(mark(X:S)) mark(U61(X1:S,X2:S)) -> a__U61(mark(X1:S),X2:S) mark(U62(X:S)) -> a__U62(mark(X:S)) mark(U71(X1:S,X2:S,X3:S)) -> a__U71(mark(X1:S),X2:S,X3:S) mark(U72(X1:S,X2:S)) -> a__U72(mark(X1:S),X2:S) mark(U81(X:S)) -> a__U81(mark(X:S)) mark(U91(X1:S,X2:S,X3:S,X4:S)) -> a__U91(mark(X1:S),X2:S,X3:S,X4:S) mark(U92(X1:S,X2:S,X3:S,X4:S)) -> a__U92(mark(X1:S),X2:S,X3:S,X4:S) mark(U93(X1:S,X2:S,X3:S,X4:S)) -> a__U93(mark(X1:S),X2:S,X3:S,X4:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S)) mark(tt) -> tt mark(zeros) -> a__zeros ->Strongly Connected Components: There is no strongly connected component The problem is finite.