/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES Problem 1: (VAR L N V V1 V2 X X1 X2 X3) (RULES a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt,V1) -> a__U13(a__isNatList(V1)) a__U12(X1,X2) -> U12(X1,X2) a__U13(tt) -> tt a__U13(X) -> U13(X) a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt,V1) -> a__U23(a__isNat(V1)) a__U22(X1,X2) -> U22(X1,X2) a__U23(tt) -> tt a__U23(X) -> U23(X) a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt,V) -> a__U33(a__isNatList(V)) a__U32(X1,X2) -> U32(X1,X2) a__U33(tt) -> tt a__U33(X) -> U33(X) a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2) a__U43(X1,X2,X3) -> U43(X1,X2,X3) a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2) a__U44(X1,X2,X3) -> U44(X1,X2,X3) a__U45(tt,V2) -> a__U46(a__isNatIList(V2)) a__U45(X1,X2) -> U45(X1,X2) a__U46(tt) -> tt a__U46(X) -> U46(X) a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt) -> tt a__U61(X) -> U61(X) a__U71(tt) -> tt a__U71(X) -> U71(X) a__U81(tt,V1,V2) -> a__U82(a__isNatKind(V1),V1,V2) a__U81(X1,X2,X3) -> U81(X1,X2,X3) a__U82(tt,V1,V2) -> a__U83(a__isNatIListKind(V2),V1,V2) a__U82(X1,X2,X3) -> U82(X1,X2,X3) a__U83(tt,V1,V2) -> a__U84(a__isNatIListKind(V2),V1,V2) a__U83(X1,X2,X3) -> U83(X1,X2,X3) a__U84(tt,V1,V2) -> a__U85(a__isNat(V1),V2) a__U84(X1,X2,X3) -> U84(X1,X2,X3) a__U85(tt,V2) -> a__U86(a__isNatList(V2)) a__U85(X1,X2) -> U85(X1,X2) a__U86(tt) -> tt a__U86(X) -> U86(X) a__U91(tt,L,N) -> a__U92(a__isNatIListKind(L),L,N) a__U91(X1,X2,X3) -> U91(X1,X2,X3) a__U92(tt,L,N) -> a__U93(a__isNat(N),L,N) a__U92(X1,X2,X3) -> U92(X1,X2,X3) a__U93(tt,L,N) -> a__U94(a__isNatKind(N),L) a__U93(X1,X2,X3) -> U93(X1,X2,X3) a__U94(tt,L) -> s(a__length(mark(L))) a__U94(X1,X2) -> U94(X1,X2) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(X) -> isNatIList(X) a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U81(a__isNatKind(V1),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U91(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X1,X2)) -> a__U32(mark(X1),X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3) mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3) mark(U45(X1,X2)) -> a__U45(mark(X1),X2) mark(U46(X)) -> a__U46(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X1,X2,X3)) -> a__U81(mark(X1),X2,X3) mark(U82(X1,X2,X3)) -> a__U82(mark(X1),X2,X3) mark(U83(X1,X2,X3)) -> a__U83(mark(X1),X2,X3) mark(U84(X1,X2,X3)) -> a__U84(mark(X1),X2,X3) mark(U85(X1,X2)) -> a__U85(mark(X1),X2) mark(U86(X)) -> a__U86(mark(X)) mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3) mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3) mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3) mark(U94(X1,X2)) -> a__U94(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(zeros) -> a__zeros ) Problem 1: Dependency Pairs Processor: -> Pairs: A__U11(tt,V1) -> A__U12(a__isNatIListKind(V1),V1) A__U11(tt,V1) -> A__ISNATILISTKIND(V1) A__U12(tt,V1) -> A__U13(a__isNatList(V1)) A__U12(tt,V1) -> A__ISNATLIST(V1) A__U21(tt,V1) -> A__U22(a__isNatKind(V1),V1) A__U21(tt,V1) -> A__ISNATKIND(V1) A__U22(tt,V1) -> A__U23(a__isNat(V1)) A__U22(tt,V1) -> A__ISNAT(V1) A__U31(tt,V) -> A__U32(a__isNatIListKind(V),V) A__U31(tt,V) -> A__ISNATILISTKIND(V) A__U32(tt,V) -> A__U33(a__isNatList(V)) A__U32(tt,V) -> A__ISNATLIST(V) A__U41(tt,V1,V2) -> A__U42(a__isNatKind(V1),V1,V2) A__U41(tt,V1,V2) -> A__ISNATKIND(V1) A__U42(tt,V1,V2) -> A__U43(a__isNatIListKind(V2),V1,V2) A__U42(tt,V1,V2) -> A__ISNATILISTKIND(V2) A__U43(tt,V1,V2) -> A__U44(a__isNatIListKind(V2),V1,V2) A__U43(tt,V1,V2) -> A__ISNATILISTKIND(V2) A__U44(tt,V1,V2) -> A__U45(a__isNat(V1),V2) A__U44(tt,V1,V2) -> A__ISNAT(V1) A__U45(tt,V2) -> A__U46(a__isNatIList(V2)) A__U45(tt,V2) -> A__ISNATILIST(V2) A__U51(tt,V2) -> A__U52(a__isNatIListKind(V2)) A__U51(tt,V2) -> A__ISNATILISTKIND(V2) A__U81(tt,V1,V2) -> A__U82(a__isNatKind(V1),V1,V2) A__U81(tt,V1,V2) -> A__ISNATKIND(V1) A__U82(tt,V1,V2) -> A__U83(a__isNatIListKind(V2),V1,V2) A__U82(tt,V1,V2) -> A__ISNATILISTKIND(V2) A__U83(tt,V1,V2) -> A__U84(a__isNatIListKind(V2),V1,V2) A__U83(tt,V1,V2) -> A__ISNATILISTKIND(V2) A__U84(tt,V1,V2) -> A__U85(a__isNat(V1),V2) A__U84(tt,V1,V2) -> A__ISNAT(V1) A__U85(tt,V2) -> A__U86(a__isNatList(V2)) A__U85(tt,V2) -> A__ISNATLIST(V2) A__U91(tt,L,N) -> A__U92(a__isNatIListKind(L),L,N) A__U91(tt,L,N) -> A__ISNATILISTKIND(L) A__U92(tt,L,N) -> A__U93(a__isNat(N),L,N) A__U92(tt,L,N) -> A__ISNAT(N) A__U93(tt,L,N) -> A__U94(a__isNatKind(N),L) A__U93(tt,L,N) -> A__ISNATKIND(N) A__U94(tt,L) -> A__LENGTH(mark(L)) A__U94(tt,L) -> MARK(L) A__ISNAT(length(V1)) -> A__U11(a__isNatIListKind(V1),V1) A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1),V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) A__ISNATILIST(cons(V1,V2)) -> A__U41(a__isNatKind(V1),V1,V2) A__ISNATILIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__U31(a__isNatIListKind(V),V) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__U51(a__isNatKind(V1),V2) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__U61(a__isNatIListKind(V1)) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__U71(a__isNatKind(V1)) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__U81(a__isNatKind(V1),V1,V2) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__U91(a__isNatList(L),L,N) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> A__U11(mark(X1),X2) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X1,X2)) -> A__U12(mark(X1),X2) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> A__U13(mark(X)) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> A__U21(mark(X1),X2) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> A__U22(mark(X1),X2) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> A__U23(mark(X)) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> A__U31(mark(X1),X2) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X1,X2)) -> A__U32(mark(X1),X2) MARK(U32(X1,X2)) -> MARK(X1) MARK(U33(X)) -> A__U33(mark(X)) MARK(U33(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> A__U41(mark(X1),X2,X3) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2,X3)) -> A__U42(mark(X1),X2,X3) MARK(U42(X1,X2,X3)) -> MARK(X1) MARK(U43(X1,X2,X3)) -> A__U43(mark(X1),X2,X3) MARK(U43(X1,X2,X3)) -> MARK(X1) MARK(U44(X1,X2,X3)) -> A__U44(mark(X1),X2,X3) MARK(U44(X1,X2,X3)) -> MARK(X1) MARK(U45(X1,X2)) -> A__U45(mark(X1),X2) MARK(U45(X1,X2)) -> MARK(X1) MARK(U46(X)) -> A__U46(mark(X)) MARK(U46(X)) -> MARK(X) MARK(U51(X1,X2)) -> A__U51(mark(X1),X2) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> A__U52(mark(X)) MARK(U52(X)) -> MARK(X) MARK(U61(X)) -> A__U61(mark(X)) MARK(U61(X)) -> MARK(X) MARK(U71(X)) -> A__U71(mark(X)) MARK(U71(X)) -> MARK(X) MARK(U81(X1,X2,X3)) -> A__U81(mark(X1),X2,X3) MARK(U81(X1,X2,X3)) -> MARK(X1) MARK(U82(X1,X2,X3)) -> A__U82(mark(X1),X2,X3) MARK(U82(X1,X2,X3)) -> MARK(X1) MARK(U83(X1,X2,X3)) -> A__U83(mark(X1),X2,X3) MARK(U83(X1,X2,X3)) -> MARK(X1) MARK(U84(X1,X2,X3)) -> A__U84(mark(X1),X2,X3) MARK(U84(X1,X2,X3)) -> MARK(X1) MARK(U85(X1,X2)) -> A__U85(mark(X1),X2) MARK(U85(X1,X2)) -> MARK(X1) MARK(U86(X)) -> A__U86(mark(X)) MARK(U86(X)) -> MARK(X) MARK(U91(X1,X2,X3)) -> A__U91(mark(X1),X2,X3) MARK(U91(X1,X2,X3)) -> MARK(X1) MARK(U92(X1,X2,X3)) -> A__U92(mark(X1),X2,X3) MARK(U92(X1,X2,X3)) -> MARK(X1) MARK(U93(X1,X2,X3)) -> A__U93(mark(X1),X2,X3) MARK(U93(X1,X2,X3)) -> MARK(X1) MARK(U94(X1,X2)) -> A__U94(mark(X1),X2) MARK(U94(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) MARK(isNatIList(X)) -> A__ISNATILIST(X) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(isNatKind(X)) -> A__ISNATKIND(X) MARK(isNatList(X)) -> A__ISNATLIST(X) MARK(length(X)) -> A__LENGTH(mark(X)) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(zeros) -> A__ZEROS -> Rules: a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt,V1) -> a__U13(a__isNatList(V1)) a__U12(X1,X2) -> U12(X1,X2) a__U13(tt) -> tt a__U13(X) -> U13(X) a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt,V1) -> a__U23(a__isNat(V1)) a__U22(X1,X2) -> U22(X1,X2) a__U23(tt) -> tt a__U23(X) -> U23(X) a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt,V) -> a__U33(a__isNatList(V)) a__U32(X1,X2) -> U32(X1,X2) a__U33(tt) -> tt a__U33(X) -> U33(X) a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2) a__U43(X1,X2,X3) -> U43(X1,X2,X3) a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2) a__U44(X1,X2,X3) -> U44(X1,X2,X3) a__U45(tt,V2) -> a__U46(a__isNatIList(V2)) a__U45(X1,X2) -> U45(X1,X2) a__U46(tt) -> tt a__U46(X) -> U46(X) a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt) -> tt a__U61(X) -> U61(X) a__U71(tt) -> tt a__U71(X) -> U71(X) a__U81(tt,V1,V2) -> a__U82(a__isNatKind(V1),V1,V2) a__U81(X1,X2,X3) -> U81(X1,X2,X3) a__U82(tt,V1,V2) -> a__U83(a__isNatIListKind(V2),V1,V2) a__U82(X1,X2,X3) -> U82(X1,X2,X3) a__U83(tt,V1,V2) -> a__U84(a__isNatIListKind(V2),V1,V2) a__U83(X1,X2,X3) -> U83(X1,X2,X3) a__U84(tt,V1,V2) -> a__U85(a__isNat(V1),V2) a__U84(X1,X2,X3) -> U84(X1,X2,X3) a__U85(tt,V2) -> a__U86(a__isNatList(V2)) a__U85(X1,X2) -> U85(X1,X2) a__U86(tt) -> tt a__U86(X) -> U86(X) a__U91(tt,L,N) -> a__U92(a__isNatIListKind(L),L,N) a__U91(X1,X2,X3) -> U91(X1,X2,X3) a__U92(tt,L,N) -> a__U93(a__isNat(N),L,N) a__U92(X1,X2,X3) -> U92(X1,X2,X3) a__U93(tt,L,N) -> a__U94(a__isNatKind(N),L) a__U93(X1,X2,X3) -> U93(X1,X2,X3) a__U94(tt,L) -> s(a__length(mark(L))) a__U94(X1,X2) -> U94(X1,X2) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(X) -> isNatIList(X) a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U81(a__isNatKind(V1),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U91(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X1,X2)) -> a__U32(mark(X1),X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3) mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3) mark(U45(X1,X2)) -> a__U45(mark(X1),X2) mark(U46(X)) -> a__U46(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X1,X2,X3)) -> a__U81(mark(X1),X2,X3) mark(U82(X1,X2,X3)) -> a__U82(mark(X1),X2,X3) mark(U83(X1,X2,X3)) -> a__U83(mark(X1),X2,X3) mark(U84(X1,X2,X3)) -> a__U84(mark(X1),X2,X3) mark(U85(X1,X2)) -> a__U85(mark(X1),X2) mark(U86(X)) -> a__U86(mark(X)) mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3) mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3) mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3) mark(U94(X1,X2)) -> a__U94(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(zeros) -> a__zeros Problem 1: SCC Processor: -> Pairs: A__U11(tt,V1) -> A__U12(a__isNatIListKind(V1),V1) A__U11(tt,V1) -> A__ISNATILISTKIND(V1) A__U12(tt,V1) -> A__U13(a__isNatList(V1)) A__U12(tt,V1) -> A__ISNATLIST(V1) A__U21(tt,V1) -> A__U22(a__isNatKind(V1),V1) A__U21(tt,V1) -> A__ISNATKIND(V1) A__U22(tt,V1) -> A__U23(a__isNat(V1)) A__U22(tt,V1) -> A__ISNAT(V1) A__U31(tt,V) -> A__U32(a__isNatIListKind(V),V) A__U31(tt,V) -> A__ISNATILISTKIND(V) A__U32(tt,V) -> A__U33(a__isNatList(V)) A__U32(tt,V) -> A__ISNATLIST(V) A__U41(tt,V1,V2) -> A__U42(a__isNatKind(V1),V1,V2) A__U41(tt,V1,V2) -> A__ISNATKIND(V1) A__U42(tt,V1,V2) -> A__U43(a__isNatIListKind(V2),V1,V2) A__U42(tt,V1,V2) -> A__ISNATILISTKIND(V2) A__U43(tt,V1,V2) -> A__U44(a__isNatIListKind(V2),V1,V2) A__U43(tt,V1,V2) -> A__ISNATILISTKIND(V2) A__U44(tt,V1,V2) -> A__U45(a__isNat(V1),V2) A__U44(tt,V1,V2) -> A__ISNAT(V1) A__U45(tt,V2) -> A__U46(a__isNatIList(V2)) A__U45(tt,V2) -> A__ISNATILIST(V2) A__U51(tt,V2) -> A__U52(a__isNatIListKind(V2)) A__U51(tt,V2) -> A__ISNATILISTKIND(V2) A__U81(tt,V1,V2) -> A__U82(a__isNatKind(V1),V1,V2) A__U81(tt,V1,V2) -> A__ISNATKIND(V1) A__U82(tt,V1,V2) -> A__U83(a__isNatIListKind(V2),V1,V2) A__U82(tt,V1,V2) -> A__ISNATILISTKIND(V2) A__U83(tt,V1,V2) -> A__U84(a__isNatIListKind(V2),V1,V2) A__U83(tt,V1,V2) -> A__ISNATILISTKIND(V2) A__U84(tt,V1,V2) -> A__U85(a__isNat(V1),V2) A__U84(tt,V1,V2) -> A__ISNAT(V1) A__U85(tt,V2) -> A__U86(a__isNatList(V2)) A__U85(tt,V2) -> A__ISNATLIST(V2) A__U91(tt,L,N) -> A__U92(a__isNatIListKind(L),L,N) A__U91(tt,L,N) -> A__ISNATILISTKIND(L) A__U92(tt,L,N) -> A__U93(a__isNat(N),L,N) A__U92(tt,L,N) -> A__ISNAT(N) A__U93(tt,L,N) -> A__U94(a__isNatKind(N),L) A__U93(tt,L,N) -> A__ISNATKIND(N) A__U94(tt,L) -> A__LENGTH(mark(L)) A__U94(tt,L) -> MARK(L) A__ISNAT(length(V1)) -> A__U11(a__isNatIListKind(V1),V1) A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1),V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) A__ISNATILIST(cons(V1,V2)) -> A__U41(a__isNatKind(V1),V1,V2) A__ISNATILIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__U31(a__isNatIListKind(V),V) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__U51(a__isNatKind(V1),V2) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__U61(a__isNatIListKind(V1)) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__U71(a__isNatKind(V1)) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__U81(a__isNatKind(V1),V1,V2) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__U91(a__isNatList(L),L,N) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> A__U11(mark(X1),X2) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X1,X2)) -> A__U12(mark(X1),X2) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> A__U13(mark(X)) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> A__U21(mark(X1),X2) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> A__U22(mark(X1),X2) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> A__U23(mark(X)) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> A__U31(mark(X1),X2) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X1,X2)) -> A__U32(mark(X1),X2) MARK(U32(X1,X2)) -> MARK(X1) MARK(U33(X)) -> A__U33(mark(X)) MARK(U33(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> A__U41(mark(X1),X2,X3) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2,X3)) -> A__U42(mark(X1),X2,X3) MARK(U42(X1,X2,X3)) -> MARK(X1) MARK(U43(X1,X2,X3)) -> A__U43(mark(X1),X2,X3) MARK(U43(X1,X2,X3)) -> MARK(X1) MARK(U44(X1,X2,X3)) -> A__U44(mark(X1),X2,X3) MARK(U44(X1,X2,X3)) -> MARK(X1) MARK(U45(X1,X2)) -> A__U45(mark(X1),X2) MARK(U45(X1,X2)) -> MARK(X1) MARK(U46(X)) -> A__U46(mark(X)) MARK(U46(X)) -> MARK(X) MARK(U51(X1,X2)) -> A__U51(mark(X1),X2) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> A__U52(mark(X)) MARK(U52(X)) -> MARK(X) MARK(U61(X)) -> A__U61(mark(X)) MARK(U61(X)) -> MARK(X) MARK(U71(X)) -> A__U71(mark(X)) MARK(U71(X)) -> MARK(X) MARK(U81(X1,X2,X3)) -> A__U81(mark(X1),X2,X3) MARK(U81(X1,X2,X3)) -> MARK(X1) MARK(U82(X1,X2,X3)) -> A__U82(mark(X1),X2,X3) MARK(U82(X1,X2,X3)) -> MARK(X1) MARK(U83(X1,X2,X3)) -> A__U83(mark(X1),X2,X3) MARK(U83(X1,X2,X3)) -> MARK(X1) MARK(U84(X1,X2,X3)) -> A__U84(mark(X1),X2,X3) MARK(U84(X1,X2,X3)) -> MARK(X1) MARK(U85(X1,X2)) -> A__U85(mark(X1),X2) MARK(U85(X1,X2)) -> MARK(X1) MARK(U86(X)) -> A__U86(mark(X)) MARK(U86(X)) -> MARK(X) MARK(U91(X1,X2,X3)) -> A__U91(mark(X1),X2,X3) MARK(U91(X1,X2,X3)) -> MARK(X1) MARK(U92(X1,X2,X3)) -> A__U92(mark(X1),X2,X3) MARK(U92(X1,X2,X3)) -> MARK(X1) MARK(U93(X1,X2,X3)) -> A__U93(mark(X1),X2,X3) MARK(U93(X1,X2,X3)) -> MARK(X1) MARK(U94(X1,X2)) -> A__U94(mark(X1),X2) MARK(U94(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) MARK(isNatIList(X)) -> A__ISNATILIST(X) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(isNatKind(X)) -> A__ISNATKIND(X) MARK(isNatList(X)) -> A__ISNATLIST(X) MARK(length(X)) -> A__LENGTH(mark(X)) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(zeros) -> A__ZEROS -> Rules: a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt,V1) -> a__U13(a__isNatList(V1)) a__U12(X1,X2) -> U12(X1,X2) a__U13(tt) -> tt a__U13(X) -> U13(X) a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt,V1) -> a__U23(a__isNat(V1)) a__U22(X1,X2) -> U22(X1,X2) a__U23(tt) -> tt a__U23(X) -> U23(X) a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt,V) -> a__U33(a__isNatList(V)) a__U32(X1,X2) -> U32(X1,X2) a__U33(tt) -> tt a__U33(X) -> U33(X) a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2) a__U43(X1,X2,X3) -> U43(X1,X2,X3) a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2) a__U44(X1,X2,X3) -> U44(X1,X2,X3) a__U45(tt,V2) -> a__U46(a__isNatIList(V2)) a__U45(X1,X2) -> U45(X1,X2) a__U46(tt) -> tt a__U46(X) -> U46(X) a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt) -> tt a__U61(X) -> U61(X) a__U71(tt) -> tt a__U71(X) -> U71(X) a__U81(tt,V1,V2) -> a__U82(a__isNatKind(V1),V1,V2) a__U81(X1,X2,X3) -> U81(X1,X2,X3) a__U82(tt,V1,V2) -> a__U83(a__isNatIListKind(V2),V1,V2) a__U82(X1,X2,X3) -> U82(X1,X2,X3) a__U83(tt,V1,V2) -> a__U84(a__isNatIListKind(V2),V1,V2) a__U83(X1,X2,X3) -> U83(X1,X2,X3) a__U84(tt,V1,V2) -> a__U85(a__isNat(V1),V2) a__U84(X1,X2,X3) -> U84(X1,X2,X3) a__U85(tt,V2) -> a__U86(a__isNatList(V2)) a__U85(X1,X2) -> U85(X1,X2) a__U86(tt) -> tt a__U86(X) -> U86(X) a__U91(tt,L,N) -> a__U92(a__isNatIListKind(L),L,N) a__U91(X1,X2,X3) -> U91(X1,X2,X3) a__U92(tt,L,N) -> a__U93(a__isNat(N),L,N) a__U92(X1,X2,X3) -> U92(X1,X2,X3) a__U93(tt,L,N) -> a__U94(a__isNatKind(N),L) a__U93(X1,X2,X3) -> U93(X1,X2,X3) a__U94(tt,L) -> s(a__length(mark(L))) a__U94(X1,X2) -> U94(X1,X2) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(X) -> isNatIList(X) a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U81(a__isNatKind(V1),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U91(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X1,X2)) -> a__U32(mark(X1),X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3) mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3) mark(U45(X1,X2)) -> a__U45(mark(X1),X2) mark(U46(X)) -> a__U46(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X1,X2,X3)) -> a__U81(mark(X1),X2,X3) mark(U82(X1,X2,X3)) -> a__U82(mark(X1),X2,X3) mark(U83(X1,X2,X3)) -> a__U83(mark(X1),X2,X3) mark(U84(X1,X2,X3)) -> a__U84(mark(X1),X2,X3) mark(U85(X1,X2)) -> a__U85(mark(X1),X2) mark(U86(X)) -> a__U86(mark(X)) mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3) mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3) mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3) mark(U94(X1,X2)) -> a__U94(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(zeros) -> a__zeros ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__U51(tt,V2) -> A__ISNATILISTKIND(V2) A__ISNATILISTKIND(cons(V1,V2)) -> A__U51(a__isNatKind(V1),V2) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) ->->-> Rules: a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt,V1) -> a__U13(a__isNatList(V1)) a__U12(X1,X2) -> U12(X1,X2) a__U13(tt) -> tt a__U13(X) -> U13(X) a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt,V1) -> a__U23(a__isNat(V1)) a__U22(X1,X2) -> U22(X1,X2) a__U23(tt) -> tt a__U23(X) -> U23(X) a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt,V) -> a__U33(a__isNatList(V)) a__U32(X1,X2) -> U32(X1,X2) a__U33(tt) -> tt a__U33(X) -> U33(X) a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2) a__U43(X1,X2,X3) -> U43(X1,X2,X3) a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2) a__U44(X1,X2,X3) -> U44(X1,X2,X3) a__U45(tt,V2) -> a__U46(a__isNatIList(V2)) a__U45(X1,X2) -> U45(X1,X2) a__U46(tt) -> tt a__U46(X) -> U46(X) a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt) -> tt a__U61(X) -> U61(X) a__U71(tt) -> tt a__U71(X) -> U71(X) a__U81(tt,V1,V2) -> a__U82(a__isNatKind(V1),V1,V2) a__U81(X1,X2,X3) -> U81(X1,X2,X3) a__U82(tt,V1,V2) -> a__U83(a__isNatIListKind(V2),V1,V2) a__U82(X1,X2,X3) -> U82(X1,X2,X3) a__U83(tt,V1,V2) -> a__U84(a__isNatIListKind(V2),V1,V2) a__U83(X1,X2,X3) -> U83(X1,X2,X3) a__U84(tt,V1,V2) -> a__U85(a__isNat(V1),V2) a__U84(X1,X2,X3) -> U84(X1,X2,X3) a__U85(tt,V2) -> a__U86(a__isNatList(V2)) a__U85(X1,X2) -> U85(X1,X2) a__U86(tt) -> tt a__U86(X) -> U86(X) a__U91(tt,L,N) -> a__U92(a__isNatIListKind(L),L,N) a__U91(X1,X2,X3) -> U91(X1,X2,X3) a__U92(tt,L,N) -> a__U93(a__isNat(N),L,N) a__U92(X1,X2,X3) -> U92(X1,X2,X3) a__U93(tt,L,N) -> a__U94(a__isNatKind(N),L) a__U93(X1,X2,X3) -> U93(X1,X2,X3) a__U94(tt,L) -> s(a__length(mark(L))) a__U94(X1,X2) -> U94(X1,X2) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(X) -> isNatIList(X) a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U81(a__isNatKind(V1),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U91(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X1,X2)) -> a__U32(mark(X1),X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3) mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3) mark(U45(X1,X2)) -> a__U45(mark(X1),X2) mark(U46(X)) -> a__U46(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X1,X2,X3)) -> a__U81(mark(X1),X2,X3) mark(U82(X1,X2,X3)) -> a__U82(mark(X1),X2,X3) mark(U83(X1,X2,X3)) -> a__U83(mark(X1),X2,X3) mark(U84(X1,X2,X3)) -> a__U84(mark(X1),X2,X3) mark(U85(X1,X2)) -> a__U85(mark(X1),X2) mark(U86(X)) -> a__U86(mark(X)) mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3) mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3) mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3) mark(U94(X1,X2)) -> a__U94(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(zeros) -> a__zeros ->->Cycle: ->->-> Pairs: A__U11(tt,V1) -> A__U12(a__isNatIListKind(V1),V1) A__U12(tt,V1) -> A__ISNATLIST(V1) A__U21(tt,V1) -> A__U22(a__isNatKind(V1),V1) A__U22(tt,V1) -> A__ISNAT(V1) A__U81(tt,V1,V2) -> A__U82(a__isNatKind(V1),V1,V2) A__U82(tt,V1,V2) -> A__U83(a__isNatIListKind(V2),V1,V2) A__U83(tt,V1,V2) -> A__U84(a__isNatIListKind(V2),V1,V2) A__U84(tt,V1,V2) -> A__U85(a__isNat(V1),V2) A__U84(tt,V1,V2) -> A__ISNAT(V1) A__U85(tt,V2) -> A__ISNATLIST(V2) A__ISNAT(length(V1)) -> A__U11(a__isNatIListKind(V1),V1) A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1),V1) A__ISNATLIST(cons(V1,V2)) -> A__U81(a__isNatKind(V1),V1,V2) ->->-> Rules: a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt,V1) -> a__U13(a__isNatList(V1)) a__U12(X1,X2) -> U12(X1,X2) a__U13(tt) -> tt a__U13(X) -> U13(X) a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt,V1) -> a__U23(a__isNat(V1)) a__U22(X1,X2) -> U22(X1,X2) a__U23(tt) -> tt a__U23(X) -> U23(X) a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt,V) -> a__U33(a__isNatList(V)) a__U32(X1,X2) -> U32(X1,X2) a__U33(tt) -> tt a__U33(X) -> U33(X) a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2) a__U43(X1,X2,X3) -> U43(X1,X2,X3) a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2) a__U44(X1,X2,X3) -> U44(X1,X2,X3) a__U45(tt,V2) -> a__U46(a__isNatIList(V2)) a__U45(X1,X2) -> U45(X1,X2) a__U46(tt) -> tt a__U46(X) -> U46(X) a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt) -> tt a__U61(X) -> U61(X) a__U71(tt) -> tt a__U71(X) -> U71(X) a__U81(tt,V1,V2) -> a__U82(a__isNatKind(V1),V1,V2) a__U81(X1,X2,X3) -> U81(X1,X2,X3) a__U82(tt,V1,V2) -> a__U83(a__isNatIListKind(V2),V1,V2) a__U82(X1,X2,X3) -> U82(X1,X2,X3) a__U83(tt,V1,V2) -> a__U84(a__isNatIListKind(V2),V1,V2) a__U83(X1,X2,X3) -> U83(X1,X2,X3) a__U84(tt,V1,V2) -> a__U85(a__isNat(V1),V2) a__U84(X1,X2,X3) -> U84(X1,X2,X3) a__U85(tt,V2) -> a__U86(a__isNatList(V2)) a__U85(X1,X2) -> U85(X1,X2) a__U86(tt) -> tt a__U86(X) -> U86(X) a__U91(tt,L,N) -> a__U92(a__isNatIListKind(L),L,N) a__U91(X1,X2,X3) -> U91(X1,X2,X3) a__U92(tt,L,N) -> a__U93(a__isNat(N),L,N) a__U92(X1,X2,X3) -> U92(X1,X2,X3) a__U93(tt,L,N) -> a__U94(a__isNatKind(N),L) a__U93(X1,X2,X3) -> U93(X1,X2,X3) a__U94(tt,L) -> s(a__length(mark(L))) a__U94(X1,X2) -> U94(X1,X2) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(X) -> isNatIList(X) a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U81(a__isNatKind(V1),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U91(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X1,X2)) -> a__U32(mark(X1),X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3) mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3) mark(U45(X1,X2)) -> a__U45(mark(X1),X2) mark(U46(X)) -> a__U46(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X1,X2,X3)) -> a__U81(mark(X1),X2,X3) mark(U82(X1,X2,X3)) -> a__U82(mark(X1),X2,X3) mark(U83(X1,X2,X3)) -> a__U83(mark(X1),X2,X3) mark(U84(X1,X2,X3)) -> a__U84(mark(X1),X2,X3) mark(U85(X1,X2)) -> a__U85(mark(X1),X2) mark(U86(X)) -> a__U86(mark(X)) mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3) mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3) mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3) mark(U94(X1,X2)) -> a__U94(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(zeros) -> a__zeros ->->Cycle: ->->-> Pairs: A__U41(tt,V1,V2) -> A__U42(a__isNatKind(V1),V1,V2) A__U42(tt,V1,V2) -> A__U43(a__isNatIListKind(V2),V1,V2) A__U43(tt,V1,V2) -> A__U44(a__isNatIListKind(V2),V1,V2) A__U44(tt,V1,V2) -> A__U45(a__isNat(V1),V2) A__U45(tt,V2) -> A__ISNATILIST(V2) A__ISNATILIST(cons(V1,V2)) -> A__U41(a__isNatKind(V1),V1,V2) ->->-> Rules: a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt,V1) -> a__U13(a__isNatList(V1)) a__U12(X1,X2) -> U12(X1,X2) a__U13(tt) -> tt a__U13(X) -> U13(X) a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt,V1) -> a__U23(a__isNat(V1)) a__U22(X1,X2) -> U22(X1,X2) a__U23(tt) -> tt a__U23(X) -> U23(X) a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt,V) -> a__U33(a__isNatList(V)) a__U32(X1,X2) -> U32(X1,X2) a__U33(tt) -> tt a__U33(X) -> U33(X) a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2) a__U43(X1,X2,X3) -> U43(X1,X2,X3) a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2) a__U44(X1,X2,X3) -> U44(X1,X2,X3) a__U45(tt,V2) -> a__U46(a__isNatIList(V2)) a__U45(X1,X2) -> U45(X1,X2) a__U46(tt) -> tt a__U46(X) -> U46(X) a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt) -> tt a__U61(X) -> U61(X) a__U71(tt) -> tt a__U71(X) -> U71(X) a__U81(tt,V1,V2) -> a__U82(a__isNatKind(V1),V1,V2) a__U81(X1,X2,X3) -> U81(X1,X2,X3) a__U82(tt,V1,V2) -> a__U83(a__isNatIListKind(V2),V1,V2) a__U82(X1,X2,X3) -> U82(X1,X2,X3) a__U83(tt,V1,V2) -> a__U84(a__isNatIListKind(V2),V1,V2) a__U83(X1,X2,X3) -> U83(X1,X2,X3) a__U84(tt,V1,V2) -> a__U85(a__isNat(V1),V2) a__U84(X1,X2,X3) -> U84(X1,X2,X3) a__U85(tt,V2) -> a__U86(a__isNatList(V2)) a__U85(X1,X2) -> U85(X1,X2) a__U86(tt) -> tt a__U86(X) -> U86(X) a__U91(tt,L,N) -> a__U92(a__isNatIListKind(L),L,N) a__U91(X1,X2,X3) -> U91(X1,X2,X3) a__U92(tt,L,N) -> a__U93(a__isNat(N),L,N) a__U92(X1,X2,X3) -> U92(X1,X2,X3) a__U93(tt,L,N) -> a__U94(a__isNatKind(N),L) a__U93(X1,X2,X3) -> U93(X1,X2,X3) a__U94(tt,L) -> s(a__length(mark(L))) a__U94(X1,X2) -> U94(X1,X2) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(X) -> isNatIList(X) a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U81(a__isNatKind(V1),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U91(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X1,X2)) -> a__U32(mark(X1),X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3) mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3) mark(U45(X1,X2)) -> a__U45(mark(X1),X2) mark(U46(X)) -> a__U46(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X1,X2,X3)) -> a__U81(mark(X1),X2,X3) mark(U82(X1,X2,X3)) -> a__U82(mark(X1),X2,X3) mark(U83(X1,X2,X3)) -> a__U83(mark(X1),X2,X3) mark(U84(X1,X2,X3)) -> a__U84(mark(X1),X2,X3) mark(U85(X1,X2)) -> a__U85(mark(X1),X2) mark(U86(X)) -> a__U86(mark(X)) mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3) mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3) mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3) mark(U94(X1,X2)) -> a__U94(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(zeros) -> a__zeros ->->Cycle: ->->-> Pairs: A__U91(tt,L,N) -> A__U92(a__isNatIListKind(L),L,N) A__U92(tt,L,N) -> A__U93(a__isNat(N),L,N) A__U93(tt,L,N) -> A__U94(a__isNatKind(N),L) A__U94(tt,L) -> A__LENGTH(mark(L)) A__U94(tt,L) -> MARK(L) A__LENGTH(cons(N,L)) -> A__U91(a__isNatList(L),L,N) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X1,X2)) -> MARK(X1) MARK(U33(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2,X3)) -> MARK(X1) MARK(U43(X1,X2,X3)) -> MARK(X1) MARK(U44(X1,X2,X3)) -> MARK(X1) MARK(U45(X1,X2)) -> MARK(X1) MARK(U46(X)) -> MARK(X) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(U61(X)) -> MARK(X) MARK(U71(X)) -> MARK(X) MARK(U81(X1,X2,X3)) -> MARK(X1) MARK(U82(X1,X2,X3)) -> MARK(X1) MARK(U83(X1,X2,X3)) -> MARK(X1) MARK(U84(X1,X2,X3)) -> MARK(X1) MARK(U85(X1,X2)) -> MARK(X1) MARK(U86(X)) -> MARK(X) MARK(U91(X1,X2,X3)) -> A__U91(mark(X1),X2,X3) MARK(U91(X1,X2,X3)) -> MARK(X1) MARK(U92(X1,X2,X3)) -> A__U92(mark(X1),X2,X3) MARK(U92(X1,X2,X3)) -> MARK(X1) MARK(U93(X1,X2,X3)) -> A__U93(mark(X1),X2,X3) MARK(U93(X1,X2,X3)) -> MARK(X1) MARK(U94(X1,X2)) -> A__U94(mark(X1),X2) MARK(U94(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(length(X)) -> A__LENGTH(mark(X)) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) ->->-> Rules: a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt,V1) -> a__U13(a__isNatList(V1)) a__U12(X1,X2) -> U12(X1,X2) a__U13(tt) -> tt a__U13(X) -> U13(X) a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt,V1) -> a__U23(a__isNat(V1)) a__U22(X1,X2) -> U22(X1,X2) a__U23(tt) -> tt a__U23(X) -> U23(X) a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt,V) -> a__U33(a__isNatList(V)) a__U32(X1,X2) -> U32(X1,X2) a__U33(tt) -> tt a__U33(X) -> U33(X) a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2) a__U43(X1,X2,X3) -> U43(X1,X2,X3) a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2) a__U44(X1,X2,X3) -> U44(X1,X2,X3) a__U45(tt,V2) -> a__U46(a__isNatIList(V2)) a__U45(X1,X2) -> U45(X1,X2) a__U46(tt) -> tt a__U46(X) -> U46(X) a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt) -> tt a__U61(X) -> U61(X) a__U71(tt) -> tt a__U71(X) -> U71(X) a__U81(tt,V1,V2) -> a__U82(a__isNatKind(V1),V1,V2) a__U81(X1,X2,X3) -> U81(X1,X2,X3) a__U82(tt,V1,V2) -> a__U83(a__isNatIListKind(V2),V1,V2) a__U82(X1,X2,X3) -> U82(X1,X2,X3) a__U83(tt,V1,V2) -> a__U84(a__isNatIListKind(V2),V1,V2) a__U83(X1,X2,X3) -> U83(X1,X2,X3) a__U84(tt,V1,V2) -> a__U85(a__isNat(V1),V2) a__U84(X1,X2,X3) -> U84(X1,X2,X3) a__U85(tt,V2) -> a__U86(a__isNatList(V2)) a__U85(X1,X2) -> U85(X1,X2) a__U86(tt) -> tt a__U86(X) -> U86(X) a__U91(tt,L,N) -> a__U92(a__isNatIListKind(L),L,N) a__U91(X1,X2,X3) -> U91(X1,X2,X3) a__U92(tt,L,N) -> a__U93(a__isNat(N),L,N) a__U92(X1,X2,X3) -> U92(X1,X2,X3) a__U93(tt,L,N) -> a__U94(a__isNatKind(N),L) a__U93(X1,X2,X3) -> U93(X1,X2,X3) a__U94(tt,L) -> s(a__length(mark(L))) a__U94(X1,X2) -> U94(X1,X2) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(X) -> isNatIList(X) a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U81(a__isNatKind(V1),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U91(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X1,X2)) -> a__U32(mark(X1),X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3) mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3) mark(U45(X1,X2)) -> a__U45(mark(X1),X2) mark(U46(X)) -> a__U46(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X1,X2,X3)) -> a__U81(mark(X1),X2,X3) mark(U82(X1,X2,X3)) -> a__U82(mark(X1),X2,X3) mark(U83(X1,X2,X3)) -> a__U83(mark(X1),X2,X3) mark(U84(X1,X2,X3)) -> a__U84(mark(X1),X2,X3) mark(U85(X1,X2)) -> a__U85(mark(X1),X2) mark(U86(X)) -> a__U86(mark(X)) mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3) mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3) mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3) mark(U94(X1,X2)) -> a__U94(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(zeros) -> a__zeros The problem is decomposed in 4 subproblems. Problem 1.1: Subterm Processor: -> Pairs: A__U51(tt,V2) -> A__ISNATILISTKIND(V2) A__ISNATILISTKIND(cons(V1,V2)) -> A__U51(a__isNatKind(V1),V2) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt,V1) -> a__U13(a__isNatList(V1)) a__U12(X1,X2) -> U12(X1,X2) a__U13(tt) -> tt a__U13(X) -> U13(X) a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt,V1) -> a__U23(a__isNat(V1)) a__U22(X1,X2) -> U22(X1,X2) a__U23(tt) -> tt a__U23(X) -> U23(X) a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt,V) -> a__U33(a__isNatList(V)) a__U32(X1,X2) -> U32(X1,X2) a__U33(tt) -> tt a__U33(X) -> U33(X) a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2) a__U43(X1,X2,X3) -> U43(X1,X2,X3) a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2) a__U44(X1,X2,X3) -> U44(X1,X2,X3) a__U45(tt,V2) -> a__U46(a__isNatIList(V2)) a__U45(X1,X2) -> U45(X1,X2) a__U46(tt) -> tt a__U46(X) -> U46(X) a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt) -> tt a__U61(X) -> U61(X) a__U71(tt) -> tt a__U71(X) -> U71(X) a__U81(tt,V1,V2) -> a__U82(a__isNatKind(V1),V1,V2) a__U81(X1,X2,X3) -> U81(X1,X2,X3) a__U82(tt,V1,V2) -> a__U83(a__isNatIListKind(V2),V1,V2) a__U82(X1,X2,X3) -> U82(X1,X2,X3) a__U83(tt,V1,V2) -> a__U84(a__isNatIListKind(V2),V1,V2) a__U83(X1,X2,X3) -> U83(X1,X2,X3) a__U84(tt,V1,V2) -> a__U85(a__isNat(V1),V2) a__U84(X1,X2,X3) -> U84(X1,X2,X3) a__U85(tt,V2) -> a__U86(a__isNatList(V2)) a__U85(X1,X2) -> U85(X1,X2) a__U86(tt) -> tt a__U86(X) -> U86(X) a__U91(tt,L,N) -> a__U92(a__isNatIListKind(L),L,N) a__U91(X1,X2,X3) -> U91(X1,X2,X3) a__U92(tt,L,N) -> a__U93(a__isNat(N),L,N) a__U92(X1,X2,X3) -> U92(X1,X2,X3) a__U93(tt,L,N) -> a__U94(a__isNatKind(N),L) a__U93(X1,X2,X3) -> U93(X1,X2,X3) a__U94(tt,L) -> s(a__length(mark(L))) a__U94(X1,X2) -> U94(X1,X2) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(X) -> isNatIList(X) a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U81(a__isNatKind(V1),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U91(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X1,X2)) -> a__U32(mark(X1),X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3) mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3) mark(U45(X1,X2)) -> a__U45(mark(X1),X2) mark(U46(X)) -> a__U46(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X1,X2,X3)) -> a__U81(mark(X1),X2,X3) mark(U82(X1,X2,X3)) -> a__U82(mark(X1),X2,X3) mark(U83(X1,X2,X3)) -> a__U83(mark(X1),X2,X3) mark(U84(X1,X2,X3)) -> a__U84(mark(X1),X2,X3) mark(U85(X1,X2)) -> a__U85(mark(X1),X2) mark(U86(X)) -> a__U86(mark(X)) mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3) mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3) mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3) mark(U94(X1,X2)) -> a__U94(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(zeros) -> a__zeros ->Projection: pi(A__U51) = 2 pi(A__ISNATILISTKIND) = 1 pi(A__ISNATKIND) = 1 Problem 1.1: SCC Processor: -> Pairs: A__U51(tt,V2) -> A__ISNATILISTKIND(V2) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt,V1) -> a__U13(a__isNatList(V1)) a__U12(X1,X2) -> U12(X1,X2) a__U13(tt) -> tt a__U13(X) -> U13(X) a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt,V1) -> a__U23(a__isNat(V1)) a__U22(X1,X2) -> U22(X1,X2) a__U23(tt) -> tt a__U23(X) -> U23(X) a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt,V) -> a__U33(a__isNatList(V)) a__U32(X1,X2) -> U32(X1,X2) a__U33(tt) -> tt a__U33(X) -> U33(X) a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2) a__U43(X1,X2,X3) -> U43(X1,X2,X3) a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2) a__U44(X1,X2,X3) -> U44(X1,X2,X3) a__U45(tt,V2) -> a__U46(a__isNatIList(V2)) a__U45(X1,X2) -> U45(X1,X2) a__U46(tt) -> tt a__U46(X) -> U46(X) a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt) -> tt a__U61(X) -> U61(X) a__U71(tt) -> tt a__U71(X) -> U71(X) a__U81(tt,V1,V2) -> a__U82(a__isNatKind(V1),V1,V2) a__U81(X1,X2,X3) -> U81(X1,X2,X3) a__U82(tt,V1,V2) -> a__U83(a__isNatIListKind(V2),V1,V2) a__U82(X1,X2,X3) -> U82(X1,X2,X3) a__U83(tt,V1,V2) -> a__U84(a__isNatIListKind(V2),V1,V2) a__U83(X1,X2,X3) -> U83(X1,X2,X3) a__U84(tt,V1,V2) -> a__U85(a__isNat(V1),V2) a__U84(X1,X2,X3) -> U84(X1,X2,X3) a__U85(tt,V2) -> a__U86(a__isNatList(V2)) a__U85(X1,X2) -> U85(X1,X2) a__U86(tt) -> tt a__U86(X) -> U86(X) a__U91(tt,L,N) -> a__U92(a__isNatIListKind(L),L,N) a__U91(X1,X2,X3) -> U91(X1,X2,X3) a__U92(tt,L,N) -> a__U93(a__isNat(N),L,N) a__U92(X1,X2,X3) -> U92(X1,X2,X3) a__U93(tt,L,N) -> a__U94(a__isNatKind(N),L) a__U93(X1,X2,X3) -> U93(X1,X2,X3) a__U94(tt,L) -> s(a__length(mark(L))) a__U94(X1,X2) -> U94(X1,X2) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(X) -> isNatIList(X) a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U81(a__isNatKind(V1),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U91(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X1,X2)) -> a__U32(mark(X1),X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3) mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3) mark(U45(X1,X2)) -> a__U45(mark(X1),X2) mark(U46(X)) -> a__U46(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X1,X2,X3)) -> a__U81(mark(X1),X2,X3) mark(U82(X1,X2,X3)) -> a__U82(mark(X1),X2,X3) mark(U83(X1,X2,X3)) -> a__U83(mark(X1),X2,X3) mark(U84(X1,X2,X3)) -> a__U84(mark(X1),X2,X3) mark(U85(X1,X2)) -> a__U85(mark(X1),X2) mark(U86(X)) -> a__U86(mark(X)) mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3) mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3) mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3) mark(U94(X1,X2)) -> a__U94(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(zeros) -> a__zeros ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Reduction Pair Processor: -> Pairs: A__U11(tt,V1) -> A__U12(a__isNatIListKind(V1),V1) A__U12(tt,V1) -> A__ISNATLIST(V1) A__U21(tt,V1) -> A__U22(a__isNatKind(V1),V1) A__U22(tt,V1) -> A__ISNAT(V1) A__U81(tt,V1,V2) -> A__U82(a__isNatKind(V1),V1,V2) A__U82(tt,V1,V2) -> A__U83(a__isNatIListKind(V2),V1,V2) A__U83(tt,V1,V2) -> A__U84(a__isNatIListKind(V2),V1,V2) A__U84(tt,V1,V2) -> A__U85(a__isNat(V1),V2) A__U84(tt,V1,V2) -> A__ISNAT(V1) A__U85(tt,V2) -> A__ISNATLIST(V2) A__ISNAT(length(V1)) -> A__U11(a__isNatIListKind(V1),V1) A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1),V1) A__ISNATLIST(cons(V1,V2)) -> A__U81(a__isNatKind(V1),V1,V2) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt,V1) -> a__U13(a__isNatList(V1)) a__U12(X1,X2) -> U12(X1,X2) a__U13(tt) -> tt a__U13(X) -> U13(X) a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt,V1) -> a__U23(a__isNat(V1)) a__U22(X1,X2) -> U22(X1,X2) a__U23(tt) -> tt a__U23(X) -> U23(X) a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt,V) -> a__U33(a__isNatList(V)) a__U32(X1,X2) -> U32(X1,X2) a__U33(tt) -> tt a__U33(X) -> U33(X) a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2) a__U43(X1,X2,X3) -> U43(X1,X2,X3) a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2) a__U44(X1,X2,X3) -> U44(X1,X2,X3) a__U45(tt,V2) -> a__U46(a__isNatIList(V2)) a__U45(X1,X2) -> U45(X1,X2) a__U46(tt) -> tt a__U46(X) -> U46(X) a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt) -> tt a__U61(X) -> U61(X) a__U71(tt) -> tt a__U71(X) -> U71(X) a__U81(tt,V1,V2) -> a__U82(a__isNatKind(V1),V1,V2) a__U81(X1,X2,X3) -> U81(X1,X2,X3) a__U82(tt,V1,V2) -> a__U83(a__isNatIListKind(V2),V1,V2) a__U82(X1,X2,X3) -> U82(X1,X2,X3) a__U83(tt,V1,V2) -> a__U84(a__isNatIListKind(V2),V1,V2) a__U83(X1,X2,X3) -> U83(X1,X2,X3) a__U84(tt,V1,V2) -> a__U85(a__isNat(V1),V2) a__U84(X1,X2,X3) -> U84(X1,X2,X3) a__U85(tt,V2) -> a__U86(a__isNatList(V2)) a__U85(X1,X2) -> U85(X1,X2) a__U86(tt) -> tt a__U86(X) -> U86(X) a__U91(tt,L,N) -> a__U92(a__isNatIListKind(L),L,N) a__U91(X1,X2,X3) -> U91(X1,X2,X3) a__U92(tt,L,N) -> a__U93(a__isNat(N),L,N) a__U92(X1,X2,X3) -> U92(X1,X2,X3) a__U93(tt,L,N) -> a__U94(a__isNatKind(N),L) a__U93(X1,X2,X3) -> U93(X1,X2,X3) a__U94(tt,L) -> s(a__length(mark(L))) a__U94(X1,X2) -> U94(X1,X2) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(X) -> isNatIList(X) a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U81(a__isNatKind(V1),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U91(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X1,X2)) -> a__U32(mark(X1),X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3) mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3) mark(U45(X1,X2)) -> a__U45(mark(X1),X2) mark(U46(X)) -> a__U46(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X1,X2,X3)) -> a__U81(mark(X1),X2,X3) mark(U82(X1,X2,X3)) -> a__U82(mark(X1),X2,X3) mark(U83(X1,X2,X3)) -> a__U83(mark(X1),X2,X3) mark(U84(X1,X2,X3)) -> a__U84(mark(X1),X2,X3) mark(U85(X1,X2)) -> a__U85(mark(X1),X2) mark(U86(X)) -> a__U86(mark(X)) mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3) mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3) mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3) mark(U94(X1,X2)) -> a__U94(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(zeros) -> a__zeros -> Usable rules: a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt,V1) -> a__U13(a__isNatList(V1)) a__U12(X1,X2) -> U12(X1,X2) a__U13(tt) -> tt a__U13(X) -> U13(X) a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt,V1) -> a__U23(a__isNat(V1)) a__U22(X1,X2) -> U22(X1,X2) a__U23(tt) -> tt a__U23(X) -> U23(X) a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt) -> tt a__U61(X) -> U61(X) a__U71(tt) -> tt a__U71(X) -> U71(X) a__U81(tt,V1,V2) -> a__U82(a__isNatKind(V1),V1,V2) a__U81(X1,X2,X3) -> U81(X1,X2,X3) a__U82(tt,V1,V2) -> a__U83(a__isNatIListKind(V2),V1,V2) a__U82(X1,X2,X3) -> U82(X1,X2,X3) a__U83(tt,V1,V2) -> a__U84(a__isNatIListKind(V2),V1,V2) a__U83(X1,X2,X3) -> U83(X1,X2,X3) a__U84(tt,V1,V2) -> a__U85(a__isNat(V1),V2) a__U84(X1,X2,X3) -> U84(X1,X2,X3) a__U85(tt,V2) -> a__U86(a__isNatList(V2)) a__U85(X1,X2) -> U85(X1,X2) a__U86(tt) -> tt a__U86(X) -> U86(X) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U81(a__isNatKind(V1),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__U11](X1,X2) = 2 [a__U12](X1,X2) = 2 [a__U13](X) = 2 [a__U21](X1,X2) = 2 [a__U22](X1,X2) = X1 [a__U23](X) = 2 [a__U51](X1,X2) = 2.X2 + 2 [a__U52](X) = X + 2 [a__U61](X) = 2 [a__U71](X) = 2 [a__U81](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [a__U82](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 [a__U83](X1,X2,X3) = 2.X1 + 2.X2 + 2 [a__U84](X1,X2,X3) = 2.X2 + 2 [a__U85](X1,X2) = 2 [a__U86](X) = 2 [a__isNat](X) = 2 [a__isNatIListKind](X) = X [a__isNatKind](X) = 2 [a__isNatList](X) = 2.X + 2 [0] = 1 [U11](X1,X2) = 1 [U12](X1,X2) = 2 [U13](X) = 2 [U21](X1,X2) = 2 [U22](X1,X2) = X1 [U23](X) = 2 [U51](X1,X2) = 2.X2 + 2 [U52](X) = X + 2 [U61](X) = 1 [U71](X) = 0 [U81](X1,X2,X3) = 2.X1 + X3 + 2 [U82](X1,X2,X3) = X1 + X2 + X3 [U83](X1,X2,X3) = 2.X1 + X2 + 2 [U84](X1,X2,X3) = 1 [U85](X1,X2) = 1 [U86](X) = 1 [cons](X1,X2) = 2.X1 + 2.X2 + 2 [isNat](X) = 0 [isNatIListKind](X) = 0 [isNatKind](X) = 2 [isNatList](X) = X + 2 [length](X) = 2.X + 2 [nil] = 2 [s](X) = 2.X + 2 [tt] = 2 [zeros] = 2 [A__U11](X1,X2) = 2.X1 + 2.X2 + 2 [A__U12](X1,X2) = X1 + X2 + 2 [A__U21](X1,X2) = 2.X1 + 2.X2 + 2 [A__U22](X1,X2) = 2.X1 + 2.X2 + 1 [A__U81](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [A__U82](X1,X2,X3) = X1 + 2.X2 + 2.X3 [A__U83](X1,X2,X3) = X1 + 2.X2 + X3 + 2 [A__U84](X1,X2,X3) = 2.X2 + X3 + 2 [A__U85](X1,X2) = X1 + X2 [A__ISNAT](X) = 2.X + 2 [A__ISNATLIST](X) = X + 2 Problem 1.2: SCC Processor: -> Pairs: A__U12(tt,V1) -> A__ISNATLIST(V1) A__U21(tt,V1) -> A__U22(a__isNatKind(V1),V1) A__U22(tt,V1) -> A__ISNAT(V1) A__U81(tt,V1,V2) -> A__U82(a__isNatKind(V1),V1,V2) A__U82(tt,V1,V2) -> A__U83(a__isNatIListKind(V2),V1,V2) A__U83(tt,V1,V2) -> A__U84(a__isNatIListKind(V2),V1,V2) A__U84(tt,V1,V2) -> A__U85(a__isNat(V1),V2) A__U84(tt,V1,V2) -> A__ISNAT(V1) A__U85(tt,V2) -> A__ISNATLIST(V2) A__ISNAT(length(V1)) -> A__U11(a__isNatIListKind(V1),V1) A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1),V1) A__ISNATLIST(cons(V1,V2)) -> A__U81(a__isNatKind(V1),V1,V2) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt,V1) -> a__U13(a__isNatList(V1)) a__U12(X1,X2) -> U12(X1,X2) a__U13(tt) -> tt a__U13(X) -> U13(X) a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt,V1) -> a__U23(a__isNat(V1)) a__U22(X1,X2) -> U22(X1,X2) a__U23(tt) -> tt a__U23(X) -> U23(X) a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt,V) -> a__U33(a__isNatList(V)) a__U32(X1,X2) -> U32(X1,X2) a__U33(tt) -> tt a__U33(X) -> U33(X) a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2) a__U43(X1,X2,X3) -> U43(X1,X2,X3) a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2) a__U44(X1,X2,X3) -> U44(X1,X2,X3) a__U45(tt,V2) -> a__U46(a__isNatIList(V2)) a__U45(X1,X2) -> U45(X1,X2) a__U46(tt) -> tt a__U46(X) -> U46(X) a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt) -> tt a__U61(X) -> U61(X) a__U71(tt) -> tt a__U71(X) -> U71(X) a__U81(tt,V1,V2) -> a__U82(a__isNatKind(V1),V1,V2) a__U81(X1,X2,X3) -> U81(X1,X2,X3) a__U82(tt,V1,V2) -> a__U83(a__isNatIListKind(V2),V1,V2) a__U82(X1,X2,X3) -> U82(X1,X2,X3) a__U83(tt,V1,V2) -> a__U84(a__isNatIListKind(V2),V1,V2) a__U83(X1,X2,X3) -> U83(X1,X2,X3) a__U84(tt,V1,V2) -> a__U85(a__isNat(V1),V2) a__U84(X1,X2,X3) -> U84(X1,X2,X3) a__U85(tt,V2) -> a__U86(a__isNatList(V2)) a__U85(X1,X2) -> U85(X1,X2) a__U86(tt) -> tt a__U86(X) -> U86(X) a__U91(tt,L,N) -> a__U92(a__isNatIListKind(L),L,N) a__U91(X1,X2,X3) -> U91(X1,X2,X3) a__U92(tt,L,N) -> a__U93(a__isNat(N),L,N) a__U92(X1,X2,X3) -> U92(X1,X2,X3) a__U93(tt,L,N) -> a__U94(a__isNatKind(N),L) a__U93(X1,X2,X3) -> U93(X1,X2,X3) a__U94(tt,L) -> s(a__length(mark(L))) a__U94(X1,X2) -> U94(X1,X2) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(X) -> isNatIList(X) a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U81(a__isNatKind(V1),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U91(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X1,X2)) -> a__U32(mark(X1),X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3) mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3) mark(U45(X1,X2)) -> a__U45(mark(X1),X2) mark(U46(X)) -> a__U46(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X1,X2,X3)) -> a__U81(mark(X1),X2,X3) mark(U82(X1,X2,X3)) -> a__U82(mark(X1),X2,X3) mark(U83(X1,X2,X3)) -> a__U83(mark(X1),X2,X3) mark(U84(X1,X2,X3)) -> a__U84(mark(X1),X2,X3) mark(U85(X1,X2)) -> a__U85(mark(X1),X2) mark(U86(X)) -> a__U86(mark(X)) mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3) mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3) mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3) mark(U94(X1,X2)) -> a__U94(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(zeros) -> a__zeros ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__U21(tt,V1) -> A__U22(a__isNatKind(V1),V1) A__U22(tt,V1) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1),V1) ->->-> Rules: a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt,V1) -> a__U13(a__isNatList(V1)) a__U12(X1,X2) -> U12(X1,X2) a__U13(tt) -> tt a__U13(X) -> U13(X) a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt,V1) -> a__U23(a__isNat(V1)) a__U22(X1,X2) -> U22(X1,X2) a__U23(tt) -> tt a__U23(X) -> U23(X) a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt,V) -> a__U33(a__isNatList(V)) a__U32(X1,X2) -> U32(X1,X2) a__U33(tt) -> tt a__U33(X) -> U33(X) a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2) a__U43(X1,X2,X3) -> U43(X1,X2,X3) a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2) a__U44(X1,X2,X3) -> U44(X1,X2,X3) a__U45(tt,V2) -> a__U46(a__isNatIList(V2)) a__U45(X1,X2) -> U45(X1,X2) a__U46(tt) -> tt a__U46(X) -> U46(X) a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt) -> tt a__U61(X) -> U61(X) a__U71(tt) -> tt a__U71(X) -> U71(X) a__U81(tt,V1,V2) -> a__U82(a__isNatKind(V1),V1,V2) a__U81(X1,X2,X3) -> U81(X1,X2,X3) a__U82(tt,V1,V2) -> a__U83(a__isNatIListKind(V2),V1,V2) a__U82(X1,X2,X3) -> U82(X1,X2,X3) a__U83(tt,V1,V2) -> a__U84(a__isNatIListKind(V2),V1,V2) a__U83(X1,X2,X3) -> U83(X1,X2,X3) a__U84(tt,V1,V2) -> a__U85(a__isNat(V1),V2) a__U84(X1,X2,X3) -> U84(X1,X2,X3) a__U85(tt,V2) -> a__U86(a__isNatList(V2)) a__U85(X1,X2) -> U85(X1,X2) a__U86(tt) -> tt a__U86(X) -> U86(X) a__U91(tt,L,N) -> a__U92(a__isNatIListKind(L),L,N) a__U91(X1,X2,X3) -> U91(X1,X2,X3) a__U92(tt,L,N) -> a__U93(a__isNat(N),L,N) a__U92(X1,X2,X3) -> U92(X1,X2,X3) a__U93(tt,L,N) -> a__U94(a__isNatKind(N),L) a__U93(X1,X2,X3) -> U93(X1,X2,X3) a__U94(tt,L) -> s(a__length(mark(L))) a__U94(X1,X2) -> U94(X1,X2) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(X) -> isNatIList(X) a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U81(a__isNatKind(V1),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U91(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X1,X2)) -> a__U32(mark(X1),X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3) mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3) mark(U45(X1,X2)) -> a__U45(mark(X1),X2) mark(U46(X)) -> a__U46(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X1,X2,X3)) -> a__U81(mark(X1),X2,X3) mark(U82(X1,X2,X3)) -> a__U82(mark(X1),X2,X3) mark(U83(X1,X2,X3)) -> a__U83(mark(X1),X2,X3) mark(U84(X1,X2,X3)) -> a__U84(mark(X1),X2,X3) mark(U85(X1,X2)) -> a__U85(mark(X1),X2) mark(U86(X)) -> a__U86(mark(X)) mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3) mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3) mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3) mark(U94(X1,X2)) -> a__U94(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(zeros) -> a__zeros ->->Cycle: ->->-> Pairs: A__U81(tt,V1,V2) -> A__U82(a__isNatKind(V1),V1,V2) A__U82(tt,V1,V2) -> A__U83(a__isNatIListKind(V2),V1,V2) A__U83(tt,V1,V2) -> A__U84(a__isNatIListKind(V2),V1,V2) A__U84(tt,V1,V2) -> A__U85(a__isNat(V1),V2) A__U85(tt,V2) -> A__ISNATLIST(V2) A__ISNATLIST(cons(V1,V2)) -> A__U81(a__isNatKind(V1),V1,V2) ->->-> Rules: a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt,V1) -> a__U13(a__isNatList(V1)) a__U12(X1,X2) -> U12(X1,X2) a__U13(tt) -> tt a__U13(X) -> U13(X) a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt,V1) -> a__U23(a__isNat(V1)) a__U22(X1,X2) -> U22(X1,X2) a__U23(tt) -> tt a__U23(X) -> U23(X) a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt,V) -> a__U33(a__isNatList(V)) a__U32(X1,X2) -> U32(X1,X2) a__U33(tt) -> tt a__U33(X) -> U33(X) a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2) a__U43(X1,X2,X3) -> U43(X1,X2,X3) a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2) a__U44(X1,X2,X3) -> U44(X1,X2,X3) a__U45(tt,V2) -> a__U46(a__isNatIList(V2)) a__U45(X1,X2) -> U45(X1,X2) a__U46(tt) -> tt a__U46(X) -> U46(X) a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt) -> tt a__U61(X) -> U61(X) a__U71(tt) -> tt a__U71(X) -> U71(X) a__U81(tt,V1,V2) -> a__U82(a__isNatKind(V1),V1,V2) a__U81(X1,X2,X3) -> U81(X1,X2,X3) a__U82(tt,V1,V2) -> a__U83(a__isNatIListKind(V2),V1,V2) a__U82(X1,X2,X3) -> U82(X1,X2,X3) a__U83(tt,V1,V2) -> a__U84(a__isNatIListKind(V2),V1,V2) a__U83(X1,X2,X3) -> U83(X1,X2,X3) a__U84(tt,V1,V2) -> a__U85(a__isNat(V1),V2) a__U84(X1,X2,X3) -> U84(X1,X2,X3) a__U85(tt,V2) -> a__U86(a__isNatList(V2)) a__U85(X1,X2) -> U85(X1,X2) a__U86(tt) -> tt a__U86(X) -> U86(X) a__U91(tt,L,N) -> a__U92(a__isNatIListKind(L),L,N) a__U91(X1,X2,X3) -> U91(X1,X2,X3) a__U92(tt,L,N) -> a__U93(a__isNat(N),L,N) a__U92(X1,X2,X3) -> U92(X1,X2,X3) a__U93(tt,L,N) -> a__U94(a__isNatKind(N),L) a__U93(X1,X2,X3) -> U93(X1,X2,X3) a__U94(tt,L) -> s(a__length(mark(L))) a__U94(X1,X2) -> U94(X1,X2) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(X) -> isNatIList(X) a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U81(a__isNatKind(V1),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U91(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X1,X2)) -> a__U32(mark(X1),X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3) mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3) mark(U45(X1,X2)) -> a__U45(mark(X1),X2) mark(U46(X)) -> a__U46(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X1,X2,X3)) -> a__U81(mark(X1),X2,X3) mark(U82(X1,X2,X3)) -> a__U82(mark(X1),X2,X3) mark(U83(X1,X2,X3)) -> a__U83(mark(X1),X2,X3) mark(U84(X1,X2,X3)) -> a__U84(mark(X1),X2,X3) mark(U85(X1,X2)) -> a__U85(mark(X1),X2) mark(U86(X)) -> a__U86(mark(X)) mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3) mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3) mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3) mark(U94(X1,X2)) -> a__U94(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(zeros) -> a__zeros The problem is decomposed in 2 subproblems. Problem 1.2.1: Subterm Processor: -> Pairs: A__U21(tt,V1) -> A__U22(a__isNatKind(V1),V1) A__U22(tt,V1) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1),V1) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt,V1) -> a__U13(a__isNatList(V1)) a__U12(X1,X2) -> U12(X1,X2) a__U13(tt) -> tt a__U13(X) -> U13(X) a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt,V1) -> a__U23(a__isNat(V1)) a__U22(X1,X2) -> U22(X1,X2) a__U23(tt) -> tt a__U23(X) -> U23(X) a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt,V) -> a__U33(a__isNatList(V)) a__U32(X1,X2) -> U32(X1,X2) a__U33(tt) -> tt a__U33(X) -> U33(X) a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2) a__U43(X1,X2,X3) -> U43(X1,X2,X3) a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2) a__U44(X1,X2,X3) -> U44(X1,X2,X3) a__U45(tt,V2) -> a__U46(a__isNatIList(V2)) a__U45(X1,X2) -> U45(X1,X2) a__U46(tt) -> tt a__U46(X) -> U46(X) a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt) -> tt a__U61(X) -> U61(X) a__U71(tt) -> tt a__U71(X) -> U71(X) a__U81(tt,V1,V2) -> a__U82(a__isNatKind(V1),V1,V2) a__U81(X1,X2,X3) -> U81(X1,X2,X3) a__U82(tt,V1,V2) -> a__U83(a__isNatIListKind(V2),V1,V2) a__U82(X1,X2,X3) -> U82(X1,X2,X3) a__U83(tt,V1,V2) -> a__U84(a__isNatIListKind(V2),V1,V2) a__U83(X1,X2,X3) -> U83(X1,X2,X3) a__U84(tt,V1,V2) -> a__U85(a__isNat(V1),V2) a__U84(X1,X2,X3) -> U84(X1,X2,X3) a__U85(tt,V2) -> a__U86(a__isNatList(V2)) a__U85(X1,X2) -> U85(X1,X2) a__U86(tt) -> tt a__U86(X) -> U86(X) a__U91(tt,L,N) -> a__U92(a__isNatIListKind(L),L,N) a__U91(X1,X2,X3) -> U91(X1,X2,X3) a__U92(tt,L,N) -> a__U93(a__isNat(N),L,N) a__U92(X1,X2,X3) -> U92(X1,X2,X3) a__U93(tt,L,N) -> a__U94(a__isNatKind(N),L) a__U93(X1,X2,X3) -> U93(X1,X2,X3) a__U94(tt,L) -> s(a__length(mark(L))) a__U94(X1,X2) -> U94(X1,X2) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(X) -> isNatIList(X) a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U81(a__isNatKind(V1),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U91(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X1,X2)) -> a__U32(mark(X1),X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3) mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3) mark(U45(X1,X2)) -> a__U45(mark(X1),X2) mark(U46(X)) -> a__U46(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X1,X2,X3)) -> a__U81(mark(X1),X2,X3) mark(U82(X1,X2,X3)) -> a__U82(mark(X1),X2,X3) mark(U83(X1,X2,X3)) -> a__U83(mark(X1),X2,X3) mark(U84(X1,X2,X3)) -> a__U84(mark(X1),X2,X3) mark(U85(X1,X2)) -> a__U85(mark(X1),X2) mark(U86(X)) -> a__U86(mark(X)) mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3) mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3) mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3) mark(U94(X1,X2)) -> a__U94(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(zeros) -> a__zeros ->Projection: pi(A__U21) = 2 pi(A__U22) = 2 pi(A__ISNAT) = 1 Problem 1.2.1: SCC Processor: -> Pairs: A__U21(tt,V1) -> A__U22(a__isNatKind(V1),V1) A__U22(tt,V1) -> A__ISNAT(V1) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt,V1) -> a__U13(a__isNatList(V1)) a__U12(X1,X2) -> U12(X1,X2) a__U13(tt) -> tt a__U13(X) -> U13(X) a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt,V1) -> a__U23(a__isNat(V1)) a__U22(X1,X2) -> U22(X1,X2) a__U23(tt) -> tt a__U23(X) -> U23(X) a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt,V) -> a__U33(a__isNatList(V)) a__U32(X1,X2) -> U32(X1,X2) a__U33(tt) -> tt a__U33(X) -> U33(X) a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2) a__U43(X1,X2,X3) -> U43(X1,X2,X3) a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2) a__U44(X1,X2,X3) -> U44(X1,X2,X3) a__U45(tt,V2) -> a__U46(a__isNatIList(V2)) a__U45(X1,X2) -> U45(X1,X2) a__U46(tt) -> tt a__U46(X) -> U46(X) a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt) -> tt a__U61(X) -> U61(X) a__U71(tt) -> tt a__U71(X) -> U71(X) a__U81(tt,V1,V2) -> a__U82(a__isNatKind(V1),V1,V2) a__U81(X1,X2,X3) -> U81(X1,X2,X3) a__U82(tt,V1,V2) -> a__U83(a__isNatIListKind(V2),V1,V2) a__U82(X1,X2,X3) -> U82(X1,X2,X3) a__U83(tt,V1,V2) -> a__U84(a__isNatIListKind(V2),V1,V2) a__U83(X1,X2,X3) -> U83(X1,X2,X3) a__U84(tt,V1,V2) -> a__U85(a__isNat(V1),V2) a__U84(X1,X2,X3) -> U84(X1,X2,X3) a__U85(tt,V2) -> a__U86(a__isNatList(V2)) a__U85(X1,X2) -> U85(X1,X2) a__U86(tt) -> tt a__U86(X) -> U86(X) a__U91(tt,L,N) -> a__U92(a__isNatIListKind(L),L,N) a__U91(X1,X2,X3) -> U91(X1,X2,X3) a__U92(tt,L,N) -> a__U93(a__isNat(N),L,N) a__U92(X1,X2,X3) -> U92(X1,X2,X3) a__U93(tt,L,N) -> a__U94(a__isNatKind(N),L) a__U93(X1,X2,X3) -> U93(X1,X2,X3) a__U94(tt,L) -> s(a__length(mark(L))) a__U94(X1,X2) -> U94(X1,X2) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(X) -> isNatIList(X) a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U81(a__isNatKind(V1),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U91(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X1,X2)) -> a__U32(mark(X1),X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3) mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3) mark(U45(X1,X2)) -> a__U45(mark(X1),X2) mark(U46(X)) -> a__U46(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X1,X2,X3)) -> a__U81(mark(X1),X2,X3) mark(U82(X1,X2,X3)) -> a__U82(mark(X1),X2,X3) mark(U83(X1,X2,X3)) -> a__U83(mark(X1),X2,X3) mark(U84(X1,X2,X3)) -> a__U84(mark(X1),X2,X3) mark(U85(X1,X2)) -> a__U85(mark(X1),X2) mark(U86(X)) -> a__U86(mark(X)) mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3) mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3) mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3) mark(U94(X1,X2)) -> a__U94(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(zeros) -> a__zeros ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2.2: Subterm Processor: -> Pairs: A__U81(tt,V1,V2) -> A__U82(a__isNatKind(V1),V1,V2) A__U82(tt,V1,V2) -> A__U83(a__isNatIListKind(V2),V1,V2) A__U83(tt,V1,V2) -> A__U84(a__isNatIListKind(V2),V1,V2) A__U84(tt,V1,V2) -> A__U85(a__isNat(V1),V2) A__U85(tt,V2) -> A__ISNATLIST(V2) A__ISNATLIST(cons(V1,V2)) -> A__U81(a__isNatKind(V1),V1,V2) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt,V1) -> a__U13(a__isNatList(V1)) a__U12(X1,X2) -> U12(X1,X2) a__U13(tt) -> tt a__U13(X) -> U13(X) a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt,V1) -> a__U23(a__isNat(V1)) a__U22(X1,X2) -> U22(X1,X2) a__U23(tt) -> tt a__U23(X) -> U23(X) a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt,V) -> a__U33(a__isNatList(V)) a__U32(X1,X2) -> U32(X1,X2) a__U33(tt) -> tt a__U33(X) -> U33(X) a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2) a__U43(X1,X2,X3) -> U43(X1,X2,X3) a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2) a__U44(X1,X2,X3) -> U44(X1,X2,X3) a__U45(tt,V2) -> a__U46(a__isNatIList(V2)) a__U45(X1,X2) -> U45(X1,X2) a__U46(tt) -> tt a__U46(X) -> U46(X) a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt) -> tt a__U61(X) -> U61(X) a__U71(tt) -> tt a__U71(X) -> U71(X) a__U81(tt,V1,V2) -> a__U82(a__isNatKind(V1),V1,V2) a__U81(X1,X2,X3) -> U81(X1,X2,X3) a__U82(tt,V1,V2) -> a__U83(a__isNatIListKind(V2),V1,V2) a__U82(X1,X2,X3) -> U82(X1,X2,X3) a__U83(tt,V1,V2) -> a__U84(a__isNatIListKind(V2),V1,V2) a__U83(X1,X2,X3) -> U83(X1,X2,X3) a__U84(tt,V1,V2) -> a__U85(a__isNat(V1),V2) a__U84(X1,X2,X3) -> U84(X1,X2,X3) a__U85(tt,V2) -> a__U86(a__isNatList(V2)) a__U85(X1,X2) -> U85(X1,X2) a__U86(tt) -> tt a__U86(X) -> U86(X) a__U91(tt,L,N) -> a__U92(a__isNatIListKind(L),L,N) a__U91(X1,X2,X3) -> U91(X1,X2,X3) a__U92(tt,L,N) -> a__U93(a__isNat(N),L,N) a__U92(X1,X2,X3) -> U92(X1,X2,X3) a__U93(tt,L,N) -> a__U94(a__isNatKind(N),L) a__U93(X1,X2,X3) -> U93(X1,X2,X3) a__U94(tt,L) -> s(a__length(mark(L))) a__U94(X1,X2) -> U94(X1,X2) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(X) -> isNatIList(X) a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U81(a__isNatKind(V1),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U91(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X1,X2)) -> a__U32(mark(X1),X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3) mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3) mark(U45(X1,X2)) -> a__U45(mark(X1),X2) mark(U46(X)) -> a__U46(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X1,X2,X3)) -> a__U81(mark(X1),X2,X3) mark(U82(X1,X2,X3)) -> a__U82(mark(X1),X2,X3) mark(U83(X1,X2,X3)) -> a__U83(mark(X1),X2,X3) mark(U84(X1,X2,X3)) -> a__U84(mark(X1),X2,X3) mark(U85(X1,X2)) -> a__U85(mark(X1),X2) mark(U86(X)) -> a__U86(mark(X)) mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3) mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3) mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3) mark(U94(X1,X2)) -> a__U94(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(zeros) -> a__zeros ->Projection: pi(A__U81) = 3 pi(A__U82) = 3 pi(A__U83) = 3 pi(A__U84) = 3 pi(A__U85) = 2 pi(A__ISNATLIST) = 1 Problem 1.2.2: SCC Processor: -> Pairs: A__U81(tt,V1,V2) -> A__U82(a__isNatKind(V1),V1,V2) A__U82(tt,V1,V2) -> A__U83(a__isNatIListKind(V2),V1,V2) A__U83(tt,V1,V2) -> A__U84(a__isNatIListKind(V2),V1,V2) A__U84(tt,V1,V2) -> A__U85(a__isNat(V1),V2) A__U85(tt,V2) -> A__ISNATLIST(V2) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt,V1) -> a__U13(a__isNatList(V1)) a__U12(X1,X2) -> U12(X1,X2) a__U13(tt) -> tt a__U13(X) -> U13(X) a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt,V1) -> a__U23(a__isNat(V1)) a__U22(X1,X2) -> U22(X1,X2) a__U23(tt) -> tt a__U23(X) -> U23(X) a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt,V) -> a__U33(a__isNatList(V)) a__U32(X1,X2) -> U32(X1,X2) a__U33(tt) -> tt a__U33(X) -> U33(X) a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2) a__U43(X1,X2,X3) -> U43(X1,X2,X3) a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2) a__U44(X1,X2,X3) -> U44(X1,X2,X3) a__U45(tt,V2) -> a__U46(a__isNatIList(V2)) a__U45(X1,X2) -> U45(X1,X2) a__U46(tt) -> tt a__U46(X) -> U46(X) a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt) -> tt a__U61(X) -> U61(X) a__U71(tt) -> tt a__U71(X) -> U71(X) a__U81(tt,V1,V2) -> a__U82(a__isNatKind(V1),V1,V2) a__U81(X1,X2,X3) -> U81(X1,X2,X3) a__U82(tt,V1,V2) -> a__U83(a__isNatIListKind(V2),V1,V2) a__U82(X1,X2,X3) -> U82(X1,X2,X3) a__U83(tt,V1,V2) -> a__U84(a__isNatIListKind(V2),V1,V2) a__U83(X1,X2,X3) -> U83(X1,X2,X3) a__U84(tt,V1,V2) -> a__U85(a__isNat(V1),V2) a__U84(X1,X2,X3) -> U84(X1,X2,X3) a__U85(tt,V2) -> a__U86(a__isNatList(V2)) a__U85(X1,X2) -> U85(X1,X2) a__U86(tt) -> tt a__U86(X) -> U86(X) a__U91(tt,L,N) -> a__U92(a__isNatIListKind(L),L,N) a__U91(X1,X2,X3) -> U91(X1,X2,X3) a__U92(tt,L,N) -> a__U93(a__isNat(N),L,N) a__U92(X1,X2,X3) -> U92(X1,X2,X3) a__U93(tt,L,N) -> a__U94(a__isNatKind(N),L) a__U93(X1,X2,X3) -> U93(X1,X2,X3) a__U94(tt,L) -> s(a__length(mark(L))) a__U94(X1,X2) -> U94(X1,X2) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(X) -> isNatIList(X) a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U81(a__isNatKind(V1),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U91(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X1,X2)) -> a__U32(mark(X1),X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3) mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3) mark(U45(X1,X2)) -> a__U45(mark(X1),X2) mark(U46(X)) -> a__U46(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X1,X2,X3)) -> a__U81(mark(X1),X2,X3) mark(U82(X1,X2,X3)) -> a__U82(mark(X1),X2,X3) mark(U83(X1,X2,X3)) -> a__U83(mark(X1),X2,X3) mark(U84(X1,X2,X3)) -> a__U84(mark(X1),X2,X3) mark(U85(X1,X2)) -> a__U85(mark(X1),X2) mark(U86(X)) -> a__U86(mark(X)) mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3) mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3) mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3) mark(U94(X1,X2)) -> a__U94(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(zeros) -> a__zeros ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Subterm Processor: -> Pairs: A__U41(tt,V1,V2) -> A__U42(a__isNatKind(V1),V1,V2) A__U42(tt,V1,V2) -> A__U43(a__isNatIListKind(V2),V1,V2) A__U43(tt,V1,V2) -> A__U44(a__isNatIListKind(V2),V1,V2) A__U44(tt,V1,V2) -> A__U45(a__isNat(V1),V2) A__U45(tt,V2) -> A__ISNATILIST(V2) A__ISNATILIST(cons(V1,V2)) -> A__U41(a__isNatKind(V1),V1,V2) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt,V1) -> a__U13(a__isNatList(V1)) a__U12(X1,X2) -> U12(X1,X2) a__U13(tt) -> tt a__U13(X) -> U13(X) a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt,V1) -> a__U23(a__isNat(V1)) a__U22(X1,X2) -> U22(X1,X2) a__U23(tt) -> tt a__U23(X) -> U23(X) a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt,V) -> a__U33(a__isNatList(V)) a__U32(X1,X2) -> U32(X1,X2) a__U33(tt) -> tt a__U33(X) -> U33(X) a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2) a__U43(X1,X2,X3) -> U43(X1,X2,X3) a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2) a__U44(X1,X2,X3) -> U44(X1,X2,X3) a__U45(tt,V2) -> a__U46(a__isNatIList(V2)) a__U45(X1,X2) -> U45(X1,X2) a__U46(tt) -> tt a__U46(X) -> U46(X) a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt) -> tt a__U61(X) -> U61(X) a__U71(tt) -> tt a__U71(X) -> U71(X) a__U81(tt,V1,V2) -> a__U82(a__isNatKind(V1),V1,V2) a__U81(X1,X2,X3) -> U81(X1,X2,X3) a__U82(tt,V1,V2) -> a__U83(a__isNatIListKind(V2),V1,V2) a__U82(X1,X2,X3) -> U82(X1,X2,X3) a__U83(tt,V1,V2) -> a__U84(a__isNatIListKind(V2),V1,V2) a__U83(X1,X2,X3) -> U83(X1,X2,X3) a__U84(tt,V1,V2) -> a__U85(a__isNat(V1),V2) a__U84(X1,X2,X3) -> U84(X1,X2,X3) a__U85(tt,V2) -> a__U86(a__isNatList(V2)) a__U85(X1,X2) -> U85(X1,X2) a__U86(tt) -> tt a__U86(X) -> U86(X) a__U91(tt,L,N) -> a__U92(a__isNatIListKind(L),L,N) a__U91(X1,X2,X3) -> U91(X1,X2,X3) a__U92(tt,L,N) -> a__U93(a__isNat(N),L,N) a__U92(X1,X2,X3) -> U92(X1,X2,X3) a__U93(tt,L,N) -> a__U94(a__isNatKind(N),L) a__U93(X1,X2,X3) -> U93(X1,X2,X3) a__U94(tt,L) -> s(a__length(mark(L))) a__U94(X1,X2) -> U94(X1,X2) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(X) -> isNatIList(X) a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U81(a__isNatKind(V1),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U91(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X1,X2)) -> a__U32(mark(X1),X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3) mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3) mark(U45(X1,X2)) -> a__U45(mark(X1),X2) mark(U46(X)) -> a__U46(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X1,X2,X3)) -> a__U81(mark(X1),X2,X3) mark(U82(X1,X2,X3)) -> a__U82(mark(X1),X2,X3) mark(U83(X1,X2,X3)) -> a__U83(mark(X1),X2,X3) mark(U84(X1,X2,X3)) -> a__U84(mark(X1),X2,X3) mark(U85(X1,X2)) -> a__U85(mark(X1),X2) mark(U86(X)) -> a__U86(mark(X)) mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3) mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3) mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3) mark(U94(X1,X2)) -> a__U94(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(zeros) -> a__zeros ->Projection: pi(A__U41) = 3 pi(A__U42) = 3 pi(A__U43) = 3 pi(A__U44) = 3 pi(A__U45) = 2 pi(A__ISNATILIST) = 1 Problem 1.3: SCC Processor: -> Pairs: A__U41(tt,V1,V2) -> A__U42(a__isNatKind(V1),V1,V2) A__U42(tt,V1,V2) -> A__U43(a__isNatIListKind(V2),V1,V2) A__U43(tt,V1,V2) -> A__U44(a__isNatIListKind(V2),V1,V2) A__U44(tt,V1,V2) -> A__U45(a__isNat(V1),V2) A__U45(tt,V2) -> A__ISNATILIST(V2) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt,V1) -> a__U13(a__isNatList(V1)) a__U12(X1,X2) -> U12(X1,X2) a__U13(tt) -> tt a__U13(X) -> U13(X) a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt,V1) -> a__U23(a__isNat(V1)) a__U22(X1,X2) -> U22(X1,X2) a__U23(tt) -> tt a__U23(X) -> U23(X) a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt,V) -> a__U33(a__isNatList(V)) a__U32(X1,X2) -> U32(X1,X2) a__U33(tt) -> tt a__U33(X) -> U33(X) a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2) a__U43(X1,X2,X3) -> U43(X1,X2,X3) a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2) a__U44(X1,X2,X3) -> U44(X1,X2,X3) a__U45(tt,V2) -> a__U46(a__isNatIList(V2)) a__U45(X1,X2) -> U45(X1,X2) a__U46(tt) -> tt a__U46(X) -> U46(X) a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt) -> tt a__U61(X) -> U61(X) a__U71(tt) -> tt a__U71(X) -> U71(X) a__U81(tt,V1,V2) -> a__U82(a__isNatKind(V1),V1,V2) a__U81(X1,X2,X3) -> U81(X1,X2,X3) a__U82(tt,V1,V2) -> a__U83(a__isNatIListKind(V2),V1,V2) a__U82(X1,X2,X3) -> U82(X1,X2,X3) a__U83(tt,V1,V2) -> a__U84(a__isNatIListKind(V2),V1,V2) a__U83(X1,X2,X3) -> U83(X1,X2,X3) a__U84(tt,V1,V2) -> a__U85(a__isNat(V1),V2) a__U84(X1,X2,X3) -> U84(X1,X2,X3) a__U85(tt,V2) -> a__U86(a__isNatList(V2)) a__U85(X1,X2) -> U85(X1,X2) a__U86(tt) -> tt a__U86(X) -> U86(X) a__U91(tt,L,N) -> a__U92(a__isNatIListKind(L),L,N) a__U91(X1,X2,X3) -> U91(X1,X2,X3) a__U92(tt,L,N) -> a__U93(a__isNat(N),L,N) a__U92(X1,X2,X3) -> U92(X1,X2,X3) a__U93(tt,L,N) -> a__U94(a__isNatKind(N),L) a__U93(X1,X2,X3) -> U93(X1,X2,X3) a__U94(tt,L) -> s(a__length(mark(L))) a__U94(X1,X2) -> U94(X1,X2) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(X) -> isNatIList(X) a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U81(a__isNatKind(V1),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U91(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X1,X2)) -> a__U32(mark(X1),X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3) mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3) mark(U45(X1,X2)) -> a__U45(mark(X1),X2) mark(U46(X)) -> a__U46(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X1,X2,X3)) -> a__U81(mark(X1),X2,X3) mark(U82(X1,X2,X3)) -> a__U82(mark(X1),X2,X3) mark(U83(X1,X2,X3)) -> a__U83(mark(X1),X2,X3) mark(U84(X1,X2,X3)) -> a__U84(mark(X1),X2,X3) mark(U85(X1,X2)) -> a__U85(mark(X1),X2) mark(U86(X)) -> a__U86(mark(X)) mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3) mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3) mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3) mark(U94(X1,X2)) -> a__U94(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(zeros) -> a__zeros ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.4: Reduction Pair Processor: -> Pairs: A__U91(tt,L,N) -> A__U92(a__isNatIListKind(L),L,N) A__U92(tt,L,N) -> A__U93(a__isNat(N),L,N) A__U93(tt,L,N) -> A__U94(a__isNatKind(N),L) A__U94(tt,L) -> A__LENGTH(mark(L)) A__U94(tt,L) -> MARK(L) A__LENGTH(cons(N,L)) -> A__U91(a__isNatList(L),L,N) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X1,X2)) -> MARK(X1) MARK(U33(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2,X3)) -> MARK(X1) MARK(U43(X1,X2,X3)) -> MARK(X1) MARK(U44(X1,X2,X3)) -> MARK(X1) MARK(U45(X1,X2)) -> MARK(X1) MARK(U46(X)) -> MARK(X) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(U61(X)) -> MARK(X) MARK(U71(X)) -> MARK(X) MARK(U81(X1,X2,X3)) -> MARK(X1) MARK(U82(X1,X2,X3)) -> MARK(X1) MARK(U83(X1,X2,X3)) -> MARK(X1) MARK(U84(X1,X2,X3)) -> MARK(X1) MARK(U85(X1,X2)) -> MARK(X1) MARK(U86(X)) -> MARK(X) MARK(U91(X1,X2,X3)) -> A__U91(mark(X1),X2,X3) MARK(U91(X1,X2,X3)) -> MARK(X1) MARK(U92(X1,X2,X3)) -> A__U92(mark(X1),X2,X3) MARK(U92(X1,X2,X3)) -> MARK(X1) MARK(U93(X1,X2,X3)) -> A__U93(mark(X1),X2,X3) MARK(U93(X1,X2,X3)) -> MARK(X1) MARK(U94(X1,X2)) -> A__U94(mark(X1),X2) MARK(U94(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(length(X)) -> A__LENGTH(mark(X)) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt,V1) -> a__U13(a__isNatList(V1)) a__U12(X1,X2) -> U12(X1,X2) a__U13(tt) -> tt a__U13(X) -> U13(X) a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt,V1) -> a__U23(a__isNat(V1)) a__U22(X1,X2) -> U22(X1,X2) a__U23(tt) -> tt a__U23(X) -> U23(X) a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt,V) -> a__U33(a__isNatList(V)) a__U32(X1,X2) -> U32(X1,X2) a__U33(tt) -> tt a__U33(X) -> U33(X) a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2) a__U43(X1,X2,X3) -> U43(X1,X2,X3) a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2) a__U44(X1,X2,X3) -> U44(X1,X2,X3) a__U45(tt,V2) -> a__U46(a__isNatIList(V2)) a__U45(X1,X2) -> U45(X1,X2) a__U46(tt) -> tt a__U46(X) -> U46(X) a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt) -> tt a__U61(X) -> U61(X) a__U71(tt) -> tt a__U71(X) -> U71(X) a__U81(tt,V1,V2) -> a__U82(a__isNatKind(V1),V1,V2) a__U81(X1,X2,X3) -> U81(X1,X2,X3) a__U82(tt,V1,V2) -> a__U83(a__isNatIListKind(V2),V1,V2) a__U82(X1,X2,X3) -> U82(X1,X2,X3) a__U83(tt,V1,V2) -> a__U84(a__isNatIListKind(V2),V1,V2) a__U83(X1,X2,X3) -> U83(X1,X2,X3) a__U84(tt,V1,V2) -> a__U85(a__isNat(V1),V2) a__U84(X1,X2,X3) -> U84(X1,X2,X3) a__U85(tt,V2) -> a__U86(a__isNatList(V2)) a__U85(X1,X2) -> U85(X1,X2) a__U86(tt) -> tt a__U86(X) -> U86(X) a__U91(tt,L,N) -> a__U92(a__isNatIListKind(L),L,N) a__U91(X1,X2,X3) -> U91(X1,X2,X3) a__U92(tt,L,N) -> a__U93(a__isNat(N),L,N) a__U92(X1,X2,X3) -> U92(X1,X2,X3) a__U93(tt,L,N) -> a__U94(a__isNatKind(N),L) a__U93(X1,X2,X3) -> U93(X1,X2,X3) a__U94(tt,L) -> s(a__length(mark(L))) a__U94(X1,X2) -> U94(X1,X2) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(X) -> isNatIList(X) a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U81(a__isNatKind(V1),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U91(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X1,X2)) -> a__U32(mark(X1),X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3) mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3) mark(U45(X1,X2)) -> a__U45(mark(X1),X2) mark(U46(X)) -> a__U46(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X1,X2,X3)) -> a__U81(mark(X1),X2,X3) mark(U82(X1,X2,X3)) -> a__U82(mark(X1),X2,X3) mark(U83(X1,X2,X3)) -> a__U83(mark(X1),X2,X3) mark(U84(X1,X2,X3)) -> a__U84(mark(X1),X2,X3) mark(U85(X1,X2)) -> a__U85(mark(X1),X2) mark(U86(X)) -> a__U86(mark(X)) mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3) mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3) mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3) mark(U94(X1,X2)) -> a__U94(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(zeros) -> a__zeros -> Usable rules: a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt,V1) -> a__U13(a__isNatList(V1)) a__U12(X1,X2) -> U12(X1,X2) a__U13(tt) -> tt a__U13(X) -> U13(X) a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt,V1) -> a__U23(a__isNat(V1)) a__U22(X1,X2) -> U22(X1,X2) a__U23(tt) -> tt a__U23(X) -> U23(X) a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt,V) -> a__U33(a__isNatList(V)) a__U32(X1,X2) -> U32(X1,X2) a__U33(tt) -> tt a__U33(X) -> U33(X) a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2) a__U43(X1,X2,X3) -> U43(X1,X2,X3) a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2) a__U44(X1,X2,X3) -> U44(X1,X2,X3) a__U45(tt,V2) -> a__U46(a__isNatIList(V2)) a__U45(X1,X2) -> U45(X1,X2) a__U46(tt) -> tt a__U46(X) -> U46(X) a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt) -> tt a__U61(X) -> U61(X) a__U71(tt) -> tt a__U71(X) -> U71(X) a__U81(tt,V1,V2) -> a__U82(a__isNatKind(V1),V1,V2) a__U81(X1,X2,X3) -> U81(X1,X2,X3) a__U82(tt,V1,V2) -> a__U83(a__isNatIListKind(V2),V1,V2) a__U82(X1,X2,X3) -> U82(X1,X2,X3) a__U83(tt,V1,V2) -> a__U84(a__isNatIListKind(V2),V1,V2) a__U83(X1,X2,X3) -> U83(X1,X2,X3) a__U84(tt,V1,V2) -> a__U85(a__isNat(V1),V2) a__U84(X1,X2,X3) -> U84(X1,X2,X3) a__U85(tt,V2) -> a__U86(a__isNatList(V2)) a__U85(X1,X2) -> U85(X1,X2) a__U86(tt) -> tt a__U86(X) -> U86(X) a__U91(tt,L,N) -> a__U92(a__isNatIListKind(L),L,N) a__U91(X1,X2,X3) -> U91(X1,X2,X3) a__U92(tt,L,N) -> a__U93(a__isNat(N),L,N) a__U92(X1,X2,X3) -> U92(X1,X2,X3) a__U93(tt,L,N) -> a__U94(a__isNatKind(N),L) a__U93(X1,X2,X3) -> U93(X1,X2,X3) a__U94(tt,L) -> s(a__length(mark(L))) a__U94(X1,X2) -> U94(X1,X2) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(X) -> isNatIList(X) a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U81(a__isNatKind(V1),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U91(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X1,X2)) -> a__U32(mark(X1),X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3) mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3) mark(U45(X1,X2)) -> a__U45(mark(X1),X2) mark(U46(X)) -> a__U46(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X1,X2,X3)) -> a__U81(mark(X1),X2,X3) mark(U82(X1,X2,X3)) -> a__U82(mark(X1),X2,X3) mark(U83(X1,X2,X3)) -> a__U83(mark(X1),X2,X3) mark(U84(X1,X2,X3)) -> a__U84(mark(X1),X2,X3) mark(U85(X1,X2)) -> a__U85(mark(X1),X2) mark(U86(X)) -> a__U86(mark(X)) mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3) mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3) mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3) mark(U94(X1,X2)) -> a__U94(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(zeros) -> a__zeros ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__U11](X1,X2) = X1 + X2 + 2 [a__U12](X1,X2) = X1 + X2 + 2 [a__U13](X) = X + 2 [a__U21](X1,X2) = X1 + X2 [a__U22](X1,X2) = X1 + X2 [a__U23](X) = X [a__U31](X1,X2) = X1 + 2.X2 [a__U32](X1,X2) = X1 + 2.X2 [a__U33](X) = X [a__U41](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [a__U42](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [a__U43](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [a__U44](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [a__U45](X1,X2) = X1 + 2.X2 + 2 [a__U46](X) = X + 2 [a__U51](X1,X2) = X1 [a__U52](X) = X [a__U61](X) = X [a__U71](X) = X [a__U81](X1,X2,X3) = X1 + X2 + 2.X3 + 1 [a__U82](X1,X2,X3) = X1 + X2 + 2.X3 + 1 [a__U83](X1,X2,X3) = X1 + X2 + 2.X3 + 1 [a__U84](X1,X2,X3) = X1 + X2 + 2.X3 + 1 [a__U85](X1,X2) = X1 + X2 + 1 [a__U86](X) = X + 2 [a__U91](X1,X2,X3) = X1 + X2 + X3 + 2 [a__U92](X1,X2,X3) = X1 + X2 + X3 + 2 [a__U93](X1,X2,X3) = X1 + X2 + 2 [a__U94](X1,X2) = X1 + X2 + 2 [a__isNat](X) = X + 2 [a__isNatIList](X) = 2.X + 2 [a__isNatIListKind](X) = 2 [a__isNatKind](X) = 2 [a__isNatList](X) = X + 1 [a__length](X) = X + 2 [a__zeros] = 2 [mark](X) = X + 2 [0] = 0 [U11](X1,X2) = X1 + X2 + 2 [U12](X1,X2) = X1 + X2 + 2 [U13](X) = X + 2 [U21](X1,X2) = X1 + X2 [U22](X1,X2) = X1 + X2 [U23](X) = X [U31](X1,X2) = X1 + 2.X2 [U32](X1,X2) = X1 + 2.X2 [U33](X) = X [U41](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U42](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U43](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U44](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U45](X1,X2) = X1 + 2.X2 + 2 [U46](X) = X + 2 [U51](X1,X2) = X1 [U52](X) = X [U61](X) = X [U71](X) = X [U81](X1,X2,X3) = X1 + X2 + 2.X3 + 1 [U82](X1,X2,X3) = X1 + X2 + 2.X3 + 1 [U83](X1,X2,X3) = X1 + X2 + 2.X3 + 1 [U84](X1,X2,X3) = X1 + X2 + 2.X3 + 1 [U85](X1,X2) = X1 + X2 + 1 [U86](X) = X + 2 [U91](X1,X2,X3) = X1 + X2 + X3 + 2 [U92](X1,X2,X3) = X1 + X2 + X3 + 2 [U93](X1,X2,X3) = X1 + X2 + 2 [U94](X1,X2) = X1 + X2 + 2 [cons](X1,X2) = X1 + 2.X2 + 2 [isNat](X) = X + 1 [isNatIList](X) = 2.X + 1 [isNatIListKind](X) = 2 [isNatKind](X) = 0 [isNatList](X) = X + 1 [length](X) = X + 2 [nil] = 2 [s](X) = X [tt] = 2 [zeros] = 0 [A__U91](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [A__U92](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1 [A__U93](X1,X2,X3) = 2.X1 + 2.X2 + 1 [A__U94](X1,X2) = 2.X1 + 2.X2 + 1 [A__LENGTH](X) = 2.X + 1 [MARK](X) = 2.X + 2 Problem 1.4: SCC Processor: -> Pairs: A__U92(tt,L,N) -> A__U93(a__isNat(N),L,N) A__U93(tt,L,N) -> A__U94(a__isNatKind(N),L) A__U94(tt,L) -> A__LENGTH(mark(L)) A__U94(tt,L) -> MARK(L) A__LENGTH(cons(N,L)) -> A__U91(a__isNatList(L),L,N) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X1,X2)) -> MARK(X1) MARK(U33(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2,X3)) -> MARK(X1) MARK(U43(X1,X2,X3)) -> MARK(X1) MARK(U44(X1,X2,X3)) -> MARK(X1) MARK(U45(X1,X2)) -> MARK(X1) MARK(U46(X)) -> MARK(X) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(U61(X)) -> MARK(X) MARK(U71(X)) -> MARK(X) MARK(U81(X1,X2,X3)) -> MARK(X1) MARK(U82(X1,X2,X3)) -> MARK(X1) MARK(U83(X1,X2,X3)) -> MARK(X1) MARK(U84(X1,X2,X3)) -> MARK(X1) MARK(U85(X1,X2)) -> MARK(X1) MARK(U86(X)) -> MARK(X) MARK(U91(X1,X2,X3)) -> A__U91(mark(X1),X2,X3) MARK(U91(X1,X2,X3)) -> MARK(X1) MARK(U92(X1,X2,X3)) -> A__U92(mark(X1),X2,X3) MARK(U92(X1,X2,X3)) -> MARK(X1) MARK(U93(X1,X2,X3)) -> A__U93(mark(X1),X2,X3) MARK(U93(X1,X2,X3)) -> MARK(X1) MARK(U94(X1,X2)) -> A__U94(mark(X1),X2) MARK(U94(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(length(X)) -> A__LENGTH(mark(X)) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt,V1) -> a__U13(a__isNatList(V1)) a__U12(X1,X2) -> U12(X1,X2) a__U13(tt) -> tt a__U13(X) -> U13(X) a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt,V1) -> a__U23(a__isNat(V1)) a__U22(X1,X2) -> U22(X1,X2) a__U23(tt) -> tt a__U23(X) -> U23(X) a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt,V) -> a__U33(a__isNatList(V)) a__U32(X1,X2) -> U32(X1,X2) a__U33(tt) -> tt a__U33(X) -> U33(X) a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2) a__U43(X1,X2,X3) -> U43(X1,X2,X3) a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2) a__U44(X1,X2,X3) -> U44(X1,X2,X3) a__U45(tt,V2) -> a__U46(a__isNatIList(V2)) a__U45(X1,X2) -> U45(X1,X2) a__U46(tt) -> tt a__U46(X) -> U46(X) a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt) -> tt a__U61(X) -> U61(X) a__U71(tt) -> tt a__U71(X) -> U71(X) a__U81(tt,V1,V2) -> a__U82(a__isNatKind(V1),V1,V2) a__U81(X1,X2,X3) -> U81(X1,X2,X3) a__U82(tt,V1,V2) -> a__U83(a__isNatIListKind(V2),V1,V2) a__U82(X1,X2,X3) -> U82(X1,X2,X3) a__U83(tt,V1,V2) -> a__U84(a__isNatIListKind(V2),V1,V2) a__U83(X1,X2,X3) -> U83(X1,X2,X3) a__U84(tt,V1,V2) -> a__U85(a__isNat(V1),V2) a__U84(X1,X2,X3) -> U84(X1,X2,X3) a__U85(tt,V2) -> a__U86(a__isNatList(V2)) a__U85(X1,X2) -> U85(X1,X2) a__U86(tt) -> tt a__U86(X) -> U86(X) a__U91(tt,L,N) -> a__U92(a__isNatIListKind(L),L,N) a__U91(X1,X2,X3) -> U91(X1,X2,X3) a__U92(tt,L,N) -> a__U93(a__isNat(N),L,N) a__U92(X1,X2,X3) -> U92(X1,X2,X3) a__U93(tt,L,N) -> a__U94(a__isNatKind(N),L) a__U93(X1,X2,X3) -> U93(X1,X2,X3) a__U94(tt,L) -> s(a__length(mark(L))) a__U94(X1,X2) -> U94(X1,X2) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(X) -> isNatIList(X) a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U81(a__isNatKind(V1),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U91(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X1,X2)) -> a__U32(mark(X1),X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3) mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3) mark(U45(X1,X2)) -> a__U45(mark(X1),X2) mark(U46(X)) -> a__U46(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X1,X2,X3)) -> a__U81(mark(X1),X2,X3) mark(U82(X1,X2,X3)) -> a__U82(mark(X1),X2,X3) mark(U83(X1,X2,X3)) -> a__U83(mark(X1),X2,X3) mark(U84(X1,X2,X3)) -> a__U84(mark(X1),X2,X3) mark(U85(X1,X2)) -> a__U85(mark(X1),X2) mark(U86(X)) -> a__U86(mark(X)) mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3) mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3) mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3) mark(U94(X1,X2)) -> a__U94(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(zeros) -> a__zeros ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__U92(tt,L,N) -> A__U93(a__isNat(N),L,N) A__U93(tt,L,N) -> A__U94(a__isNatKind(N),L) A__U94(tt,L) -> MARK(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X1,X2)) -> MARK(X1) MARK(U33(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2,X3)) -> MARK(X1) MARK(U43(X1,X2,X3)) -> MARK(X1) MARK(U44(X1,X2,X3)) -> MARK(X1) MARK(U45(X1,X2)) -> MARK(X1) MARK(U46(X)) -> MARK(X) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(U61(X)) -> MARK(X) MARK(U71(X)) -> MARK(X) MARK(U81(X1,X2,X3)) -> MARK(X1) MARK(U82(X1,X2,X3)) -> MARK(X1) MARK(U83(X1,X2,X3)) -> MARK(X1) MARK(U84(X1,X2,X3)) -> MARK(X1) MARK(U85(X1,X2)) -> MARK(X1) MARK(U86(X)) -> MARK(X) MARK(U91(X1,X2,X3)) -> MARK(X1) MARK(U92(X1,X2,X3)) -> A__U92(mark(X1),X2,X3) MARK(U92(X1,X2,X3)) -> MARK(X1) MARK(U93(X1,X2,X3)) -> A__U93(mark(X1),X2,X3) MARK(U93(X1,X2,X3)) -> MARK(X1) MARK(U94(X1,X2)) -> A__U94(mark(X1),X2) MARK(U94(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) ->->-> Rules: a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt,V1) -> a__U13(a__isNatList(V1)) a__U12(X1,X2) -> U12(X1,X2) a__U13(tt) -> tt a__U13(X) -> U13(X) a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt,V1) -> a__U23(a__isNat(V1)) a__U22(X1,X2) -> U22(X1,X2) a__U23(tt) -> tt a__U23(X) -> U23(X) a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt,V) -> a__U33(a__isNatList(V)) a__U32(X1,X2) -> U32(X1,X2) a__U33(tt) -> tt a__U33(X) -> U33(X) a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2) a__U43(X1,X2,X3) -> U43(X1,X2,X3) a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2) a__U44(X1,X2,X3) -> U44(X1,X2,X3) a__U45(tt,V2) -> a__U46(a__isNatIList(V2)) a__U45(X1,X2) -> U45(X1,X2) a__U46(tt) -> tt a__U46(X) -> U46(X) a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt) -> tt a__U61(X) -> U61(X) a__U71(tt) -> tt a__U71(X) -> U71(X) a__U81(tt,V1,V2) -> a__U82(a__isNatKind(V1),V1,V2) a__U81(X1,X2,X3) -> U81(X1,X2,X3) a__U82(tt,V1,V2) -> a__U83(a__isNatIListKind(V2),V1,V2) a__U82(X1,X2,X3) -> U82(X1,X2,X3) a__U83(tt,V1,V2) -> a__U84(a__isNatIListKind(V2),V1,V2) a__U83(X1,X2,X3) -> U83(X1,X2,X3) a__U84(tt,V1,V2) -> a__U85(a__isNat(V1),V2) a__U84(X1,X2,X3) -> U84(X1,X2,X3) a__U85(tt,V2) -> a__U86(a__isNatList(V2)) a__U85(X1,X2) -> U85(X1,X2) a__U86(tt) -> tt a__U86(X) -> U86(X) a__U91(tt,L,N) -> a__U92(a__isNatIListKind(L),L,N) a__U91(X1,X2,X3) -> U91(X1,X2,X3) a__U92(tt,L,N) -> a__U93(a__isNat(N),L,N) a__U92(X1,X2,X3) -> U92(X1,X2,X3) a__U93(tt,L,N) -> a__U94(a__isNatKind(N),L) a__U93(X1,X2,X3) -> U93(X1,X2,X3) a__U94(tt,L) -> s(a__length(mark(L))) a__U94(X1,X2) -> U94(X1,X2) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(X) -> isNatIList(X) a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U81(a__isNatKind(V1),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U91(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X1,X2)) -> a__U32(mark(X1),X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3) mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3) mark(U45(X1,X2)) -> a__U45(mark(X1),X2) mark(U46(X)) -> a__U46(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X1,X2,X3)) -> a__U81(mark(X1),X2,X3) mark(U82(X1,X2,X3)) -> a__U82(mark(X1),X2,X3) mark(U83(X1,X2,X3)) -> a__U83(mark(X1),X2,X3) mark(U84(X1,X2,X3)) -> a__U84(mark(X1),X2,X3) mark(U85(X1,X2)) -> a__U85(mark(X1),X2) mark(U86(X)) -> a__U86(mark(X)) mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3) mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3) mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3) mark(U94(X1,X2)) -> a__U94(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(zeros) -> a__zeros Problem 1.4: Subterm Processor: -> Pairs: A__U92(tt,L,N) -> A__U93(a__isNat(N),L,N) A__U93(tt,L,N) -> A__U94(a__isNatKind(N),L) A__U94(tt,L) -> MARK(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X1,X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X1,X2)) -> MARK(X1) MARK(U33(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2,X3)) -> MARK(X1) MARK(U43(X1,X2,X3)) -> MARK(X1) MARK(U44(X1,X2,X3)) -> MARK(X1) MARK(U45(X1,X2)) -> MARK(X1) MARK(U46(X)) -> MARK(X) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(U61(X)) -> MARK(X) MARK(U71(X)) -> MARK(X) MARK(U81(X1,X2,X3)) -> MARK(X1) MARK(U82(X1,X2,X3)) -> MARK(X1) MARK(U83(X1,X2,X3)) -> MARK(X1) MARK(U84(X1,X2,X3)) -> MARK(X1) MARK(U85(X1,X2)) -> MARK(X1) MARK(U86(X)) -> MARK(X) MARK(U91(X1,X2,X3)) -> MARK(X1) MARK(U92(X1,X2,X3)) -> A__U92(mark(X1),X2,X3) MARK(U92(X1,X2,X3)) -> MARK(X1) MARK(U93(X1,X2,X3)) -> A__U93(mark(X1),X2,X3) MARK(U93(X1,X2,X3)) -> MARK(X1) MARK(U94(X1,X2)) -> A__U94(mark(X1),X2) MARK(U94(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt,V1) -> a__U13(a__isNatList(V1)) a__U12(X1,X2) -> U12(X1,X2) a__U13(tt) -> tt a__U13(X) -> U13(X) a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt,V1) -> a__U23(a__isNat(V1)) a__U22(X1,X2) -> U22(X1,X2) a__U23(tt) -> tt a__U23(X) -> U23(X) a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt,V) -> a__U33(a__isNatList(V)) a__U32(X1,X2) -> U32(X1,X2) a__U33(tt) -> tt a__U33(X) -> U33(X) a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2) a__U43(X1,X2,X3) -> U43(X1,X2,X3) a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2) a__U44(X1,X2,X3) -> U44(X1,X2,X3) a__U45(tt,V2) -> a__U46(a__isNatIList(V2)) a__U45(X1,X2) -> U45(X1,X2) a__U46(tt) -> tt a__U46(X) -> U46(X) a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt) -> tt a__U61(X) -> U61(X) a__U71(tt) -> tt a__U71(X) -> U71(X) a__U81(tt,V1,V2) -> a__U82(a__isNatKind(V1),V1,V2) a__U81(X1,X2,X3) -> U81(X1,X2,X3) a__U82(tt,V1,V2) -> a__U83(a__isNatIListKind(V2),V1,V2) a__U82(X1,X2,X3) -> U82(X1,X2,X3) a__U83(tt,V1,V2) -> a__U84(a__isNatIListKind(V2),V1,V2) a__U83(X1,X2,X3) -> U83(X1,X2,X3) a__U84(tt,V1,V2) -> a__U85(a__isNat(V1),V2) a__U84(X1,X2,X3) -> U84(X1,X2,X3) a__U85(tt,V2) -> a__U86(a__isNatList(V2)) a__U85(X1,X2) -> U85(X1,X2) a__U86(tt) -> tt a__U86(X) -> U86(X) a__U91(tt,L,N) -> a__U92(a__isNatIListKind(L),L,N) a__U91(X1,X2,X3) -> U91(X1,X2,X3) a__U92(tt,L,N) -> a__U93(a__isNat(N),L,N) a__U92(X1,X2,X3) -> U92(X1,X2,X3) a__U93(tt,L,N) -> a__U94(a__isNatKind(N),L) a__U93(X1,X2,X3) -> U93(X1,X2,X3) a__U94(tt,L) -> s(a__length(mark(L))) a__U94(X1,X2) -> U94(X1,X2) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(X) -> isNatIList(X) a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U81(a__isNatKind(V1),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U91(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X1,X2)) -> a__U32(mark(X1),X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3) mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3) mark(U45(X1,X2)) -> a__U45(mark(X1),X2) mark(U46(X)) -> a__U46(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X1,X2,X3)) -> a__U81(mark(X1),X2,X3) mark(U82(X1,X2,X3)) -> a__U82(mark(X1),X2,X3) mark(U83(X1,X2,X3)) -> a__U83(mark(X1),X2,X3) mark(U84(X1,X2,X3)) -> a__U84(mark(X1),X2,X3) mark(U85(X1,X2)) -> a__U85(mark(X1),X2) mark(U86(X)) -> a__U86(mark(X)) mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3) mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3) mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3) mark(U94(X1,X2)) -> a__U94(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(zeros) -> a__zeros ->Projection: pi(A__U92) = 2 pi(A__U93) = 2 pi(A__U94) = 2 pi(MARK) = 1 Problem 1.4: SCC Processor: -> Pairs: A__U92(tt,L,N) -> A__U93(a__isNat(N),L,N) A__U93(tt,L,N) -> A__U94(a__isNatKind(N),L) A__U94(tt,L) -> MARK(L) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt,V1) -> a__U13(a__isNatList(V1)) a__U12(X1,X2) -> U12(X1,X2) a__U13(tt) -> tt a__U13(X) -> U13(X) a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt,V1) -> a__U23(a__isNat(V1)) a__U22(X1,X2) -> U22(X1,X2) a__U23(tt) -> tt a__U23(X) -> U23(X) a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt,V) -> a__U33(a__isNatList(V)) a__U32(X1,X2) -> U32(X1,X2) a__U33(tt) -> tt a__U33(X) -> U33(X) a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2) a__U43(X1,X2,X3) -> U43(X1,X2,X3) a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2) a__U44(X1,X2,X3) -> U44(X1,X2,X3) a__U45(tt,V2) -> a__U46(a__isNatIList(V2)) a__U45(X1,X2) -> U45(X1,X2) a__U46(tt) -> tt a__U46(X) -> U46(X) a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt) -> tt a__U61(X) -> U61(X) a__U71(tt) -> tt a__U71(X) -> U71(X) a__U81(tt,V1,V2) -> a__U82(a__isNatKind(V1),V1,V2) a__U81(X1,X2,X3) -> U81(X1,X2,X3) a__U82(tt,V1,V2) -> a__U83(a__isNatIListKind(V2),V1,V2) a__U82(X1,X2,X3) -> U82(X1,X2,X3) a__U83(tt,V1,V2) -> a__U84(a__isNatIListKind(V2),V1,V2) a__U83(X1,X2,X3) -> U83(X1,X2,X3) a__U84(tt,V1,V2) -> a__U85(a__isNat(V1),V2) a__U84(X1,X2,X3) -> U84(X1,X2,X3) a__U85(tt,V2) -> a__U86(a__isNatList(V2)) a__U85(X1,X2) -> U85(X1,X2) a__U86(tt) -> tt a__U86(X) -> U86(X) a__U91(tt,L,N) -> a__U92(a__isNatIListKind(L),L,N) a__U91(X1,X2,X3) -> U91(X1,X2,X3) a__U92(tt,L,N) -> a__U93(a__isNat(N),L,N) a__U92(X1,X2,X3) -> U92(X1,X2,X3) a__U93(tt,L,N) -> a__U94(a__isNatKind(N),L) a__U93(X1,X2,X3) -> U93(X1,X2,X3) a__U94(tt,L) -> s(a__length(mark(L))) a__U94(X1,X2) -> U94(X1,X2) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(X) -> isNatIList(X) a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U81(a__isNatKind(V1),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U91(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(U13(X)) -> a__U13(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X1,X2)) -> a__U22(mark(X1),X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X1,X2)) -> a__U32(mark(X1),X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3) mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3) mark(U45(X1,X2)) -> a__U45(mark(X1),X2) mark(U46(X)) -> a__U46(mark(X)) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X1,X2,X3)) -> a__U81(mark(X1),X2,X3) mark(U82(X1,X2,X3)) -> a__U82(mark(X1),X2,X3) mark(U83(X1,X2,X3)) -> a__U83(mark(X1),X2,X3) mark(U84(X1,X2,X3)) -> a__U84(mark(X1),X2,X3) mark(U85(X1,X2)) -> a__U85(mark(X1),X2) mark(U86(X)) -> a__U86(mark(X)) mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3) mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3) mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3) mark(U94(X1,X2)) -> a__U94(mark(X1),X2) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(tt) -> tt mark(zeros) -> a__zeros ->Strongly Connected Components: There is no strongly connected component The problem is finite.