/export/starexec/sandbox/solver/bin/starexec_run_default /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES Problem 1: (VAR L N V V1 V2 X X1 X2 X3) (RULES a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__isNatList(V1)) A__U11(tt,V1) -> A__ISNATLIST(V1) A__U21(tt,V1) -> A__U22(a__isNat(V1)) A__U21(tt,V1) -> A__ISNAT(V1) A__U31(tt,V) -> A__U32(a__isNatList(V)) A__U31(tt,V) -> A__ISNATLIST(V) A__U41(tt,V1,V2) -> A__U42(a__isNat(V1),V2) A__U41(tt,V1,V2) -> A__ISNAT(V1) A__U42(tt,V2) -> A__U43(a__isNatIList(V2)) A__U42(tt,V2) -> A__ISNATILIST(V2) A__U51(tt,V1,V2) -> A__U52(a__isNat(V1),V2) A__U51(tt,V1,V2) -> A__ISNAT(V1) A__U52(tt,V2) -> A__U53(a__isNatList(V2)) A__U52(tt,V2) -> A__ISNATLIST(V2) A__U61(tt,L) -> A__LENGTH(mark(L)) A__U61(tt,L) -> MARK(L) A__AND(tt,X) -> MARK(X) 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__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(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__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) 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(X)) -> A__U12(mark(X)) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> A__U21(mark(X1),X2) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> A__U22(mark(X)) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> A__U31(mark(X1),X2) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> A__U32(mark(X)) MARK(U32(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)) -> A__U42(mark(X1),X2) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> A__U43(mark(X)) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> A__U51(mark(X1),X2,X3) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> A__U52(mark(X1),X2) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> A__U53(mark(X)) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> A__U61(mark(X1),X2) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__isNatList(V1)) A__U11(tt,V1) -> A__ISNATLIST(V1) A__U21(tt,V1) -> A__U22(a__isNat(V1)) A__U21(tt,V1) -> A__ISNAT(V1) A__U31(tt,V) -> A__U32(a__isNatList(V)) A__U31(tt,V) -> A__ISNATLIST(V) A__U41(tt,V1,V2) -> A__U42(a__isNat(V1),V2) A__U41(tt,V1,V2) -> A__ISNAT(V1) A__U42(tt,V2) -> A__U43(a__isNatIList(V2)) A__U42(tt,V2) -> A__ISNATILIST(V2) A__U51(tt,V1,V2) -> A__U52(a__isNat(V1),V2) A__U51(tt,V1,V2) -> A__ISNAT(V1) A__U52(tt,V2) -> A__U53(a__isNatList(V2)) A__U52(tt,V2) -> A__ISNATLIST(V2) A__U61(tt,L) -> A__LENGTH(mark(L)) A__U61(tt,L) -> MARK(L) A__AND(tt,X) -> MARK(X) 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__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(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__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) 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(X)) -> A__U12(mark(X)) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> A__U21(mark(X1),X2) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> A__U22(mark(X)) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> A__U31(mark(X1),X2) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> A__U32(mark(X)) MARK(U32(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)) -> A__U42(mark(X1),X2) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> A__U43(mark(X)) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> A__U51(mark(X1),X2,X3) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> A__U52(mark(X1),X2) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> A__U53(mark(X)) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> A__U61(mark(X1),X2) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__U11(tt,V1) -> A__ISNATLIST(V1) A__U21(tt,V1) -> A__ISNAT(V1) A__U31(tt,V) -> A__ISNATLIST(V) A__U41(tt,V1,V2) -> A__U42(a__isNat(V1),V2) A__U41(tt,V1,V2) -> A__ISNAT(V1) A__U42(tt,V2) -> A__ISNATILIST(V2) A__U51(tt,V1,V2) -> A__U52(a__isNat(V1),V2) A__U51(tt,V1,V2) -> A__ISNAT(V1) A__U52(tt,V2) -> A__ISNATLIST(V2) A__U61(tt,L) -> A__LENGTH(mark(L)) A__U61(tt,L) -> MARK(L) A__AND(tt,X) -> MARK(X) 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__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(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__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) 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(X)) -> MARK(X) MARK(U21(X1,X2)) -> A__U21(mark(X1),X2) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> A__U31(mark(X1),X2) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(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)) -> A__U42(mark(X1),X2) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> A__U51(mark(X1),X2,X3) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> A__U52(mark(X1),X2) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> A__U61(mark(X1),X2) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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) ->->-> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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: Reduction Pair Processor: -> Pairs: A__U11(tt,V1) -> A__ISNATLIST(V1) A__U21(tt,V1) -> A__ISNAT(V1) A__U31(tt,V) -> A__ISNATLIST(V) A__U41(tt,V1,V2) -> A__U42(a__isNat(V1),V2) A__U41(tt,V1,V2) -> A__ISNAT(V1) A__U42(tt,V2) -> A__ISNATILIST(V2) A__U51(tt,V1,V2) -> A__U52(a__isNat(V1),V2) A__U51(tt,V1,V2) -> A__ISNAT(V1) A__U52(tt,V2) -> A__ISNATLIST(V2) A__U61(tt,L) -> A__LENGTH(mark(L)) A__U61(tt,L) -> MARK(L) A__AND(tt,X) -> MARK(X) 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__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(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__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) 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(X)) -> MARK(X) MARK(U21(X1,X2)) -> A__U21(mark(X1),X2) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> A__U31(mark(X1),X2) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(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)) -> A__U42(mark(X1),X2) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> A__U51(mark(X1),X2,X3) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> A__U52(mark(X1),X2) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> A__U61(mark(X1),X2) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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 + 1 [a__U12](X) = X + 2 [a__U21](X1,X2) = X1 + X2 + 1 [a__U22](X) = X [a__U31](X1,X2) = X1 + X2 [a__U32](X) = X + 2 [a__U41](X1,X2,X3) = X1 + X2 + 2.X3 + 2 [a__U42](X1,X2) = X1 + 2.X2 + 2 [a__U43](X) = X + 1 [a__U51](X1,X2,X3) = X1 + X2 + 2.X3 [a__U52](X1,X2) = X1 + X2 [a__U53](X) = X + 1 [a__U61](X1,X2) = X1 + X2 + 2 [a__and](X1,X2) = X1 + X2 [a__isNat](X) = X + 2 [a__isNatIList](X) = 2.X + 2 [a__isNatIListKind](X) = 2 [a__isNatKind](X) = 2 [a__isNatList](X) = X [a__length](X) = X + 1 [a__zeros] = 2 [mark](X) = X + 2 [0] = 0 [U11](X1,X2) = X1 + X2 + 1 [U12](X) = X + 2 [U21](X1,X2) = X1 + X2 + 1 [U22](X) = X [U31](X1,X2) = X1 + X2 [U32](X) = X + 2 [U41](X1,X2,X3) = X1 + X2 + 2.X3 + 2 [U42](X1,X2) = X1 + 2.X2 + 2 [U43](X) = X + 1 [U51](X1,X2,X3) = X1 + X2 + 2.X3 [U52](X1,X2) = X1 + X2 [U53](X) = X + 1 [U61](X1,X2) = X1 + X2 + 2 [and](X1,X2) = X1 + X2 [cons](X1,X2) = X1 + 2.X2 + 2 [isNat](X) = X [isNatIList](X) = 2.X + 2 [isNatIListKind](X) = 0 [isNatKind](X) = 0 [isNatList](X) = X [length](X) = X + 1 [nil] = 2 [s](X) = X + 1 [tt] = 2 [zeros] = 0 [A__U11](X1,X2) = X1 + 2.X2 + 2 [A__U21](X1,X2) = X1 + 2.X2 + 1 [A__U31](X1,X2) = 2.X2 + 2 [A__U41](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [A__U42](X1,X2) = 2.X1 + 2.X2 [A__U51](X1,X2,X3) = X1 + 2.X2 + 2.X3 [A__U52](X1,X2) = 2.X2 + 2 [A__U61](X1,X2) = 2.X1 + 2.X2 + 2 [A__AND](X1,X2) = 2.X2 + 2 [A__ISNAT](X) = 2.X + 2 [A__ISNATILIST](X) = 2.X + 2 [A__ISNATILISTKIND](X) = 2 [A__ISNATKIND](X) = 2 [A__ISNATLIST](X) = 2.X + 2 [A__LENGTH](X) = 2.X [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__U21(tt,V1) -> A__ISNAT(V1) A__U31(tt,V) -> A__ISNATLIST(V) A__U41(tt,V1,V2) -> A__U42(a__isNat(V1),V2) A__U41(tt,V1,V2) -> A__ISNAT(V1) A__U42(tt,V2) -> A__ISNATILIST(V2) A__U51(tt,V1,V2) -> A__U52(a__isNat(V1),V2) A__U51(tt,V1,V2) -> A__ISNAT(V1) A__U52(tt,V2) -> A__ISNATLIST(V2) A__U61(tt,L) -> A__LENGTH(mark(L)) A__U61(tt,L) -> MARK(L) A__AND(tt,X) -> MARK(X) 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__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(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__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) 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(X)) -> MARK(X) MARK(U21(X1,X2)) -> A__U21(mark(X1),X2) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> A__U31(mark(X1),X2) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(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)) -> A__U42(mark(X1),X2) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> A__U51(mark(X1),X2,X3) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> A__U52(mark(X1),X2) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> A__U61(mark(X1),X2) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__ISNAT(V1) A__U31(tt,V) -> A__ISNATLIST(V) A__U41(tt,V1,V2) -> A__U42(a__isNat(V1),V2) A__U41(tt,V1,V2) -> A__ISNAT(V1) A__U42(tt,V2) -> A__ISNATILIST(V2) A__U51(tt,V1,V2) -> A__U52(a__isNat(V1),V2) A__U51(tt,V1,V2) -> A__ISNAT(V1) A__U52(tt,V2) -> A__ISNATLIST(V2) A__U61(tt,L) -> A__LENGTH(mark(L)) A__U61(tt,L) -> MARK(L) A__AND(tt,X) -> MARK(X) 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__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(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__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> A__U21(mark(X1),X2) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> A__U31(mark(X1),X2) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(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)) -> A__U42(mark(X1),X2) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> A__U51(mark(X1),X2,X3) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> A__U52(mark(X1),X2) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> A__U61(mark(X1),X2) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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) ->->-> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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: Reduction Pair Processor: -> Pairs: A__U21(tt,V1) -> A__ISNAT(V1) A__U31(tt,V) -> A__ISNATLIST(V) A__U41(tt,V1,V2) -> A__U42(a__isNat(V1),V2) A__U41(tt,V1,V2) -> A__ISNAT(V1) A__U42(tt,V2) -> A__ISNATILIST(V2) A__U51(tt,V1,V2) -> A__U52(a__isNat(V1),V2) A__U51(tt,V1,V2) -> A__ISNAT(V1) A__U52(tt,V2) -> A__ISNATLIST(V2) A__U61(tt,L) -> A__LENGTH(mark(L)) A__U61(tt,L) -> MARK(L) A__AND(tt,X) -> MARK(X) 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__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(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__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> A__U21(mark(X1),X2) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> A__U31(mark(X1),X2) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(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)) -> A__U42(mark(X1),X2) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> A__U51(mark(X1),X2,X3) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> A__U52(mark(X1),X2) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> A__U61(mark(X1),X2) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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 [a__U12](X) = X + 2 [a__U21](X1,X2) = X1 + X2 + 2 [a__U22](X) = X [a__U31](X1,X2) = X1 + 2.X2 [a__U32](X) = X + 2 [a__U41](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 1 [a__U42](X1,X2) = X1 + 2.X2 + 1 [a__U43](X) = X + 1 [a__U51](X1,X2,X3) = X1 + X2 + 2.X3 [a__U52](X1,X2) = X1 + 2.X2 [a__U53](X) = X + 2 [a__U61](X1,X2) = X1 + X2 + 2 [a__and](X1,X2) = X1 + X2 [a__isNat](X) = X + 2 [a__isNatIList](X) = 2.X + 2 [a__isNatIListKind](X) = 2 [a__isNatKind](X) = 2 [a__isNatList](X) = X [a__length](X) = X [a__zeros] = 2 [mark](X) = X + 2 [0] = 0 [U11](X1,X2) = X1 + X2 [U12](X) = X + 2 [U21](X1,X2) = X1 + X2 + 2 [U22](X) = X [U31](X1,X2) = X1 + 2.X2 [U32](X) = X + 2 [U41](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 1 [U42](X1,X2) = X1 + 2.X2 + 1 [U43](X) = X + 1 [U51](X1,X2,X3) = X1 + X2 + 2.X3 [U52](X1,X2) = X1 + 2.X2 [U53](X) = X + 2 [U61](X1,X2) = X1 + X2 + 2 [and](X1,X2) = X1 + X2 [cons](X1,X2) = X1 + 2.X2 + 2 [isNat](X) = X [isNatIList](X) = 2.X + 1 [isNatIListKind](X) = 0 [isNatKind](X) = 0 [isNatList](X) = X [length](X) = X [nil] = 2 [s](X) = X + 2 [tt] = 2 [zeros] = 0 [A__U21](X1,X2) = X1 + X2 + 2 [A__U31](X1,X2) = 2.X2 + 2 [A__U41](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 1 [A__U42](X1,X2) = X1 + 2.X2 + 1 [A__U51](X1,X2,X3) = X2 + 2.X3 + 2 [A__U52](X1,X2) = 2.X2 + 2 [A__U61](X1,X2) = X1 + X2 [A__AND](X1,X2) = X1 + X2 [A__ISNAT](X) = X + 2 [A__ISNATILIST](X) = 2.X + 2 [A__ISNATILISTKIND](X) = 2 [A__ISNATKIND](X) = 2 [A__ISNATLIST](X) = X + 2 [A__LENGTH](X) = X [MARK](X) = X + 2 Problem 1: SCC Processor: -> Pairs: A__U31(tt,V) -> A__ISNATLIST(V) A__U41(tt,V1,V2) -> A__U42(a__isNat(V1),V2) A__U41(tt,V1,V2) -> A__ISNAT(V1) A__U42(tt,V2) -> A__ISNATILIST(V2) A__U51(tt,V1,V2) -> A__U52(a__isNat(V1),V2) A__U51(tt,V1,V2) -> A__ISNAT(V1) A__U52(tt,V2) -> A__ISNATLIST(V2) A__U61(tt,L) -> A__LENGTH(mark(L)) A__U61(tt,L) -> MARK(L) A__AND(tt,X) -> MARK(X) 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__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(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__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> A__U21(mark(X1),X2) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> A__U31(mark(X1),X2) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(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)) -> A__U42(mark(X1),X2) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> A__U51(mark(X1),X2,X3) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> A__U52(mark(X1),X2) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> A__U61(mark(X1),X2) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__U31(tt,V) -> A__ISNATLIST(V) A__U41(tt,V1,V2) -> A__U42(a__isNat(V1),V2) A__U41(tt,V1,V2) -> A__ISNAT(V1) A__U42(tt,V2) -> A__ISNATILIST(V2) A__U51(tt,V1,V2) -> A__U52(a__isNat(V1),V2) A__U51(tt,V1,V2) -> A__ISNAT(V1) A__U52(tt,V2) -> A__ISNATLIST(V2) A__U61(tt,L) -> A__LENGTH(mark(L)) A__U61(tt,L) -> MARK(L) A__AND(tt,X) -> MARK(X) A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) A__ISNATILIST(cons(V1,V2)) -> A__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(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__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> A__U31(mark(X1),X2) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(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)) -> A__U42(mark(X1),X2) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> A__U51(mark(X1),X2,X3) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> A__U52(mark(X1),X2) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> A__U61(mark(X1),X2) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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) ->->-> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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: Reduction Pair Processor: -> Pairs: A__U31(tt,V) -> A__ISNATLIST(V) A__U41(tt,V1,V2) -> A__U42(a__isNat(V1),V2) A__U41(tt,V1,V2) -> A__ISNAT(V1) A__U42(tt,V2) -> A__ISNATILIST(V2) A__U51(tt,V1,V2) -> A__U52(a__isNat(V1),V2) A__U51(tt,V1,V2) -> A__ISNAT(V1) A__U52(tt,V2) -> A__ISNATLIST(V2) A__U61(tt,L) -> A__LENGTH(mark(L)) A__U61(tt,L) -> MARK(L) A__AND(tt,X) -> MARK(X) A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) A__ISNATILIST(cons(V1,V2)) -> A__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(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__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> A__U31(mark(X1),X2) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(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)) -> A__U42(mark(X1),X2) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> A__U51(mark(X1),X2,X3) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> A__U52(mark(X1),X2) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> A__U61(mark(X1),X2) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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 + 1 [a__U12](X) = X + 2 [a__U21](X1,X2) = X1 + X2 + 1 [a__U22](X) = X + 1 [a__U31](X1,X2) = X1 + 2.X2 [a__U32](X) = X + 1 [a__U41](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [a__U42](X1,X2) = X1 + 2.X2 + 2 [a__U43](X) = X + 2 [a__U51](X1,X2,X3) = X1 + X2 + X3 + 1 [a__U52](X1,X2) = X1 + X2 + 1 [a__U53](X) = X + 2 [a__U61](X1,X2) = X1 + X2 + 2 [a__and](X1,X2) = X1 + X2 [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 + 1 [a__zeros] = 2 [mark](X) = X + 2 [0] = 0 [U11](X1,X2) = X1 + X2 + 1 [U12](X) = X + 2 [U21](X1,X2) = X1 + X2 + 1 [U22](X) = X + 1 [U31](X1,X2) = X1 + 2.X2 [U32](X) = X + 1 [U41](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U42](X1,X2) = X1 + 2.X2 + 2 [U43](X) = X + 2 [U51](X1,X2,X3) = X1 + X2 + X3 + 1 [U52](X1,X2) = X1 + X2 + 1 [U53](X) = X + 2 [U61](X1,X2) = X1 + X2 + 2 [and](X1,X2) = X1 + X2 [cons](X1,X2) = X1 + 2.X2 + 2 [isNat](X) = X [isNatIList](X) = 2.X + 2 [isNatIListKind](X) = 0 [isNatKind](X) = 0 [isNatList](X) = X + 1 [length](X) = X + 1 [nil] = 2 [s](X) = X + 1 [tt] = 2 [zeros] = 0 [A__U31](X1,X2) = 2.X2 + 2 [A__U41](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 [A__U42](X1,X2) = X1 + 2.X2 + 2 [A__U51](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [A__U52](X1,X2) = X1 + 2.X2 + 2 [A__U61](X1,X2) = 2.X1 + 2.X2 + 2 [A__AND](X1,X2) = 2.X2 + 2 [A__ISNAT](X) = 2.X + 2 [A__ISNATILIST](X) = 2.X + 2 [A__ISNATILISTKIND](X) = 2 [A__ISNATKIND](X) = 2 [A__ISNATLIST](X) = 2.X [A__LENGTH](X) = 2.X [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__U41(tt,V1,V2) -> A__U42(a__isNat(V1),V2) A__U41(tt,V1,V2) -> A__ISNAT(V1) A__U42(tt,V2) -> A__ISNATILIST(V2) A__U51(tt,V1,V2) -> A__U52(a__isNat(V1),V2) A__U51(tt,V1,V2) -> A__ISNAT(V1) A__U52(tt,V2) -> A__ISNATLIST(V2) A__U61(tt,L) -> A__LENGTH(mark(L)) A__U61(tt,L) -> MARK(L) A__AND(tt,X) -> MARK(X) A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) A__ISNATILIST(cons(V1,V2)) -> A__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(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__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> A__U31(mark(X1),X2) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(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)) -> A__U42(mark(X1),X2) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> A__U51(mark(X1),X2,X3) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> A__U52(mark(X1),X2) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> A__U61(mark(X1),X2) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__U41(tt,V1,V2) -> A__U42(a__isNat(V1),V2) A__U41(tt,V1,V2) -> A__ISNAT(V1) A__U42(tt,V2) -> A__ISNATILIST(V2) A__U51(tt,V1,V2) -> A__U52(a__isNat(V1),V2) A__U51(tt,V1,V2) -> A__ISNAT(V1) A__U52(tt,V2) -> A__ISNATLIST(V2) A__U61(tt,L) -> A__LENGTH(mark(L)) A__U61(tt,L) -> MARK(L) A__AND(tt,X) -> MARK(X) A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) A__ISNATILIST(cons(V1,V2)) -> A__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(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)) -> A__U42(mark(X1),X2) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> A__U51(mark(X1),X2,X3) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> A__U52(mark(X1),X2) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> A__U61(mark(X1),X2) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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) ->->-> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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: Reduction Pair Processor: -> Pairs: A__U41(tt,V1,V2) -> A__U42(a__isNat(V1),V2) A__U41(tt,V1,V2) -> A__ISNAT(V1) A__U42(tt,V2) -> A__ISNATILIST(V2) A__U51(tt,V1,V2) -> A__U52(a__isNat(V1),V2) A__U51(tt,V1,V2) -> A__ISNAT(V1) A__U52(tt,V2) -> A__ISNATLIST(V2) A__U61(tt,L) -> A__LENGTH(mark(L)) A__U61(tt,L) -> MARK(L) A__AND(tt,X) -> MARK(X) A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) A__ISNATILIST(cons(V1,V2)) -> A__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(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)) -> A__U42(mark(X1),X2) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> A__U51(mark(X1),X2,X3) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> A__U52(mark(X1),X2) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> A__U61(mark(X1),X2) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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 [a__U12](X) = X [a__U21](X1,X2) = X1 [a__U22](X) = X [a__U31](X1,X2) = X1 + 2.X2 [a__U32](X) = X [a__U41](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [a__U42](X1,X2) = 2.X1 + 2.X2 + 2 [a__U43](X) = X + 1 [a__U51](X1,X2,X3) = X1 [a__U52](X1,X2) = X1 [a__U53](X) = X [a__U61](X1,X2) = X1 + 2.X2 + 2 [a__and](X1,X2) = X1 + 2.X2 [a__isNat](X) = 2 [a__isNatIList](X) = 2.X + 2 [a__isNatIListKind](X) = 2 [a__isNatKind](X) = 2 [a__isNatList](X) = 2 [a__length](X) = X + 2 [a__zeros] = 2 [mark](X) = 2.X + 2 [0] = 0 [U11](X1,X2) = X1 [U12](X) = X [U21](X1,X2) = X1 [U22](X) = X [U31](X1,X2) = X1 + X2 [U32](X) = X [U41](X1,X2,X3) = 2.X1 + X2 + X3 + 2 [U42](X1,X2) = 2.X1 + 2.X2 + 2 [U43](X) = X + 1 [U51](X1,X2,X3) = X1 [U52](X1,X2) = X1 [U53](X) = X [U61](X1,X2) = X1 + X2 + 2 [and](X1,X2) = X1 + 2.X2 [cons](X1,X2) = 2.X1 + 2.X2 + 2 [isNat](X) = 0 [isNatIList](X) = X + 1 [isNatIListKind](X) = 0 [isNatKind](X) = 0 [isNatList](X) = 1 [length](X) = X + 2 [nil] = 2 [s](X) = X [tt] = 2 [zeros] = 0 [A__U41](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [A__U42](X1,X2) = X1 + 2.X2 + 2 [A__U51](X1,X2,X3) = 2 [A__U52](X1,X2) = 2 [A__U61](X1,X2) = X1 + 2.X2 + 2 [A__AND](X1,X2) = 2.X2 + 2 [A__ISNAT](X) = 2 [A__ISNATILIST](X) = 2.X + 2 [A__ISNATILISTKIND](X) = 2 [A__ISNATKIND](X) = 2 [A__ISNATLIST](X) = 2 [A__LENGTH](X) = X + 2 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__U41(tt,V1,V2) -> A__ISNAT(V1) A__U42(tt,V2) -> A__ISNATILIST(V2) A__U51(tt,V1,V2) -> A__U52(a__isNat(V1),V2) A__U51(tt,V1,V2) -> A__ISNAT(V1) A__U52(tt,V2) -> A__ISNATLIST(V2) A__U61(tt,L) -> A__LENGTH(mark(L)) A__U61(tt,L) -> MARK(L) A__AND(tt,X) -> MARK(X) A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) A__ISNATILIST(cons(V1,V2)) -> A__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(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)) -> A__U42(mark(X1),X2) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> A__U51(mark(X1),X2,X3) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> A__U52(mark(X1),X2) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> A__U61(mark(X1),X2) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__U41(tt,V1,V2) -> A__ISNAT(V1) A__U42(tt,V2) -> A__ISNATILIST(V2) A__U51(tt,V1,V2) -> A__U52(a__isNat(V1),V2) A__U51(tt,V1,V2) -> A__ISNAT(V1) A__U52(tt,V2) -> A__ISNATLIST(V2) A__U61(tt,L) -> A__LENGTH(mark(L)) A__U61(tt,L) -> MARK(L) A__AND(tt,X) -> MARK(X) A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) A__ISNATILIST(cons(V1,V2)) -> A__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(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)) -> A__U42(mark(X1),X2) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> A__U51(mark(X1),X2,X3) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> A__U52(mark(X1),X2) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> A__U61(mark(X1),X2) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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) ->->-> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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: Reduction Pair Processor: -> Pairs: A__U41(tt,V1,V2) -> A__ISNAT(V1) A__U42(tt,V2) -> A__ISNATILIST(V2) A__U51(tt,V1,V2) -> A__U52(a__isNat(V1),V2) A__U51(tt,V1,V2) -> A__ISNAT(V1) A__U52(tt,V2) -> A__ISNATLIST(V2) A__U61(tt,L) -> A__LENGTH(mark(L)) A__U61(tt,L) -> MARK(L) A__AND(tt,X) -> MARK(X) A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) A__ISNATILIST(cons(V1,V2)) -> A__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(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)) -> A__U42(mark(X1),X2) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> A__U51(mark(X1),X2,X3) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> A__U52(mark(X1),X2) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> A__U61(mark(X1),X2) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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 [a__U12](X) = X [a__U21](X1,X2) = X1 [a__U22](X) = X [a__U31](X1,X2) = X1 [a__U32](X) = X [a__U41](X1,X2,X3) = 2.X1 + X2 + 2.X3 + 2 [a__U42](X1,X2) = 2.X1 + 2.X2 + 2 [a__U43](X) = X + 2 [a__U51](X1,X2,X3) = X1 [a__U52](X1,X2) = X1 [a__U53](X) = X [a__U61](X1,X2) = X1 + 2.X2 + 2 [a__and](X1,X2) = X1 + 2.X2 [a__isNat](X) = 2 [a__isNatIList](X) = 2.X + 2 [a__isNatIListKind](X) = 2 [a__isNatKind](X) = 2 [a__isNatList](X) = 2 [a__length](X) = X + 2 [a__zeros] = 2 [mark](X) = 2.X + 2 [0] = 0 [U11](X1,X2) = X1 [U12](X) = X [U21](X1,X2) = X1 [U22](X) = X [U31](X1,X2) = X1 [U32](X) = X [U41](X1,X2,X3) = 2.X1 + X2 + 2.X3 + 2 [U42](X1,X2) = 2.X1 + X2 + 2 [U43](X) = X + 2 [U51](X1,X2,X3) = X1 [U52](X1,X2) = X1 [U53](X) = X [U61](X1,X2) = X1 + 2.X2 + 2 [and](X1,X2) = X1 + 2.X2 [cons](X1,X2) = 2.X1 + 2.X2 + 2 [isNat](X) = 0 [isNatIList](X) = X + 1 [isNatIListKind](X) = 0 [isNatKind](X) = 0 [isNatList](X) = 2 [length](X) = X + 2 [nil] = 1 [s](X) = X [tt] = 2 [zeros] = 0 [A__U41](X1,X2,X3) = X1 + 2.X2 + X3 + 2 [A__U42](X1,X2) = 2.X1 + 2.X2 + 1 [A__U51](X1,X2,X3) = 2 [A__U52](X1,X2) = 2 [A__U61](X1,X2) = X1 + 2.X2 + 2 [A__AND](X1,X2) = X1 + 2.X2 [A__ISNAT](X) = 2 [A__ISNATILIST](X) = 2.X + 2 [A__ISNATILISTKIND](X) = 2 [A__ISNATKIND](X) = 2 [A__ISNATLIST](X) = 2 [A__LENGTH](X) = X + 2 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__U42(tt,V2) -> A__ISNATILIST(V2) A__U51(tt,V1,V2) -> A__U52(a__isNat(V1),V2) A__U51(tt,V1,V2) -> A__ISNAT(V1) A__U52(tt,V2) -> A__ISNATLIST(V2) A__U61(tt,L) -> A__LENGTH(mark(L)) A__U61(tt,L) -> MARK(L) A__AND(tt,X) -> MARK(X) A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) A__ISNATILIST(cons(V1,V2)) -> A__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(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)) -> A__U42(mark(X1),X2) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> A__U51(mark(X1),X2,X3) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> A__U52(mark(X1),X2) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> A__U61(mark(X1),X2) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__U42(tt,V2) -> A__ISNATILIST(V2) A__U51(tt,V1,V2) -> A__U52(a__isNat(V1),V2) A__U51(tt,V1,V2) -> A__ISNAT(V1) A__U52(tt,V2) -> A__ISNATLIST(V2) A__U61(tt,L) -> A__LENGTH(mark(L)) A__U61(tt,L) -> MARK(L) A__AND(tt,X) -> MARK(X) A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> A__U42(mark(X1),X2) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> A__U51(mark(X1),X2,X3) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> A__U52(mark(X1),X2) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> A__U61(mark(X1),X2) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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) ->->-> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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: Reduction Pair Processor: -> Pairs: A__U42(tt,V2) -> A__ISNATILIST(V2) A__U51(tt,V1,V2) -> A__U52(a__isNat(V1),V2) A__U51(tt,V1,V2) -> A__ISNAT(V1) A__U52(tt,V2) -> A__ISNATLIST(V2) A__U61(tt,L) -> A__LENGTH(mark(L)) A__U61(tt,L) -> MARK(L) A__AND(tt,X) -> MARK(X) A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> A__U42(mark(X1),X2) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> A__U51(mark(X1),X2,X3) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> A__U52(mark(X1),X2) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> A__U61(mark(X1),X2) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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 [a__U12](X) = X [a__U21](X1,X2) = X1 [a__U22](X) = X [a__U31](X1,X2) = X1 + 2.X2 [a__U32](X) = X [a__U41](X1,X2,X3) = 2.X1 + 2.X3 + 2 [a__U42](X1,X2) = 2.X1 + 2.X2 + 2 [a__U43](X) = X [a__U51](X1,X2,X3) = X1 [a__U52](X1,X2) = X1 [a__U53](X) = X [a__U61](X1,X2) = X1 + 2.X2 + 2 [a__and](X1,X2) = X1 + 2.X2 [a__isNat](X) = 2 [a__isNatIList](X) = 2.X + 2 [a__isNatIListKind](X) = 2 [a__isNatKind](X) = 2 [a__isNatList](X) = 2 [a__length](X) = X + 2 [a__zeros] = 2 [mark](X) = 2.X + 2 [0] = 0 [U11](X1,X2) = X1 [U12](X) = X [U21](X1,X2) = X1 [U22](X) = X [U31](X1,X2) = X1 + 2.X2 [U32](X) = X [U41](X1,X2,X3) = 2.X1 + 2.X3 + 2 [U42](X1,X2) = 2.X1 + 2.X2 + 2 [U43](X) = X [U51](X1,X2,X3) = X1 [U52](X1,X2) = X1 [U53](X) = X [U61](X1,X2) = X1 + 2.X2 + 1 [and](X1,X2) = X1 + 2.X2 [cons](X1,X2) = 2.X1 + 2.X2 + 2 [isNat](X) = 0 [isNatIList](X) = X + 2 [isNatIListKind](X) = 0 [isNatKind](X) = 0 [isNatList](X) = 1 [length](X) = X + 1 [nil] = 2 [s](X) = X [tt] = 2 [zeros] = 0 [A__U42](X1,X2) = 2.X1 + 2.X2 + 2 [A__U51](X1,X2,X3) = 2 [A__U52](X1,X2) = 2 [A__U61](X1,X2) = X1 + 2.X2 + 2 [A__AND](X1,X2) = X1 + 2.X2 [A__ISNAT](X) = 2 [A__ISNATILIST](X) = X + 2 [A__ISNATILISTKIND](X) = 2 [A__ISNATKIND](X) = 2 [A__ISNATLIST](X) = 2 [A__LENGTH](X) = X + 2 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__U51(tt,V1,V2) -> A__U52(a__isNat(V1),V2) A__U51(tt,V1,V2) -> A__ISNAT(V1) A__U52(tt,V2) -> A__ISNATLIST(V2) A__U61(tt,L) -> A__LENGTH(mark(L)) A__U61(tt,L) -> MARK(L) A__AND(tt,X) -> MARK(X) A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> A__U42(mark(X1),X2) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> A__U51(mark(X1),X2,X3) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> A__U52(mark(X1),X2) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> A__U61(mark(X1),X2) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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,V1,V2) -> A__U52(a__isNat(V1),V2) A__U51(tt,V1,V2) -> A__ISNAT(V1) A__U52(tt,V2) -> A__ISNATLIST(V2) A__U61(tt,L) -> A__LENGTH(mark(L)) A__U61(tt,L) -> MARK(L) A__AND(tt,X) -> MARK(X) A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> A__U51(mark(X1),X2,X3) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> A__U52(mark(X1),X2) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> A__U61(mark(X1),X2) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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) ->->-> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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: Reduction Pair Processor: -> Pairs: A__U51(tt,V1,V2) -> A__U52(a__isNat(V1),V2) A__U51(tt,V1,V2) -> A__ISNAT(V1) A__U52(tt,V2) -> A__ISNATLIST(V2) A__U61(tt,L) -> A__LENGTH(mark(L)) A__U61(tt,L) -> MARK(L) A__AND(tt,X) -> MARK(X) A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> A__U51(mark(X1),X2,X3) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> A__U52(mark(X1),X2) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> A__U61(mark(X1),X2) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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](X) = X + 1 [a__U21](X1,X2) = X1 + X2 [a__U22](X) = X [a__U31](X1,X2) = X1 + X2 [a__U32](X) = X + 1 [a__U41](X1,X2,X3) = X1 + X2 + 2.X3 + 2 [a__U42](X1,X2) = X1 + 2.X2 + 2 [a__U43](X) = X + 2 [a__U51](X1,X2,X3) = X1 + X2 + 2.X3 [a__U52](X1,X2) = X1 + 2.X2 [a__U53](X) = X + 2 [a__U61](X1,X2) = X1 + X2 + 2 [a__and](X1,X2) = X1 + X2 [a__isNat](X) = X + 2 [a__isNatIList](X) = 2.X + 2 [a__isNatIListKind](X) = 2 [a__isNatKind](X) = 2 [a__isNatList](X) = X [a__length](X) = X + 2 [a__zeros] = 2 [mark](X) = X + 2 [0] = 0 [U11](X1,X2) = X1 + X2 + 2 [U12](X) = X + 1 [U21](X1,X2) = X1 + X2 [U22](X) = X [U31](X1,X2) = X1 + X2 [U32](X) = X + 1 [U41](X1,X2,X3) = X1 + X2 + 2.X3 + 2 [U42](X1,X2) = X1 + 2.X2 + 2 [U43](X) = X + 2 [U51](X1,X2,X3) = X1 + X2 + 2.X3 [U52](X1,X2) = X1 + 2.X2 [U53](X) = X + 2 [U61](X1,X2) = X1 + X2 + 2 [and](X1,X2) = X1 + X2 [cons](X1,X2) = X1 + 2.X2 + 2 [isNat](X) = X + 2 [isNatIList](X) = 2.X + 2 [isNatIListKind](X) = 0 [isNatKind](X) = 0 [isNatList](X) = X [length](X) = X + 2 [nil] = 2 [s](X) = X [tt] = 2 [zeros] = 0 [A__U51](X1,X2,X3) = X2 + 2.X3 + 2 [A__U52](X1,X2) = 2.X2 [A__U61](X1,X2) = 2.X1 + 2.X2 + 2 [A__AND](X1,X2) = 2.X2 + 2 [A__ISNAT](X) = X + 2 [A__ISNATILIST](X) = 2 [A__ISNATILISTKIND](X) = 2 [A__ISNATKIND](X) = 2 [A__ISNATLIST](X) = X [A__LENGTH](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__U51(tt,V1,V2) -> A__ISNAT(V1) A__U52(tt,V2) -> A__ISNATLIST(V2) A__U61(tt,L) -> A__LENGTH(mark(L)) A__U61(tt,L) -> MARK(L) A__AND(tt,X) -> MARK(X) A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> A__U51(mark(X1),X2,X3) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> A__U52(mark(X1),X2) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> A__U61(mark(X1),X2) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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,V1,V2) -> A__ISNAT(V1) A__U52(tt,V2) -> A__ISNATLIST(V2) A__U61(tt,L) -> A__LENGTH(mark(L)) A__U61(tt,L) -> MARK(L) A__AND(tt,X) -> MARK(X) A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> A__U51(mark(X1),X2,X3) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> A__U52(mark(X1),X2) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> A__U61(mark(X1),X2) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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) ->->-> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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: Reduction Pair Processor: -> Pairs: A__U51(tt,V1,V2) -> A__ISNAT(V1) A__U52(tt,V2) -> A__ISNATLIST(V2) A__U61(tt,L) -> A__LENGTH(mark(L)) A__U61(tt,L) -> MARK(L) A__AND(tt,X) -> MARK(X) A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> A__U51(mark(X1),X2,X3) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> A__U52(mark(X1),X2) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> A__U61(mark(X1),X2) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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](X) = X + 2 [a__U21](X1,X2) = X1 + X2 [a__U22](X) = X [a__U31](X1,X2) = X1 + 2.X2 [a__U32](X) = X [a__U41](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [a__U42](X1,X2) = X1 + 2.X2 + 2 [a__U43](X) = X + 1 [a__U51](X1,X2,X3) = X1 + X2 + 2.X3 + 1 [a__U52](X1,X2) = X1 + 2.X2 + 1 [a__U53](X) = X + 2 [a__U61](X1,X2) = X1 + X2 + 2 [a__and](X1,X2) = X1 + X2 [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](X) = X + 2 [U21](X1,X2) = X1 + X2 [U22](X) = X [U31](X1,X2) = X1 + 2.X2 [U32](X) = X [U41](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U42](X1,X2) = X1 + 2.X2 + 2 [U43](X) = X + 1 [U51](X1,X2,X3) = X1 + X2 + 2.X3 + 1 [U52](X1,X2) = X1 + 2.X2 + 1 [U53](X) = X + 2 [U61](X1,X2) = X1 + X2 + 2 [and](X1,X2) = X1 + X2 [cons](X1,X2) = X1 + 2.X2 + 2 [isNat](X) = X + 1 [isNatIList](X) = 2.X + 2 [isNatIListKind](X) = 0 [isNatKind](X) = 0 [isNatList](X) = X + 1 [length](X) = X + 2 [nil] = 1 [s](X) = X [tt] = 2 [zeros] = 0 [A__U51](X1,X2,X3) = X1 + 2.X2 + X3 + 1 [A__U52](X1,X2) = X1 + 2.X2 + 2 [A__U61](X1,X2) = 2.X1 + 2.X2 + 2 [A__AND](X1,X2) = X1 + 2.X2 [A__ISNAT](X) = 2.X + 2 [A__ISNATILIST](X) = 2.X + 2 [A__ISNATILISTKIND](X) = 2 [A__ISNATKIND](X) = 2 [A__ISNATLIST](X) = 2.X + 1 [A__LENGTH](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__U52(tt,V2) -> A__ISNATLIST(V2) A__U61(tt,L) -> A__LENGTH(mark(L)) A__U61(tt,L) -> MARK(L) A__AND(tt,X) -> MARK(X) A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> A__U51(mark(X1),X2,X3) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> A__U52(mark(X1),X2) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> A__U61(mark(X1),X2) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__U52(tt,V2) -> A__ISNATLIST(V2) A__U61(tt,L) -> A__LENGTH(mark(L)) A__U61(tt,L) -> MARK(L) A__AND(tt,X) -> MARK(X) A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> A__U52(mark(X1),X2) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> A__U61(mark(X1),X2) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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) ->->-> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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: Reduction Pair Processor: -> Pairs: A__U52(tt,V2) -> A__ISNATLIST(V2) A__U61(tt,L) -> A__LENGTH(mark(L)) A__U61(tt,L) -> MARK(L) A__AND(tt,X) -> MARK(X) A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> A__U52(mark(X1),X2) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> A__U61(mark(X1),X2) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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](X) = X + 2 [a__U21](X1,X2) = X1 + X2 [a__U22](X) = X [a__U31](X1,X2) = X1 + 2.X2 [a__U32](X) = X [a__U41](X1,X2,X3) = X1 + X2 + 2.X3 + 2 [a__U42](X1,X2) = X1 + 2.X2 + 2 [a__U43](X) = X + 2 [a__U51](X1,X2,X3) = X1 + X2 + X3 + 2 [a__U52](X1,X2) = X1 + X2 + 2 [a__U53](X) = X + 2 [a__U61](X1,X2) = X1 + X2 + 2 [a__and](X1,X2) = X1 + X2 [a__isNat](X) = X + 2 [a__isNatIList](X) = 2.X + 2 [a__isNatIListKind](X) = 2 [a__isNatKind](X) = 2 [a__isNatList](X) = X + 2 [a__length](X) = X + 2 [a__zeros] = 2 [mark](X) = X + 2 [0] = 0 [U11](X1,X2) = X1 + X2 + 2 [U12](X) = X + 2 [U21](X1,X2) = X1 + X2 [U22](X) = X [U31](X1,X2) = X1 + 2.X2 [U32](X) = X [U41](X1,X2,X3) = X1 + X2 + 2.X3 + 2 [U42](X1,X2) = X1 + 2.X2 + 2 [U43](X) = X + 2 [U51](X1,X2,X3) = X1 + X2 + X3 + 2 [U52](X1,X2) = X1 + X2 + 2 [U53](X) = X + 2 [U61](X1,X2) = X1 + X2 + 2 [and](X1,X2) = X1 + X2 [cons](X1,X2) = X1 + 2.X2 + 2 [isNat](X) = X [isNatIList](X) = 2.X + 2 [isNatIListKind](X) = 0 [isNatKind](X) = 0 [isNatList](X) = X + 2 [length](X) = X + 2 [nil] = 1 [s](X) = X [tt] = 2 [zeros] = 0 [A__U52](X1,X2) = X1 + 2.X2 + 1 [A__U61](X1,X2) = 2.X1 + 2.X2 + 2 [A__AND](X1,X2) = 2.X2 + 2 [A__ISNAT](X) = 2.X + 2 [A__ISNATILIST](X) = 2 [A__ISNATILISTKIND](X) = 2 [A__ISNATKIND](X) = 2 [A__ISNATLIST](X) = X [A__LENGTH](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__U61(tt,L) -> A__LENGTH(mark(L)) A__U61(tt,L) -> MARK(L) A__AND(tt,X) -> MARK(X) A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> A__U52(mark(X1),X2) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> A__U61(mark(X1),X2) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__U61(tt,L) -> A__LENGTH(mark(L)) A__U61(tt,L) -> MARK(L) A__AND(tt,X) -> MARK(X) A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> A__U61(mark(X1),X2) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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) ->->-> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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: Reduction Pair Processor: -> Pairs: A__U61(tt,L) -> A__LENGTH(mark(L)) A__U61(tt,L) -> MARK(L) A__AND(tt,X) -> MARK(X) A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> A__U61(mark(X1),X2) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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 [a__U12](X) = X + 1 [a__U21](X1,X2) = X1 + X2 + 1 [a__U22](X) = X [a__U31](X1,X2) = X1 + 2.X2 [a__U32](X) = X + 1 [a__U41](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [a__U42](X1,X2) = X1 + 2.X2 + 2 [a__U43](X) = X + 2 [a__U51](X1,X2,X3) = X1 + X2 + 2.X3 + 1 [a__U52](X1,X2) = X1 + X2 + 1 [a__U53](X) = X + 1 [a__U61](X1,X2) = X1 + X2 + 1 [a__and](X1,X2) = X1 + X2 [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 [a__zeros] = 2 [mark](X) = X + 2 [0] = 0 [U11](X1,X2) = X1 + X2 [U12](X) = X + 1 [U21](X1,X2) = X1 + X2 + 1 [U22](X) = X [U31](X1,X2) = X1 + 2.X2 [U32](X) = X + 1 [U41](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U42](X1,X2) = X1 + 2.X2 + 2 [U43](X) = X + 2 [U51](X1,X2,X3) = X1 + X2 + 2.X3 + 1 [U52](X1,X2) = X1 + X2 + 1 [U53](X) = X + 1 [U61](X1,X2) = X1 + X2 + 1 [and](X1,X2) = X1 + X2 [cons](X1,X2) = X1 + 2.X2 + 2 [isNat](X) = X [isNatIList](X) = 2.X + 2 [isNatIListKind](X) = 0 [isNatKind](X) = 0 [isNatList](X) = X + 1 [length](X) = X [nil] = 2 [s](X) = X + 1 [tt] = 2 [zeros] = 0 [A__U61](X1,X2) = X1 + X2 + 1 [A__AND](X1,X2) = X2 + 2 [A__ISNAT](X) = X + 2 [A__ISNATILIST](X) = 2 [A__ISNATILISTKIND](X) = 2 [A__ISNATKIND](X) = 2 [A__ISNATLIST](X) = X + 2 [A__LENGTH](X) = X [MARK](X) = X + 2 Problem 1: SCC Processor: -> Pairs: A__U61(tt,L) -> MARK(L) A__AND(tt,X) -> MARK(X) A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> A__U61(mark(X1),X2) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__U61(tt,L) -> MARK(L) A__AND(tt,X) -> MARK(X) A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> A__U61(mark(X1),X2) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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) ->->-> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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: Reduction Pair Processor: -> Pairs: A__U61(tt,L) -> MARK(L) A__AND(tt,X) -> MARK(X) A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> A__U61(mark(X1),X2) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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 [a__U12](X) = X [a__U21](X1,X2) = X1 [a__U22](X) = X [a__U31](X1,X2) = X1 + 2.X2 [a__U32](X) = X [a__U41](X1,X2,X3) = 2.X1 + X2 + 2.X3 + 2 [a__U42](X1,X2) = 2.X1 + 2.X2 + 2 [a__U43](X) = X + 1 [a__U51](X1,X2,X3) = X1 [a__U52](X1,X2) = X1 [a__U53](X) = X [a__U61](X1,X2) = X1 + 2.X2 + 2 [a__and](X1,X2) = X1 + 2.X2 [a__isNat](X) = 2 [a__isNatIList](X) = 2.X + 2 [a__isNatIListKind](X) = 2 [a__isNatKind](X) = 2 [a__isNatList](X) = 2 [a__length](X) = X + 2 [a__zeros] = 2 [mark](X) = 2.X + 2 [0] = 0 [U11](X1,X2) = X1 [U12](X) = X [U21](X1,X2) = X1 [U22](X) = X [U31](X1,X2) = X1 + X2 [U32](X) = X [U41](X1,X2,X3) = 2.X1 + X2 + 2.X3 + 2 [U42](X1,X2) = 2.X1 + 2.X2 + 2 [U43](X) = X + 1 [U51](X1,X2,X3) = X1 [U52](X1,X2) = X1 [U53](X) = X [U61](X1,X2) = X1 + 2.X2 + 2 [and](X1,X2) = X1 + 2.X2 [cons](X1,X2) = 2.X1 + 2.X2 + 2 [isNat](X) = 0 [isNatIList](X) = 2.X + 2 [isNatIListKind](X) = 0 [isNatKind](X) = 0 [isNatList](X) = 2 [length](X) = X + 1 [nil] = 2 [s](X) = X [tt] = 2 [zeros] = 0 [A__U61](X1,X2) = X1 + 2.X2 + 2 [A__AND](X1,X2) = 2.X2 + 2 [A__ISNAT](X) = 2 [A__ISNATILIST](X) = 2.X + 2 [A__ISNATILISTKIND](X) = 2 [A__ISNATKIND](X) = 2 [A__ISNATLIST](X) = 2 [A__LENGTH](X) = X + 2 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> A__U61(mark(X1),X2) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__AND(tt,X) -> MARK(X) A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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) ->->-> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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: Reduction Pair Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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](X) = X + 2 [a__U21](X1,X2) = X1 + X2 [a__U22](X) = X [a__U31](X1,X2) = X1 + X2 [a__U32](X) = X [a__U41](X1,X2,X3) = X1 + X2 + 2.X3 + 2 [a__U42](X1,X2) = X1 + 2.X2 + 2 [a__U43](X) = X + 2 [a__U51](X1,X2,X3) = X1 + X2 + 2.X3 + 2 [a__U52](X1,X2) = X1 + 2.X2 + 2 [a__U53](X) = X + 2 [a__U61](X1,X2) = X1 + X2 + 2 [a__and](X1,X2) = X1 + X2 [a__isNat](X) = X + 2 [a__isNatIList](X) = X + 2 [a__isNatIListKind](X) = 2 [a__isNatKind](X) = 2 [a__isNatList](X) = X + 2 [a__length](X) = X + 2 [a__zeros] = 2 [mark](X) = X + 2 [0] = 0 [U11](X1,X2) = X1 + X2 + 2 [U12](X) = X + 2 [U21](X1,X2) = X1 + X2 [U22](X) = X [U31](X1,X2) = X1 + X2 [U32](X) = X [U41](X1,X2,X3) = X1 + X2 + 2.X3 + 2 [U42](X1,X2) = X1 + 2.X2 + 2 [U43](X) = X + 2 [U51](X1,X2,X3) = X1 + X2 + 2.X3 + 2 [U52](X1,X2) = X1 + 2.X2 + 2 [U53](X) = X + 2 [U61](X1,X2) = X1 + X2 + 2 [and](X1,X2) = X1 + X2 [cons](X1,X2) = X1 + 2.X2 + 2 [isNat](X) = X [isNatIList](X) = X + 2 [isNatIListKind](X) = 0 [isNatKind](X) = 0 [isNatList](X) = X + 2 [length](X) = X + 2 [nil] = 2 [s](X) = X [tt] = 2 [zeros] = 0 [A__AND](X1,X2) = 2.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 + 2 [A__LENGTH](X) = 2.X + 1 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__AND(tt,X) -> MARK(X) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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) ->->-> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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: Reduction Pair Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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 + 1 [a__U12](X) = X [a__U21](X1,X2) = X1 + X2 + 1 [a__U22](X) = X + 1 [a__U31](X1,X2) = X1 + 2.X2 [a__U32](X) = X + 1 [a__U41](X1,X2,X3) = X1 + X2 + 2.X3 + 2 [a__U42](X1,X2) = X1 + 2.X2 + 2 [a__U43](X) = X + 2 [a__U51](X1,X2,X3) = X1 + X2 + 2.X3 + 1 [a__U52](X1,X2) = X1 + 2.X2 + 1 [a__U53](X) = X + 2 [a__U61](X1,X2) = X1 + X2 + 2 [a__and](X1,X2) = X1 + X2 [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 + 1 [a__zeros] = 2 [mark](X) = X + 2 [0] = 0 [U11](X1,X2) = X1 + X2 + 1 [U12](X) = X [U21](X1,X2) = X1 + X2 + 1 [U22](X) = X + 1 [U31](X1,X2) = X1 + 2.X2 [U32](X) = X + 1 [U41](X1,X2,X3) = X1 + X2 + 2.X3 + 2 [U42](X1,X2) = X1 + 2.X2 + 2 [U43](X) = X + 2 [U51](X1,X2,X3) = X1 + X2 + 2.X3 + 1 [U52](X1,X2) = X1 + 2.X2 + 1 [U53](X) = X + 2 [U61](X1,X2) = X1 + X2 + 2 [and](X1,X2) = X1 + X2 [cons](X1,X2) = X1 + 2.X2 + 2 [isNat](X) = X [isNatIList](X) = 2.X + 2 [isNatIListKind](X) = 0 [isNatKind](X) = 0 [isNatList](X) = X + 1 [length](X) = X + 1 [nil] = 2 [s](X) = X + 1 [tt] = 2 [zeros] = 0 [A__AND](X1,X2) = X1 + 2.X2 [A__ISNAT](X) = 2.X + 2 [A__ISNATILIST](X) = 2 [A__ISNATILISTKIND](X) = 2 [A__ISNATKIND](X) = 2 [A__ISNATLIST](X) = 2.X [A__LENGTH](X) = 2.X [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__AND(tt,X) -> MARK(X) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) 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) ->->-> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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: Reduction Pair Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNATILIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) 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) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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 [a__U12](X) = X [a__U21](X1,X2) = X1 [a__U22](X) = X [a__U31](X1,X2) = X1 + 2.X2 [a__U32](X) = X [a__U41](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [a__U42](X1,X2) = 2.X1 + 2.X2 + 2 [a__U43](X) = X [a__U51](X1,X2,X3) = X1 [a__U52](X1,X2) = X1 [a__U53](X) = X [a__U61](X1,X2) = X1 + 2.X2 + 2 [a__and](X1,X2) = X1 + 2.X2 [a__isNat](X) = 2 [a__isNatIList](X) = 2.X + 2 [a__isNatIListKind](X) = 2 [a__isNatKind](X) = 2 [a__isNatList](X) = 2 [a__length](X) = X + 2 [a__zeros] = 2 [mark](X) = 2.X + 2 [0] = 0 [U11](X1,X2) = X1 [U12](X) = X [U21](X1,X2) = X1 [U22](X) = X [U31](X1,X2) = X1 + 2.X2 [U32](X) = X [U41](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2 [U42](X1,X2) = 2.X1 + 2.X2 + 2 [U43](X) = X [U51](X1,X2,X3) = X1 [U52](X1,X2) = X1 [U53](X) = X [U61](X1,X2) = X1 + 2.X2 + 2 [and](X1,X2) = X1 + 2.X2 [cons](X1,X2) = 2.X1 + 2.X2 + 2 [isNat](X) = 0 [isNatIList](X) = 2.X + 2 [isNatIListKind](X) = 0 [isNatKind](X) = 0 [isNatList](X) = 2 [length](X) = X + 2 [nil] = 1 [s](X) = X [tt] = 2 [zeros] = 0 [A__AND](X1,X2) = 2.X2 + 2 [A__ISNATILIST](X) = 2.X + 2 [A__ISNATILISTKIND](X) = 2 [A__ISNATKIND](X) = 2 [A__ISNATLIST](X) = 2 [A__LENGTH](X) = X + 2 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNATILIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) 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) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__AND(tt,X) -> MARK(X) A__ISNATILIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) 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) ->->-> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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: Reduction Pair Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNATILIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) 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) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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 [a__U12](X) = X [a__U21](X1,X2) = X1 [a__U22](X) = X [a__U31](X1,X2) = X1 + 2.X2 [a__U32](X) = X [a__U41](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [a__U42](X1,X2) = 2.X1 + 2.X2 + 2 [a__U43](X) = X + 2 [a__U51](X1,X2,X3) = X1 [a__U52](X1,X2) = X1 [a__U53](X) = X [a__U61](X1,X2) = X1 + 2.X2 + 1 [a__and](X1,X2) = X1 + 2.X2 [a__isNat](X) = 2 [a__isNatIList](X) = 2.X + 2 [a__isNatIListKind](X) = 2 [a__isNatKind](X) = 2 [a__isNatList](X) = 2 [a__length](X) = X + 1 [a__zeros] = 2 [mark](X) = 2.X + 2 [0] = 0 [U11](X1,X2) = X1 [U12](X) = X [U21](X1,X2) = X1 [U22](X) = X [U31](X1,X2) = X1 + 2.X2 [U32](X) = X [U41](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2 [U42](X1,X2) = 2.X1 + X2 + 2 [U43](X) = X + 1 [U51](X1,X2,X3) = X1 [U52](X1,X2) = X1 [U53](X) = X [U61](X1,X2) = X1 + X2 + 1 [and](X1,X2) = X1 + 2.X2 [cons](X1,X2) = 2.X1 + 2.X2 + 2 [isNat](X) = 0 [isNatIList](X) = 2.X + 2 [isNatIListKind](X) = 0 [isNatKind](X) = 0 [isNatList](X) = 1 [length](X) = X + 1 [nil] = 2 [s](X) = X [tt] = 2 [zeros] = 0 [A__AND](X1,X2) = X1 + 2.X2 [A__ISNATILIST](X) = X + 2 [A__ISNATILISTKIND](X) = 2 [A__ISNATKIND](X) = 2 [A__ISNATLIST](X) = 2 [A__LENGTH](X) = X + 2 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) 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) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__AND(tt,X) -> MARK(X) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) 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) ->->-> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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: Reduction Pair Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) 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) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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 [a__U12](X) = X [a__U21](X1,X2) = X1 [a__U22](X) = X [a__U31](X1,X2) = X1 [a__U32](X) = X [a__U41](X1,X2,X3) = X1 [a__U42](X1,X2) = X1 [a__U43](X) = X [a__U51](X1,X2,X3) = X1 [a__U52](X1,X2) = X1 [a__U53](X) = X [a__U61](X1,X2) = X1 + 2.X2 + 2 [a__and](X1,X2) = X1 + 2.X2 [a__isNat](X) = 2 [a__isNatIList](X) = 2 [a__isNatIListKind](X) = 2 [a__isNatKind](X) = 2 [a__isNatList](X) = 2 [a__length](X) = X + 2 [a__zeros] = 2 [mark](X) = 2.X + 2 [0] = 0 [U11](X1,X2) = X1 [U12](X) = X [U21](X1,X2) = X1 [U22](X) = X [U31](X1,X2) = X1 [U32](X) = X [U41](X1,X2,X3) = X1 [U42](X1,X2) = X1 [U43](X) = X [U51](X1,X2,X3) = X1 [U52](X1,X2) = X1 [U53](X) = X [U61](X1,X2) = X1 + X2 + 2 [and](X1,X2) = X1 + 2.X2 [cons](X1,X2) = 2.X1 + 2.X2 + 2 [isNat](X) = 0 [isNatIList](X) = 2 [isNatIListKind](X) = 0 [isNatKind](X) = 0 [isNatList](X) = 2 [length](X) = X + 2 [nil] = 1 [s](X) = X [tt] = 2 [zeros] = 0 [A__AND](X1,X2) = 2.X2 + 1 [A__ISNATILIST](X) = 2 [A__ISNATILISTKIND](X) = 1 [A__ISNATKIND](X) = 1 [A__ISNATLIST](X) = 2 [A__LENGTH](X) = X + 1 [MARK](X) = 2.X + 1 Problem 1: SCC Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) 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) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) 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) ->->-> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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: Reduction Pair Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) 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) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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](X) = X + 2 [a__U21](X1,X2) = X1 + X2 [a__U22](X) = X [a__U31](X1,X2) = X1 + 2.X2 [a__U32](X) = X [a__U41](X1,X2,X3) = X1 + X2 + 2.X3 + 2 [a__U42](X1,X2) = X1 + 2.X2 + 2 [a__U43](X) = X + 2 [a__U51](X1,X2,X3) = X1 + X2 + 2.X3 + 2 [a__U52](X1,X2) = X1 + 2.X2 + 2 [a__U53](X) = X [a__U61](X1,X2) = X1 + X2 + 2 [a__and](X1,X2) = X1 + X2 [a__isNat](X) = X + 2 [a__isNatIList](X) = 2.X + 2 [a__isNatIListKind](X) = 2 [a__isNatKind](X) = 2 [a__isNatList](X) = X + 2 [a__length](X) = X + 2 [a__zeros] = 2 [mark](X) = X + 2 [0] = 0 [U11](X1,X2) = X1 + X2 + 2 [U12](X) = X + 2 [U21](X1,X2) = X1 + X2 [U22](X) = X [U31](X1,X2) = X1 + 2.X2 [U32](X) = X [U41](X1,X2,X3) = X1 + X2 + 2.X3 + 2 [U42](X1,X2) = X1 + 2.X2 + 2 [U43](X) = X + 2 [U51](X1,X2,X3) = X1 + X2 + 2.X3 + 2 [U52](X1,X2) = X1 + 2.X2 + 2 [U53](X) = X [U61](X1,X2) = X1 + X2 + 2 [and](X1,X2) = X1 + X2 [cons](X1,X2) = X1 + 2.X2 + 2 [isNat](X) = X [isNatIList](X) = 2.X [isNatIListKind](X) = 0 [isNatKind](X) = 0 [isNatList](X) = X + 2 [length](X) = X + 2 [nil] = 2 [s](X) = X [tt] = 2 [zeros] = 0 [A__AND](X1,X2) = 2.X2 + 2 [A__ISNATILISTKIND](X) = 2 [A__ISNATKIND](X) = 2 [A__ISNATLIST](X) = 2.X + 2 [A__LENGTH](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) 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) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) 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) ->->-> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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: Reduction Pair Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) 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) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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 [a__U12](X) = X [a__U21](X1,X2) = X1 [a__U22](X) = X [a__U31](X1,X2) = X1 + 2.X2 [a__U32](X) = X [a__U41](X1,X2,X3) = 2.X1 + 2.X3 + 2 [a__U42](X1,X2) = 2.X1 + 2.X2 + 2 [a__U43](X) = X [a__U51](X1,X2,X3) = X1 [a__U52](X1,X2) = X1 [a__U53](X) = X [a__U61](X1,X2) = X1 + 2.X2 + 2 [a__and](X1,X2) = X1 + 2.X2 [a__isNat](X) = 2 [a__isNatIList](X) = 2.X + 2 [a__isNatIListKind](X) = 2 [a__isNatKind](X) = 2 [a__isNatList](X) = 2 [a__length](X) = X + 2 [a__zeros] = 2 [mark](X) = 2.X + 2 [0] = 0 [U11](X1,X2) = X1 [U12](X) = X [U21](X1,X2) = X1 [U22](X) = X [U31](X1,X2) = X1 + 2.X2 [U32](X) = X [U41](X1,X2,X3) = 2.X1 + X3 + 2 [U42](X1,X2) = 2.X1 + 2.X2 + 2 [U43](X) = X [U51](X1,X2,X3) = X1 [U52](X1,X2) = X1 [U53](X) = X [U61](X1,X2) = X1 + 2.X2 + 2 [and](X1,X2) = X1 + 2.X2 [cons](X1,X2) = 2.X1 + 2.X2 + 2 [isNat](X) = 0 [isNatIList](X) = 2.X + 1 [isNatIListKind](X) = 0 [isNatKind](X) = 0 [isNatList](X) = 1 [length](X) = X + 2 [nil] = 1 [s](X) = X [tt] = 2 [zeros] = 0 [A__AND](X1,X2) = 2.X2 + 1 [A__ISNATILISTKIND](X) = 1 [A__ISNATKIND](X) = 1 [A__ISNATLIST](X) = 2 [A__LENGTH](X) = X [MARK](X) = 2.X + 1 Problem 1: SCC Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) 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) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(isNatKind(X)) -> A__ISNATKIND(X) 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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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: Reduction Pair Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__AND(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(isNatKind(X)) -> A__ISNATKIND(X) 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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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 [a__U12](X) = X [a__U21](X1,X2) = X1 [a__U22](X) = X [a__U31](X1,X2) = X1 + 2.X2 [a__U32](X) = X [a__U41](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [a__U42](X1,X2) = 2.X1 + 2.X2 + 2 [a__U43](X) = X + 2 [a__U51](X1,X2,X3) = X1 [a__U52](X1,X2) = X1 [a__U53](X) = X [a__U61](X1,X2) = X1 + 2.X2 + 2 [a__and](X1,X2) = X1 + 2.X2 [a__isNat](X) = 2 [a__isNatIList](X) = 2.X + 2 [a__isNatIListKind](X) = 2 [a__isNatKind](X) = 2 [a__isNatList](X) = 2 [a__length](X) = X + 2 [a__zeros] = 2 [mark](X) = 2.X + 2 [0] = 0 [U11](X1,X2) = X1 [U12](X) = X [U21](X1,X2) = X1 [U22](X) = X [U31](X1,X2) = X1 + X2 [U32](X) = X [U41](X1,X2,X3) = 2.X1 + X2 + 2.X3 + 2 [U42](X1,X2) = 2.X1 + 2.X2 + 2 [U43](X) = X + 2 [U51](X1,X2,X3) = X1 [U52](X1,X2) = X1 [U53](X) = X [U61](X1,X2) = X1 + 2.X2 + 1 [and](X1,X2) = X1 + 2.X2 [cons](X1,X2) = 2.X1 + 2.X2 + 2 [isNat](X) = 0 [isNatIList](X) = 2.X + 1 [isNatIListKind](X) = 0 [isNatKind](X) = 0 [isNatList](X) = 0 [length](X) = X + 1 [nil] = 0 [s](X) = X [tt] = 2 [zeros] = 0 [A__AND](X1,X2) = 2.X2 + 2 [A__ISNATILISTKIND](X) = 2 [A__ISNATKIND](X) = 2 [A__LENGTH](X) = X + 2 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(isNatKind(X)) -> A__ISNATKIND(X) 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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(isNatKind(X)) -> A__ISNATKIND(X) 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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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: Reduction Pair Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__LENGTH(cons(N,L)) -> A__AND(a__isNatList(L),isNatIListKind(L)) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(isNatKind(X)) -> A__ISNATKIND(X) 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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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](X) = X + 2 [a__U21](X1,X2) = X1 + X2 [a__U22](X) = X [a__U31](X1,X2) = X1 + 2.X2 [a__U32](X) = X [a__U41](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [a__U42](X1,X2) = X1 + 2.X2 + 2 [a__U43](X) = X + 2 [a__U51](X1,X2,X3) = X1 + X2 + 2.X3 + 2 [a__U52](X1,X2) = X1 + 2.X2 + 2 [a__U53](X) = X + 1 [a__U61](X1,X2) = X1 + X2 + 2 [a__and](X1,X2) = X1 + X2 [a__isNat](X) = X + 2 [a__isNatIList](X) = 2.X + 2 [a__isNatIListKind](X) = 2 [a__isNatKind](X) = 2 [a__isNatList](X) = X + 2 [a__length](X) = X + 2 [a__zeros] = 2 [mark](X) = X + 2 [0] = 0 [U11](X1,X2) = X1 + X2 + 2 [U12](X) = X + 2 [U21](X1,X2) = X1 + X2 [U22](X) = X [U31](X1,X2) = X1 + 2.X2 [U32](X) = X [U41](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U42](X1,X2) = X1 + 2.X2 + 2 [U43](X) = X + 2 [U51](X1,X2,X3) = X1 + X2 + 2.X3 + 2 [U52](X1,X2) = X1 + 2.X2 + 2 [U53](X) = X + 1 [U61](X1,X2) = X1 + X2 + 2 [and](X1,X2) = X1 + X2 [cons](X1,X2) = X1 + 2.X2 + 2 [isNat](X) = X [isNatIList](X) = 2.X [isNatIListKind](X) = 0 [isNatKind](X) = 0 [isNatList](X) = X + 1 [length](X) = X + 2 [nil] = 1 [s](X) = X [tt] = 2 [zeros] = 0 [A__AND](X1,X2) = X1 + 2.X2 [A__ISNATILISTKIND](X) = 2 [A__ISNATKIND](X) = 2 [A__LENGTH](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(isNatKind(X)) -> A__ISNATKIND(X) 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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(isNatKind(X)) -> A__ISNATKIND(X) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) ->->-> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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: Reduction Pair Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) MARK(U11(X1,X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(isNatKind(X)) -> A__ISNATKIND(X) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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](X) = X + 2 [a__U21](X1,X2) = X1 + X2 [a__U22](X) = X [a__U31](X1,X2) = X1 + X2 [a__U32](X) = X [a__U41](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [a__U42](X1,X2) = X1 + 2.X2 + 2 [a__U43](X) = X + 2 [a__U51](X1,X2,X3) = X1 + X2 + 2.X3 + 2 [a__U52](X1,X2) = X1 + 2.X2 [a__U53](X) = X [a__U61](X1,X2) = X1 + X2 + 2 [a__and](X1,X2) = X1 + X2 [a__isNat](X) = X + 2 [a__isNatIList](X) = 2.X + 2 [a__isNatIListKind](X) = 2 [a__isNatKind](X) = 2 [a__isNatList](X) = X + 2 [a__length](X) = X + 2 [a__zeros] = 2 [mark](X) = X + 2 [0] = 0 [U11](X1,X2) = X1 + X2 + 2 [U12](X) = X + 2 [U21](X1,X2) = X1 + X2 [U22](X) = X [U31](X1,X2) = X1 + X2 [U32](X) = X [U41](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U42](X1,X2) = X1 + 2.X2 + 2 [U43](X) = X + 2 [U51](X1,X2,X3) = X1 + X2 + 2.X3 + 2 [U52](X1,X2) = X1 + 2.X2 [U53](X) = X [U61](X1,X2) = X1 + X2 + 2 [and](X1,X2) = X1 + X2 [cons](X1,X2) = X1 + 2.X2 + 2 [isNat](X) = X [isNatIList](X) = 2.X + 2 [isNatIListKind](X) = 0 [isNatKind](X) = 0 [isNatList](X) = X + 2 [length](X) = X + 2 [nil] = 1 [s](X) = X [tt] = 2 [zeros] = 0 [A__AND](X1,X2) = 2.X2 + 2 [A__ISNATILISTKIND](X) = 2 [A__ISNATKIND](X) = 2 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(isNatKind(X)) -> A__ISNATKIND(X) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(isNatKind(X)) -> A__ISNATKIND(X) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) ->->-> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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: Reduction Pair Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) MARK(U12(X)) -> MARK(X) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(isNatKind(X)) -> A__ISNATKIND(X) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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](X) = X + 2 [a__U21](X1,X2) = X1 + X2 [a__U22](X) = X [a__U31](X1,X2) = X1 + 2.X2 [a__U32](X) = X [a__U41](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [a__U42](X1,X2) = X1 + 2.X2 + 2 [a__U43](X) = X + 2 [a__U51](X1,X2,X3) = X1 + X2 + 2.X3 + 2 [a__U52](X1,X2) = X1 + X2 + 2 [a__U53](X) = X + 2 [a__U61](X1,X2) = X1 + X2 + 2 [a__and](X1,X2) = X1 + X2 [a__isNat](X) = X + 2 [a__isNatIList](X) = 2.X + 2 [a__isNatIListKind](X) = 2 [a__isNatKind](X) = 2 [a__isNatList](X) = X + 2 [a__length](X) = X + 2 [a__zeros] = 2 [mark](X) = X + 2 [0] = 0 [U11](X1,X2) = X1 + X2 + 2 [U12](X) = X + 2 [U21](X1,X2) = X1 + X2 [U22](X) = X [U31](X1,X2) = X1 + 2.X2 [U32](X) = X [U41](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U42](X1,X2) = X1 + 2.X2 + 2 [U43](X) = X + 2 [U51](X1,X2,X3) = X1 + X2 + 2.X3 + 2 [U52](X1,X2) = X1 + X2 + 2 [U53](X) = X + 2 [U61](X1,X2) = X1 + X2 + 2 [and](X1,X2) = X1 + X2 [cons](X1,X2) = X1 + 2.X2 + 2 [isNat](X) = X [isNatIList](X) = 2.X + 1 [isNatIListKind](X) = 0 [isNatKind](X) = 0 [isNatList](X) = X [length](X) = X + 2 [nil] = 1 [s](X) = X [tt] = 2 [zeros] = 0 [A__AND](X1,X2) = X1 + 2.X2 [A__ISNATILISTKIND](X) = 2 [A__ISNATKIND](X) = 2 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(isNatKind(X)) -> A__ISNATKIND(X) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(isNatKind(X)) -> A__ISNATKIND(X) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) ->->-> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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: Reduction Pair Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(isNatKind(X)) -> A__ISNATKIND(X) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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) = X2 + 2 [a__U12](X) = X + 2 [a__U21](X1,X2) = X1 + X2 + 2 [a__U22](X) = X + 2 [a__U31](X1,X2) = X1 + 2.X2 [a__U32](X) = X + 2 [a__U41](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [a__U42](X1,X2) = X1 + 2.X2 + 2 [a__U43](X) = X + 2 [a__U51](X1,X2,X3) = X1 + X2 + 2.X3 [a__U52](X1,X2) = X1 + X2 [a__U53](X) = X + 2 [a__U61](X1,X2) = X1 + X2 + 2 [a__and](X1,X2) = X1 + X2 [a__isNat](X) = X + 2 [a__isNatIList](X) = 2.X + 2 [a__isNatIListKind](X) = 2 [a__isNatKind](X) = 2 [a__isNatList](X) = X [a__length](X) = X [a__zeros] = 2 [mark](X) = X + 2 [0] = 0 [U11](X1,X2) = X2 [U12](X) = X + 2 [U21](X1,X2) = X1 + X2 + 2 [U22](X) = X + 2 [U31](X1,X2) = X1 + 2.X2 [U32](X) = X + 2 [U41](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U42](X1,X2) = X1 + 2.X2 + 2 [U43](X) = X + 2 [U51](X1,X2,X3) = X1 + X2 + 2.X3 [U52](X1,X2) = X1 + X2 [U53](X) = X + 2 [U61](X1,X2) = X1 + X2 + 2 [and](X1,X2) = X1 + X2 [cons](X1,X2) = X1 + 2.X2 + 2 [isNat](X) = X [isNatIList](X) = 2.X + 1 [isNatIListKind](X) = 0 [isNatKind](X) = 0 [isNatList](X) = X [length](X) = X [nil] = 2 [s](X) = X + 2 [tt] = 2 [zeros] = 0 [A__AND](X1,X2) = X1 + 2.X2 [A__ISNATILISTKIND](X) = 2 [A__ISNATKIND](X) = 2 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(isNatKind(X)) -> A__ISNATKIND(X) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(isNatKind(X)) -> A__ISNATKIND(X) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) ->->-> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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: Reduction Pair Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(isNatKind(X)) -> A__ISNATKIND(X) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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) = 2.X2 + 2 [a__U12](X) = 2.X + 2 [a__U21](X1,X2) = X2 [a__U22](X) = X [a__U31](X1,X2) = X1 + X2 + 2 [a__U32](X) = X + 2 [a__U41](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [a__U42](X1,X2) = 2.X1 + 2.X2 + 2 [a__U43](X) = X [a__U51](X1,X2,X3) = X1 + X2 + X3 [a__U52](X1,X2) = X1 + X2 [a__U53](X) = X [a__U61](X1,X2) = X1 + 2.X2 + 2 [a__and](X1,X2) = X1 + X2 [a__isNat](X) = X [a__isNatIList](X) = 2.X + 2 [a__isNatIListKind](X) = X [a__isNatKind](X) = X [a__isNatList](X) = X [a__length](X) = 2.X + 2 [a__zeros] = 0 [mark](X) = X [0] = 0 [U11](X1,X2) = 2.X2 + 2 [U12](X) = 2.X + 2 [U21](X1,X2) = X2 [U22](X) = X [U31](X1,X2) = X1 + X2 + 2 [U32](X) = X + 2 [U41](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U42](X1,X2) = 2.X1 + 2.X2 + 2 [U43](X) = X [U51](X1,X2,X3) = X1 + X2 + X3 [U52](X1,X2) = X1 + X2 [U53](X) = X [U61](X1,X2) = X1 + 2.X2 + 2 [and](X1,X2) = X1 + X2 [cons](X1,X2) = 2.X1 + 2.X2 [isNat](X) = X [isNatIList](X) = 2.X + 2 [isNatIListKind](X) = X [isNatKind](X) = X [isNatList](X) = X [length](X) = 2.X + 2 [nil] = 1 [s](X) = X [tt] = 0 [zeros] = 0 [A__AND](X1,X2) = 2.X2 + 2 [A__ISNATILISTKIND](X) = 2.X + 2 [A__ISNATKIND](X) = 2.X + 1 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(isNatKind(X)) -> A__ISNATKIND(X) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(isNatKind(X)) -> A__ISNATKIND(X) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) ->->-> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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: Reduction Pair Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(isNatKind(X)) -> A__ISNATKIND(X) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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) = 0 [a__U12](X) = 0 [a__U21](X1,X2) = 0 [a__U22](X) = 2.X [a__U31](X1,X2) = X1 + X2 + 2 [a__U32](X) = X + 2 [a__U41](X1,X2,X3) = 2.X1 + X2 + 2.X3 + 2 [a__U42](X1,X2) = 2.X1 + 2.X2 + 2 [a__U43](X) = X [a__U51](X1,X2,X3) = X1 + X2 + X3 [a__U52](X1,X2) = 2.X1 + X2 [a__U53](X) = X [a__U61](X1,X2) = X1 + 2.X2 + 2 [a__and](X1,X2) = X1 + X2 [a__isNat](X) = 0 [a__isNatIList](X) = 2.X + 2 [a__isNatIListKind](X) = X [a__isNatKind](X) = X [a__isNatList](X) = X [a__length](X) = 2.X + 2 [a__zeros] = 0 [mark](X) = X [0] = 0 [U11](X1,X2) = 0 [U12](X) = 0 [U21](X1,X2) = 0 [U22](X) = 2.X [U31](X1,X2) = X1 + X2 + 2 [U32](X) = X + 2 [U41](X1,X2,X3) = 2.X1 + X2 + 2.X3 + 2 [U42](X1,X2) = 2.X1 + 2.X2 + 2 [U43](X) = X [U51](X1,X2,X3) = X1 + X2 + X3 [U52](X1,X2) = 2.X1 + X2 [U53](X) = X [U61](X1,X2) = X1 + 2.X2 + 2 [and](X1,X2) = X1 + X2 [cons](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 0 [isNatIList](X) = 2.X + 2 [isNatIListKind](X) = X [isNatKind](X) = X [isNatList](X) = X [length](X) = 2.X + 2 [nil] = 2 [s](X) = X [tt] = 0 [zeros] = 0 [A__AND](X1,X2) = X1 + 2.X2 + 2 [A__ISNATILISTKIND](X) = 2.X + 2 [A__ISNATKIND](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(isNatKind(X)) -> A__ISNATKIND(X) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) ->->-> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) ->->-> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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 1.1: Subterm Processor: -> Pairs: A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__ISNATKIND) = 1 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) MARK(U22(X)) -> MARK(X) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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) = X2 + 2 [a__U12](X) = X + 1 [a__U21](X1,X2) = X1 + X2 + 1 [a__U22](X) = X + 1 [a__U31](X1,X2) = X1 + X2 [a__U32](X) = X + 1 [a__U41](X1,X2,X3) = X1 + X2 + 2.X3 + 2 [a__U42](X1,X2) = X1 + 2.X2 + 2 [a__U43](X) = X + 2 [a__U51](X1,X2,X3) = X1 + X2 + 2.X3 + 1 [a__U52](X1,X2) = X1 + 2.X2 + 1 [a__U53](X) = X + 1 [a__U61](X1,X2) = X1 + X2 + 2 [a__and](X1,X2) = X1 + X2 [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 + 1 [a__zeros] = 2 [mark](X) = X + 2 [0] = 0 [U11](X1,X2) = X2 + 2 [U12](X) = X + 1 [U21](X1,X2) = X1 + X2 + 1 [U22](X) = X + 1 [U31](X1,X2) = X1 + X2 [U32](X) = X + 1 [U41](X1,X2,X3) = X1 + X2 + 2.X3 + 2 [U42](X1,X2) = X1 + 2.X2 + 2 [U43](X) = X + 2 [U51](X1,X2,X3) = X1 + X2 + 2.X3 + 1 [U52](X1,X2) = X1 + 2.X2 + 1 [U53](X) = X + 1 [U61](X1,X2) = X1 + X2 + 2 [and](X1,X2) = X1 + X2 [cons](X1,X2) = X1 + 2.X2 + 2 [isNat](X) = X [isNatIList](X) = 2.X + 1 [isNatIListKind](X) = 0 [isNatKind](X) = 0 [isNatList](X) = X + 1 [length](X) = X + 1 [nil] = 2 [s](X) = X + 1 [tt] = 2 [zeros] = 0 [A__AND](X1,X2) = X1 + 2.X2 [A__ISNATILISTKIND](X) = 2 [MARK](X) = 2.X + 2 Problem 1.2: SCC Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) ->->-> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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 1.2: Reduction Pair Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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) = 1 [a__U12](X) = 1 [a__U21](X1,X2) = 1 [a__U22](X) = X [a__U31](X1,X2) = X1 + 1 [a__U32](X) = X + 1 [a__U41](X1,X2,X3) = 2.X1 + 2.X3 + 1 [a__U42](X1,X2) = X1 + 2.X2 + 2 [a__U43](X) = X + 1 [a__U51](X1,X2,X3) = X1 [a__U52](X1,X2) = X1 [a__U53](X) = X [a__U61](X1,X2) = 2.X1 + 2.X2 + 1 [a__and](X1,X2) = X1 + 2.X2 [a__isNat](X) = 1 [a__isNatIList](X) = 2.X + 2 [a__isNatIListKind](X) = 1 [a__isNatKind](X) = 1 [a__isNatList](X) = 1 [a__length](X) = X + 2 [a__zeros] = 1 [mark](X) = 2.X + 1 [0] = 0 [U11](X1,X2) = 0 [U12](X) = 1 [U21](X1,X2) = 1 [U22](X) = X [U31](X1,X2) = X1 + 1 [U32](X) = X + 1 [U41](X1,X2,X3) = 2.X1 + 2.X3 + 1 [U42](X1,X2) = X1 + X2 + 2 [U43](X) = X + 1 [U51](X1,X2,X3) = X1 [U52](X1,X2) = X1 [U53](X) = X [U61](X1,X2) = 2.X1 + 2.X2 + 1 [and](X1,X2) = X1 + X2 [cons](X1,X2) = 2.X1 + 2.X2 + 1 [isNat](X) = 0 [isNatIList](X) = 2.X + 2 [isNatIListKind](X) = 0 [isNatKind](X) = 0 [isNatList](X) = 0 [length](X) = X + 2 [nil] = 0 [s](X) = X [tt] = 1 [zeros] = 0 [A__AND](X1,X2) = X1 + 2.X2 + 1 [A__ISNATILISTKIND](X) = 2 [MARK](X) = 2.X + 2 Problem 1.2: SCC Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) ->->-> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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 1.2: Reduction Pair Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) MARK(U32(X)) -> MARK(X) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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) = 2.X1 [a__U12](X) = X [a__U21](X1,X2) = 2.X1 [a__U22](X) = 2.X [a__U31](X1,X2) = 2.X1 + 2 [a__U32](X) = 2.X + 2 [a__U41](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [a__U42](X1,X2) = 2.X1 + 2.X2 + 2 [a__U43](X) = X [a__U51](X1,X2,X3) = 2.X1 [a__U52](X1,X2) = 2.X1 [a__U53](X) = 2.X [a__U61](X1,X2) = 2.X1 + 2.X2 + 2 [a__and](X1,X2) = 2.X1 + 2.X2 [a__isNat](X) = 0 [a__isNatIList](X) = 2.X + 2 [a__isNatIListKind](X) = 0 [a__isNatKind](X) = 0 [a__isNatList](X) = 0 [a__length](X) = X + 2 [a__zeros] = 2 [mark](X) = 2.X [0] = 0 [U11](X1,X2) = 2.X1 [U12](X) = X [U21](X1,X2) = 2.X1 [U22](X) = 2.X [U31](X1,X2) = 2.X1 + 1 [U32](X) = 2.X + 2 [U41](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U42](X1,X2) = 2.X1 + X2 + 2 [U43](X) = X [U51](X1,X2,X3) = 2.X1 [U52](X1,X2) = 2.X1 [U53](X) = 2.X [U61](X1,X2) = 2.X1 + X2 + 2 [and](X1,X2) = 2.X1 + X2 [cons](X1,X2) = 2.X1 + 2.X2 [isNat](X) = 0 [isNatIList](X) = X + 2 [isNatIListKind](X) = 0 [isNatKind](X) = 0 [isNatList](X) = 0 [length](X) = X + 2 [nil] = 1 [s](X) = X [tt] = 0 [zeros] = 1 [A__AND](X1,X2) = X1 + 2.X2 + 2 [A__ISNATILISTKIND](X) = 2 [MARK](X) = 2.X + 2 Problem 1.2: SCC Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) ->->-> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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 1.2: Reduction Pair Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) MARK(U41(X1,X2,X3)) -> MARK(X1) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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) = 2 [a__U12](X) = 2 [a__U21](X1,X2) = X1 [a__U22](X) = 2 [a__U31](X1,X2) = X1 + 2.X2 [a__U32](X) = X [a__U41](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [a__U42](X1,X2) = X1 + 2.X2 + 2 [a__U43](X) = X + 2 [a__U51](X1,X2,X3) = X1 [a__U52](X1,X2) = X1 [a__U53](X) = X [a__U61](X1,X2) = X1 + 2.X2 + 1 [a__and](X1,X2) = X1 + 2.X2 [a__isNat](X) = 2 [a__isNatIList](X) = 2.X + 2 [a__isNatIListKind](X) = 2 [a__isNatKind](X) = 2 [a__isNatList](X) = 2 [a__length](X) = X + 1 [a__zeros] = 2 [mark](X) = 2.X + 2 [0] = 0 [U11](X1,X2) = 2 [U12](X) = 2 [U21](X1,X2) = X1 [U22](X) = 1 [U31](X1,X2) = X1 + 2.X2 [U32](X) = X [U41](X1,X2,X3) = X1 + 2.X2 + X3 + 2 [U42](X1,X2) = X1 + 2.X2 + 2 [U43](X) = X + 2 [U51](X1,X2,X3) = X1 [U52](X1,X2) = X1 [U53](X) = X [U61](X1,X2) = X1 + X2 + 1 [and](X1,X2) = X1 + 2.X2 [cons](X1,X2) = 2.X1 + 2.X2 + 2 [isNat](X) = 0 [isNatIList](X) = X + 1 [isNatIListKind](X) = 0 [isNatKind](X) = 0 [isNatList](X) = 1 [length](X) = X + 1 [nil] = 1 [s](X) = X [tt] = 2 [zeros] = 0 [A__AND](X1,X2) = 2.X2 + 2 [A__ISNATILISTKIND](X) = 2 [MARK](X) = 2.X + 2 Problem 1.2: SCC Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) ->->-> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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 1.2: Reduction Pair Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) MARK(U42(X1,X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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) = 2 [a__U12](X) = 2 [a__U21](X1,X2) = 2 [a__U22](X) = 2 [a__U31](X1,X2) = X1 + 2.X2 [a__U32](X) = X [a__U41](X1,X2,X3) = 2.X1 + 2.X3 + 2 [a__U42](X1,X2) = 2.X1 + 2.X2 + 2 [a__U43](X) = X [a__U51](X1,X2,X3) = X1 [a__U52](X1,X2) = X1 [a__U53](X) = X [a__U61](X1,X2) = X1 + 2.X2 + 2 [a__and](X1,X2) = X1 + 2.X2 [a__isNat](X) = 2 [a__isNatIList](X) = 2.X + 2 [a__isNatIListKind](X) = 2 [a__isNatKind](X) = 2 [a__isNatList](X) = 2 [a__length](X) = X + 2 [a__zeros] = 2 [mark](X) = 2.X + 2 [0] = 0 [U11](X1,X2) = 1 [U12](X) = 2 [U21](X1,X2) = 0 [U22](X) = 0 [U31](X1,X2) = X1 + 2.X2 [U32](X) = X [U41](X1,X2,X3) = 2.X1 + 2.X3 + 2 [U42](X1,X2) = 2.X1 + X2 + 2 [U43](X) = X [U51](X1,X2,X3) = X1 [U52](X1,X2) = X1 [U53](X) = X [U61](X1,X2) = X1 + 2.X2 + 2 [and](X1,X2) = X1 + X2 [cons](X1,X2) = 2.X1 + 2.X2 + 2 [isNat](X) = 0 [isNatIList](X) = X + 2 [isNatIListKind](X) = 0 [isNatKind](X) = 0 [isNatList](X) = 2 [length](X) = X + 1 [nil] = 2 [s](X) = X [tt] = 2 [zeros] = 0 [A__AND](X1,X2) = X1 + 2.X2 [A__ISNATILISTKIND](X) = 2 [MARK](X) = 2.X + 2 Problem 1.2: SCC Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) ->->-> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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 1.2: Reduction Pair Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) MARK(U43(X)) -> MARK(X) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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 [a__U12](X) = X [a__U21](X1,X2) = 2 [a__U22](X) = X [a__U31](X1,X2) = 2.X2 + 2 [a__U32](X) = X [a__U41](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [a__U42](X1,X2) = 2.X1 + 2.X2 + 2 [a__U43](X) = X + 2 [a__U51](X1,X2,X3) = X1 [a__U52](X1,X2) = X1 [a__U53](X) = X [a__U61](X1,X2) = X1 + 2.X2 + 2 [a__and](X1,X2) = X1 + 2.X2 [a__isNat](X) = 2 [a__isNatIList](X) = 2.X + 2 [a__isNatIListKind](X) = 2 [a__isNatKind](X) = 2 [a__isNatList](X) = 2 [a__length](X) = X + 2 [a__zeros] = 2 [mark](X) = 2.X + 2 [0] = 0 [U11](X1,X2) = X1 [U12](X) = X [U21](X1,X2) = 2 [U22](X) = X [U31](X1,X2) = 2.X2 + 2 [U32](X) = X [U41](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2 [U42](X1,X2) = 2.X1 + 2.X2 + 2 [U43](X) = X + 2 [U51](X1,X2,X3) = X1 [U52](X1,X2) = X1 [U53](X) = X [U61](X1,X2) = X1 + 2.X2 + 2 [and](X1,X2) = X1 + 2.X2 [cons](X1,X2) = X1 + 2.X2 + 2 [isNat](X) = 0 [isNatIList](X) = X [isNatIListKind](X) = 0 [isNatKind](X) = 0 [isNatList](X) = 1 [length](X) = X + 2 [nil] = 0 [s](X) = X [tt] = 2 [zeros] = 0 [A__AND](X1,X2) = 2.X2 + 2 [A__ISNATILISTKIND](X) = 2 [MARK](X) = 2.X + 2 Problem 1.2: SCC Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) ->->-> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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 1.2: Reduction Pair Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) MARK(U51(X1,X2,X3)) -> MARK(X1) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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 + 1 [a__U12](X) = X + 1 [a__U21](X1,X2) = X1 + X2 + 1 [a__U22](X) = X + 1 [a__U31](X1,X2) = X1 + 2.X2 [a__U32](X) = X + 1 [a__U41](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [a__U42](X1,X2) = X1 + 2.X2 + 2 [a__U43](X) = X + 2 [a__U51](X1,X2,X3) = X1 + X2 + 2.X3 + 1 [a__U52](X1,X2) = X1 + 2.X2 + 1 [a__U53](X) = X + 2 [a__U61](X1,X2) = X1 + X2 + 2 [a__and](X1,X2) = X1 + X2 [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 + 1 [a__zeros] = 2 [mark](X) = X + 2 [0] = 0 [U11](X1,X2) = X1 + X2 + 1 [U12](X) = X + 1 [U21](X1,X2) = X1 + X2 + 1 [U22](X) = X + 1 [U31](X1,X2) = X1 + 2.X2 [U32](X) = X + 1 [U41](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U42](X1,X2) = X1 + 2.X2 + 2 [U43](X) = X + 2 [U51](X1,X2,X3) = X1 + X2 + 2.X3 + 1 [U52](X1,X2) = X1 + 2.X2 + 1 [U53](X) = X + 2 [U61](X1,X2) = X1 + X2 + 2 [and](X1,X2) = X1 + X2 [cons](X1,X2) = X1 + 2.X2 + 2 [isNat](X) = X [isNatIList](X) = 2.X + 2 [isNatIListKind](X) = 0 [isNatKind](X) = 0 [isNatList](X) = X + 1 [length](X) = X + 1 [nil] = 1 [s](X) = X + 1 [tt] = 2 [zeros] = 0 [A__AND](X1,X2) = 2.X2 + 2 [A__ISNATILISTKIND](X) = 2 [MARK](X) = 2.X + 2 Problem 1.2: SCC Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) ->->-> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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 1.2: Reduction Pair Processor: -> Pairs: A__AND(tt,X) -> MARK(X) A__ISNATILISTKIND(cons(V1,V2)) -> A__AND(a__isNatKind(V1),isNatIListKind(V2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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) = 2 [a__U12](X) = X [a__U21](X1,X2) = 2 [a__U22](X) = 2 [a__U31](X1,X2) = X1 [a__U32](X) = 2 [a__U41](X1,X2,X3) = X1 + X2 + 2.X3 + 2 [a__U42](X1,X2) = X1 + 2.X2 + 2 [a__U43](X) = X + 1 [a__U51](X1,X2,X3) = 2 [a__U52](X1,X2) = X1 [a__U53](X) = X [a__U61](X1,X2) = X1 + X2 + 2 [a__and](X1,X2) = X1 + X2 [a__isNat](X) = 2 [a__isNatIList](X) = 2.X + 2 [a__isNatIListKind](X) = X + 2 [a__isNatKind](X) = X + 2 [a__isNatList](X) = 2 [a__length](X) = X + 2 [a__zeros] = 2 [mark](X) = X + 2 [0] = 0 [U11](X1,X2) = 1 [U12](X) = X [U21](X1,X2) = 0 [U22](X) = 2 [U31](X1,X2) = X1 [U32](X) = 2 [U41](X1,X2,X3) = X1 + X2 + 2.X3 + 2 [U42](X1,X2) = X1 + 2.X2 + 2 [U43](X) = X + 1 [U51](X1,X2,X3) = 1 [U52](X1,X2) = X1 [U53](X) = X [U61](X1,X2) = X1 + X2 + 2 [and](X1,X2) = X1 + X2 [cons](X1,X2) = X1 + 2.X2 + 2 [isNat](X) = 0 [isNatIList](X) = 2.X + 1 [isNatIListKind](X) = X [isNatKind](X) = X [isNatList](X) = 1 [length](X) = X + 2 [nil] = 2 [s](X) = X [tt] = 2 [zeros] = 0 [A__AND](X1,X2) = 2.X2 [A__ISNATILISTKIND](X) = 2.X [MARK](X) = 2.X Problem 1.2: SCC Processor: -> Pairs: A__AND(tt,X) -> MARK(X) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__AND(tt,X) -> MARK(X) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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 1.2: Subterm Processor: -> Pairs: A__AND(tt,X) -> MARK(X) MARK(U52(X1,X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> A__AND(mark(X1),X2) MARK(and(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__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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__AND) = 2 pi(MARK) = 1 Problem 1.2: SCC Processor: -> Pairs: A__AND(tt,X) -> MARK(X) -> Rules: a__U11(tt,V1) -> a__U12(a__isNatList(V1)) a__U11(X1,X2) -> U11(X1,X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt,V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U31(tt,V) -> a__U32(a__isNatList(V)) a__U31(X1,X2) -> U31(X1,X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(tt,V1,V2) -> a__U42(a__isNat(V1),V2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(tt,V2) -> a__U43(a__isNatIList(V2)) a__U42(X1,X2) -> U42(X1,X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__U51(tt,V1,V2) -> a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(tt,V2) -> a__U53(a__isNatList(V2)) a__U52(X1,X2) -> U52(X1,X2) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(tt,L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__and(tt,X) -> mark(X) a__and(X1,X2) -> and(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__and(a__isNatKind(V1),isNatIListKind(V2)),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__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) 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(X)) -> a__U12(mark(X)) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(and(X1,X2)) -> a__and(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.