YES Problem: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) -> tt() a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(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(isNatIList(X)) -> a__isNatIList(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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2) -> U61(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Proof: Matrix Interpretation Processor: dim=1 interpretation: [U61](x0, x1) = x0 + 6x1, [U53](x0) = x0, [U52](x0, x1) = x0 + x1, [U51](x0, x1, x2) = x0 + x1 + x2, [isNatIList](x0) = 3x0, [U43](x0) = 2x0, [U42](x0, x1) = x0 + 6x1, [U41](x0, x1, x2) = x0 + x1 + 6x2, [U32](x0) = x0, [U31](x0, x1) = x0 + x1, [U22](x0) = x0, [U21](x0, x1) = x0 + x1, [isNatList](x0) = x0, [U12](x0) = x0, [U11](x0, x1) = x0 + x1, [and](x0, x1) = 2x0 + 2x1, [isNatKind](x0) = x0, [isNat](x0) = x0, [nil] = 2, [isNatIListKind](x0) = 2x0, [a__isNatKind](x0) = x0, [a__isNatIListKind](x0) = 2x0, [length](x0) = 3x0, [a__and](x0, x1) = 2x0 + 2x1, [s](x0) = 2x0, [a__length](x0) = 3x0, [mark](x0) = x0, [a__U61](x0, x1) = x0 + 6x1, [a__U53](x0) = x0, [a__U52](x0, x1) = x0 + x1, [a__U51](x0, x1, x2) = x0 + x1 + x2, [a__U43](x0) = 2x0, [a__isNatIList](x0) = 3x0, [a__U42](x0, x1) = x0 + 6x1, [a__U41](x0, x1, x2) = x0 + x1 + 6x2, [a__U32](x0) = x0, [a__U31](x0, x1) = x0 + x1, [a__U22](x0) = x0, [a__isNat](x0) = x0, [a__U21](x0, x1) = x0 + x1, [a__U12](x0) = x0, [a__isNatList](x0) = x0, [a__U11](x0, x1) = x0 + x1, [tt] = 0, [cons](x0, x1) = 3x0 + 7x1, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),V1) = V1 >= V1 = a__U12(a__isNatList(V1)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = V1 >= V1 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U31(tt(),V) = V >= V = a__U32(a__isNatList(V)) a__U32(tt()) = 0 >= 0 = tt() a__U41(tt(),V1,V2) = V1 + 6V2 >= V1 + 6V2 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = 6V2 >= 6V2 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = V1 + V2 >= V1 + V2 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = V2 >= V2 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),L) = 6L >= 6L = s(a__length(mark(L))) a__and(tt(),X) = 2X >= X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 3V1 >= 3V1 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 2V1 >= 2V1 = a__U21(a__isNatKind(V1),V1) a__isNatIList(V) = 3V >= 3V = a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(V1,V2)) = 9V1 + 21V2 >= 3V1 + 10V2 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) = 4 >= 0 = tt() a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 6V1 + 14V2 >= 2V1 + 4V2 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = 3V1 >= 2V1 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 2V1 >= V1 = a__isNatKind(V1) a__isNatList(nil()) = 2 >= 0 = tt() a__isNatList(cons(V1,V2)) = 3V1 + 7V2 >= 3V1 + 5V2 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) = 6 >= 0 = 0() a__length(cons(N,L)) = 21L + 9N >= 18L + 8N = a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = X1 + X2 >= X1 + X2 = a__U11(mark(X1),X2) mark(U12(X)) = X >= X = a__U12(mark(X)) mark(isNatList(X)) = X >= X = a__isNatList(X) mark(U21(X1,X2)) = X1 + X2 >= X1 + X2 = a__U21(mark(X1),X2) mark(U22(X)) = X >= X = a__U22(mark(X)) mark(isNat(X)) = X >= X = a__isNat(X) mark(U31(X1,X2)) = X1 + X2 >= X1 + X2 = a__U31(mark(X1),X2) mark(U32(X)) = X >= X = a__U32(mark(X)) mark(U41(X1,X2,X3)) = X1 + X2 + 6X3 >= X1 + X2 + 6X3 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = X1 + 6X2 >= X1 + 6X2 = a__U42(mark(X1),X2) mark(U43(X)) = 2X >= 2X = a__U43(mark(X)) mark(isNatIList(X)) = 3X >= 3X = a__isNatIList(X) mark(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = X1 + X2 >= X1 + X2 = a__U52(mark(X1),X2) mark(U53(X)) = X >= X = a__U53(mark(X)) mark(U61(X1,X2)) = X1 + 6X2 >= X1 + 6X2 = a__U61(mark(X1),X2) mark(length(X)) = 3X >= 3X = a__length(mark(X)) mark(and(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 2X >= 2X = a__isNatIListKind(X) mark(isNatKind(X)) = X >= X = a__isNatKind(X) mark(cons(X1,X2)) = 3X1 + 7X2 >= 3X1 + 7X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 2X >= 2X = s(mark(X)) mark(nil()) = 2 >= 2 = nil() a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = X1 + X2 >= X1 + X2 = U11(X1,X2) a__U12(X) = X >= X = U12(X) a__isNatList(X) = X >= X = isNatList(X) a__U21(X1,X2) = X1 + X2 >= X1 + X2 = U21(X1,X2) a__U22(X) = X >= X = U22(X) a__isNat(X) = X >= X = isNat(X) a__U31(X1,X2) = X1 + X2 >= X1 + X2 = U31(X1,X2) a__U32(X) = X >= X = U32(X) a__U41(X1,X2,X3) = X1 + X2 + 6X3 >= X1 + X2 + 6X3 = U41(X1,X2,X3) a__U42(X1,X2) = X1 + 6X2 >= X1 + 6X2 = U42(X1,X2) a__U43(X) = 2X >= 2X = U43(X) a__isNatIList(X) = 3X >= 3X = isNatIList(X) a__U51(X1,X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) a__U52(X1,X2) = X1 + X2 >= X1 + X2 = U52(X1,X2) a__U53(X) = X >= X = U53(X) a__U61(X1,X2) = X1 + 6X2 >= X1 + 6X2 = U61(X1,X2) a__length(X) = 3X >= 3X = length(X) a__and(X1,X2) = 2X1 + 2X2 >= 2X1 + 2X2 = and(X1,X2) a__isNatIListKind(X) = 2X >= 2X = isNatIListKind(X) a__isNatKind(X) = X >= X = isNatKind(X) problem: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() 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__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(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(isNatIList(X)) -> a__isNatIList(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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2) -> U61(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dim=1 interpretation: [U61](x0, x1) = x0 + 4x1, [U53](x0) = x0, [U52](x0, x1) = x0 + x1, [U51](x0, x1, x2) = x0 + x1 + x2, [isNatIList](x0) = 2x0, [U43](x0) = x0, [U42](x0, x1) = x0 + 3x1, [U41](x0, x1, x2) = x0 + x1 + 3x2, [U32](x0) = x0, [U31](x0, x1) = x0 + x1, [U22](x0) = x0, [U21](x0, x1) = x0 + 3x1, [isNatList](x0) = x0, [U12](x0) = x0, [U11](x0, x1) = x0 + x1, [and](x0, x1) = x0 + x1, [isNatKind](x0) = x0, [isNat](x0) = x0, [nil] = 6, [isNatIListKind](x0) = x0, [a__isNatKind](x0) = x0, [a__isNatIListKind](x0) = x0, [length](x0) = x0, [a__and](x0, x1) = x0 + 2x1, [s](x0) = 2x0, [a__length](x0) = x0, [mark](x0) = 2x0, [a__U61](x0, x1) = x0 + 4x1, [a__U53](x0) = x0, [a__U52](x0, x1) = x0 + x1, [a__U51](x0, x1, x2) = x0 + 2x1 + x2, [a__U43](x0) = x0, [a__isNatIList](x0) = 2x0, [a__U42](x0, x1) = x0 + 3x1, [a__U41](x0, x1, x2) = x0 + 2x1 + 3x2, [a__U32](x0) = x0, [a__U31](x0, x1) = x0 + x1, [a__U22](x0) = x0, [a__isNat](x0) = 2x0, [a__U21](x0, x1) = x0 + 3x1, [a__U12](x0) = x0, [a__isNatList](x0) = x0, [a__U11](x0, x1) = x0 + x1, [tt] = 0, [cons](x0, x1) = 4x0 + 7x1, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),V1) = V1 >= V1 = a__U12(a__isNatList(V1)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = 3V1 >= 2V1 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U31(tt(),V) = V >= V = a__U32(a__isNatList(V)) a__U32(tt()) = 0 >= 0 = tt() a__U41(tt(),V1,V2) = 2V1 + 3V2 >= 2V1 + 3V2 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = 3V2 >= 2V2 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = 2V1 + V2 >= 2V1 + V2 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = V2 >= V2 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),L) = 4L >= 4L = s(a__length(mark(L))) a__and(tt(),X) = 2X >= 2X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 2V1 >= 2V1 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 4V1 >= 4V1 = a__U21(a__isNatKind(V1),V1) a__isNatIList(V) = 2V >= 2V = a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(V1,V2)) = 8V1 + 14V2 >= 3V1 + 5V2 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 4V1 + 7V2 >= V1 + 2V2 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = V1 >= V1 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 2V1 >= V1 = a__isNatKind(V1) a__isNatList(cons(V1,V2)) = 4V1 + 7V2 >= 3V1 + 3V2 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(cons(N,L)) = 7L + 4N >= 7L + 4N = a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = 2X1 + 2X2 >= 2X1 + X2 = a__U11(mark(X1),X2) mark(U12(X)) = 2X >= 2X = a__U12(mark(X)) mark(isNatList(X)) = 2X >= X = a__isNatList(X) mark(U21(X1,X2)) = 2X1 + 6X2 >= 2X1 + 3X2 = a__U21(mark(X1),X2) mark(U22(X)) = 2X >= 2X = a__U22(mark(X)) mark(isNat(X)) = 2X >= 2X = a__isNat(X) mark(U31(X1,X2)) = 2X1 + 2X2 >= 2X1 + X2 = a__U31(mark(X1),X2) mark(U32(X)) = 2X >= 2X = a__U32(mark(X)) mark(U41(X1,X2,X3)) = 2X1 + 2X2 + 6X3 >= 2X1 + 2X2 + 3X3 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = 2X1 + 6X2 >= 2X1 + 3X2 = a__U42(mark(X1),X2) mark(U43(X)) = 2X >= 2X = a__U43(mark(X)) mark(isNatIList(X)) = 4X >= 2X = a__isNatIList(X) mark(U51(X1,X2,X3)) = 2X1 + 2X2 + 2X3 >= 2X1 + 2X2 + X3 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = 2X1 + 2X2 >= 2X1 + X2 = a__U52(mark(X1),X2) mark(U53(X)) = 2X >= 2X = a__U53(mark(X)) mark(U61(X1,X2)) = 2X1 + 8X2 >= 2X1 + 4X2 = a__U61(mark(X1),X2) mark(length(X)) = 2X >= 2X = a__length(mark(X)) mark(and(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 2X >= X = a__isNatIListKind(X) mark(isNatKind(X)) = 2X >= X = a__isNatKind(X) mark(cons(X1,X2)) = 8X1 + 14X2 >= 8X1 + 7X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 4X >= 4X = s(mark(X)) mark(nil()) = 12 >= 6 = nil() a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = X1 + X2 >= X1 + X2 = U11(X1,X2) a__U12(X) = X >= X = U12(X) a__isNatList(X) = X >= X = isNatList(X) a__U21(X1,X2) = X1 + 3X2 >= X1 + 3X2 = U21(X1,X2) a__U22(X) = X >= X = U22(X) a__isNat(X) = 2X >= X = isNat(X) a__U31(X1,X2) = X1 + X2 >= X1 + X2 = U31(X1,X2) a__U32(X) = X >= X = U32(X) a__U41(X1,X2,X3) = X1 + 2X2 + 3X3 >= X1 + X2 + 3X3 = U41(X1,X2,X3) a__U42(X1,X2) = X1 + 3X2 >= X1 + 3X2 = U42(X1,X2) a__U43(X) = X >= X = U43(X) a__isNatIList(X) = 2X >= 2X = isNatIList(X) a__U51(X1,X2,X3) = X1 + 2X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) a__U52(X1,X2) = X1 + X2 >= X1 + X2 = U52(X1,X2) a__U53(X) = X >= X = U53(X) a__U61(X1,X2) = X1 + 4X2 >= X1 + 4X2 = U61(X1,X2) a__length(X) = X >= X = length(X) a__and(X1,X2) = X1 + 2X2 >= X1 + X2 = and(X1,X2) a__isNatIListKind(X) = X >= X = isNatIListKind(X) a__isNatKind(X) = X >= X = isNatKind(X) problem: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() 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__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(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(isNatIList(X)) -> a__isNatIList(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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2) -> U61(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dim=1 interpretation: [U61](x0, x1) = x0 + 6x1, [U53](x0) = x0, [U52](x0, x1) = x0 + x1, [U51](x0, x1, x2) = x0 + x1 + x2, [isNatIList](x0) = 2x0 + 1, [U43](x0) = x0, [U42](x0, x1) = 2x0 + 2x1 + 1, [U41](x0, x1, x2) = 2x0 + 2x1 + 2x2 + 1, [U32](x0) = x0, [U31](x0, x1) = x0 + x1 + 1, [U22](x0) = x0, [U21](x0, x1) = x0 + x1, [isNatList](x0) = x0, [U12](x0) = x0, [U11](x0, x1) = x0 + x1, [and](x0, x1) = 2x0 + 2x1, [isNatKind](x0) = 2x0, [isNat](x0) = x0, [isNatIListKind](x0) = x0, [a__isNatKind](x0) = 2x0, [a__isNatIListKind](x0) = x0, [length](x0) = 2x0, [a__and](x0, x1) = 2x0 + 2x1, [s](x0) = 3x0, [a__length](x0) = 2x0, [mark](x0) = x0, [a__U61](x0, x1) = x0 + 6x1, [a__U53](x0) = x0, [a__U52](x0, x1) = x0 + x1, [a__U51](x0, x1, x2) = x0 + x1 + x2, [a__U43](x0) = x0, [a__isNatIList](x0) = 2x0 + 1, [a__U42](x0, x1) = 2x0 + 2x1 + 1, [a__U41](x0, x1, x2) = 2x0 + 2x1 + 2x2 + 1, [a__U32](x0) = x0, [a__U31](x0, x1) = x0 + x1 + 1, [a__U22](x0) = x0, [a__isNat](x0) = x0, [a__U21](x0, x1) = x0 + x1, [a__U12](x0) = x0, [a__isNatList](x0) = x0, [a__U11](x0, x1) = x0 + x1, [tt] = 0, [cons](x0, x1) = 6x0 + 7x1, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),V1) = V1 >= V1 = a__U12(a__isNatList(V1)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = V1 >= V1 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U31(tt(),V) = V + 1 >= V = a__U32(a__isNatList(V)) a__U32(tt()) = 0 >= 0 = tt() a__U41(tt(),V1,V2) = 2V1 + 2V2 + 1 >= 2V1 + 2V2 + 1 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = 2V2 + 1 >= 2V2 + 1 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = V1 + V2 >= V1 + V2 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = V2 >= V2 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),L) = 6L >= 6L = s(a__length(mark(L))) a__and(tt(),X) = 2X >= X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 2V1 >= 2V1 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 3V1 >= 3V1 = a__U21(a__isNatKind(V1),V1) a__isNatIList(V) = 2V + 1 >= 2V + 1 = a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) = 1 >= 0 = tt() a__isNatIList(cons(V1,V2)) = 12V1 + 14V2 + 1 >= 10V1 + 6V2 + 1 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 6V1 + 7V2 >= 4V1 + 2V2 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = 4V1 >= V1 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 6V1 >= 2V1 = a__isNatKind(V1) a__isNatList(cons(V1,V2)) = 6V1 + 7V2 >= 5V1 + 3V2 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(cons(N,L)) = 14L + 12N >= 14L + 12N = a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = X1 + X2 >= X1 + X2 = a__U11(mark(X1),X2) mark(U12(X)) = X >= X = a__U12(mark(X)) mark(isNatList(X)) = X >= X = a__isNatList(X) mark(U21(X1,X2)) = X1 + X2 >= X1 + X2 = a__U21(mark(X1),X2) mark(U22(X)) = X >= X = a__U22(mark(X)) mark(isNat(X)) = X >= X = a__isNat(X) mark(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__U31(mark(X1),X2) mark(U32(X)) = X >= X = a__U32(mark(X)) mark(U41(X1,X2,X3)) = 2X1 + 2X2 + 2X3 + 1 >= 2X1 + 2X2 + 2X3 + 1 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = 2X1 + 2X2 + 1 >= 2X1 + 2X2 + 1 = a__U42(mark(X1),X2) mark(U43(X)) = X >= X = a__U43(mark(X)) mark(isNatIList(X)) = 2X + 1 >= 2X + 1 = a__isNatIList(X) mark(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = X1 + X2 >= X1 + X2 = a__U52(mark(X1),X2) mark(U53(X)) = X >= X = a__U53(mark(X)) mark(U61(X1,X2)) = X1 + 6X2 >= X1 + 6X2 = a__U61(mark(X1),X2) mark(length(X)) = 2X >= 2X = a__length(mark(X)) mark(and(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = X >= X = a__isNatIListKind(X) mark(isNatKind(X)) = 2X >= 2X = a__isNatKind(X) mark(cons(X1,X2)) = 6X1 + 7X2 >= 6X1 + 7X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 3X >= 3X = s(mark(X)) a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = X1 + X2 >= X1 + X2 = U11(X1,X2) a__U12(X) = X >= X = U12(X) a__isNatList(X) = X >= X = isNatList(X) a__U21(X1,X2) = X1 + X2 >= X1 + X2 = U21(X1,X2) a__U22(X) = X >= X = U22(X) a__isNat(X) = X >= X = isNat(X) a__U31(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(X1,X2) a__U32(X) = X >= X = U32(X) a__U41(X1,X2,X3) = 2X1 + 2X2 + 2X3 + 1 >= 2X1 + 2X2 + 2X3 + 1 = U41(X1,X2,X3) a__U42(X1,X2) = 2X1 + 2X2 + 1 >= 2X1 + 2X2 + 1 = U42(X1,X2) a__U43(X) = X >= X = U43(X) a__isNatIList(X) = 2X + 1 >= 2X + 1 = isNatIList(X) a__U51(X1,X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) a__U52(X1,X2) = X1 + X2 >= X1 + X2 = U52(X1,X2) a__U53(X) = X >= X = U53(X) a__U61(X1,X2) = X1 + 6X2 >= X1 + 6X2 = U61(X1,X2) a__length(X) = 2X >= 2X = length(X) a__and(X1,X2) = 2X1 + 2X2 >= 2X1 + 2X2 = and(X1,X2) a__isNatIListKind(X) = X >= X = isNatIListKind(X) a__isNatKind(X) = 2X >= 2X = isNatKind(X) problem: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() 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__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(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(isNatIList(X)) -> a__isNatIList(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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2) -> U61(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dim=1 interpretation: [U61](x0, x1) = x0 + 6x1, [U53](x0) = x0, [U52](x0, x1) = x0 + x1, [U51](x0, x1, x2) = x0 + x1 + 2x2, [isNatIList](x0) = 4x0, [U43](x0) = x0, [U42](x0, x1) = x0 + 5x1, [U41](x0, x1, x2) = x0 + x1 + 5x2, [U32](x0) = x0 + 4, [U31](x0, x1) = x0 + x1, [U22](x0) = x0, [U21](x0, x1) = x0 + x1, [isNatList](x0) = x0, [U12](x0) = x0, [U11](x0, x1) = x0 + x1, [and](x0, x1) = 2x0 + 2x1, [isNatKind](x0) = x0, [isNat](x0) = x0, [isNatIListKind](x0) = 2x0, [a__isNatKind](x0) = x0, [a__isNatIListKind](x0) = 2x0, [length](x0) = 3x0, [a__and](x0, x1) = 2x0 + 2x1, [s](x0) = 2x0, [a__length](x0) = 3x0, [mark](x0) = x0, [a__U61](x0, x1) = x0 + 6x1, [a__U53](x0) = x0, [a__U52](x0, x1) = x0 + x1, [a__U51](x0, x1, x2) = x0 + x1 + 2x2, [a__U43](x0) = x0, [a__isNatIList](x0) = 4x0, [a__U42](x0, x1) = x0 + 5x1, [a__U41](x0, x1, x2) = x0 + x1 + 5x2, [a__U32](x0) = x0 + 4, [a__U31](x0, x1) = x0 + x1, [a__U22](x0) = x0, [a__isNat](x0) = x0, [a__U21](x0, x1) = x0 + x1, [a__U12](x0) = x0, [a__isNatList](x0) = x0, [a__U11](x0, x1) = x0 + x1, [tt] = 0, [cons](x0, x1) = 3x0 + 6x1, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),V1) = V1 >= V1 = a__U12(a__isNatList(V1)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = V1 >= V1 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U32(tt()) = 4 >= 0 = tt() a__U41(tt(),V1,V2) = V1 + 5V2 >= V1 + 5V2 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = 5V2 >= 4V2 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = V1 + 2V2 >= V1 + V2 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = V2 >= V2 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),L) = 6L >= 6L = s(a__length(mark(L))) a__and(tt(),X) = 2X >= X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 3V1 >= 3V1 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 2V1 >= 2V1 = a__U21(a__isNatKind(V1),V1) a__isNatIList(V) = 4V >= 3V = a__U31(a__isNatIListKind(V),V) a__isNatIList(cons(V1,V2)) = 12V1 + 24V2 >= 3V1 + 9V2 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 6V1 + 12V2 >= 2V1 + 4V2 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = 3V1 >= 2V1 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 2V1 >= V1 = a__isNatKind(V1) a__isNatList(cons(V1,V2)) = 3V1 + 6V2 >= 3V1 + 6V2 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(cons(N,L)) = 18L + 9N >= 18L + 8N = a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = X1 + X2 >= X1 + X2 = a__U11(mark(X1),X2) mark(U12(X)) = X >= X = a__U12(mark(X)) mark(isNatList(X)) = X >= X = a__isNatList(X) mark(U21(X1,X2)) = X1 + X2 >= X1 + X2 = a__U21(mark(X1),X2) mark(U22(X)) = X >= X = a__U22(mark(X)) mark(isNat(X)) = X >= X = a__isNat(X) mark(U31(X1,X2)) = X1 + X2 >= X1 + X2 = a__U31(mark(X1),X2) mark(U32(X)) = X + 4 >= X + 4 = a__U32(mark(X)) mark(U41(X1,X2,X3)) = X1 + X2 + 5X3 >= X1 + X2 + 5X3 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = X1 + 5X2 >= X1 + 5X2 = a__U42(mark(X1),X2) mark(U43(X)) = X >= X = a__U43(mark(X)) mark(isNatIList(X)) = 4X >= 4X = a__isNatIList(X) mark(U51(X1,X2,X3)) = X1 + X2 + 2X3 >= X1 + X2 + 2X3 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = X1 + X2 >= X1 + X2 = a__U52(mark(X1),X2) mark(U53(X)) = X >= X = a__U53(mark(X)) mark(U61(X1,X2)) = X1 + 6X2 >= X1 + 6X2 = a__U61(mark(X1),X2) mark(length(X)) = 3X >= 3X = a__length(mark(X)) mark(and(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 2X >= 2X = a__isNatIListKind(X) mark(isNatKind(X)) = X >= X = a__isNatKind(X) mark(cons(X1,X2)) = 3X1 + 6X2 >= 3X1 + 6X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 2X >= 2X = s(mark(X)) a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = X1 + X2 >= X1 + X2 = U11(X1,X2) a__U12(X) = X >= X = U12(X) a__isNatList(X) = X >= X = isNatList(X) a__U21(X1,X2) = X1 + X2 >= X1 + X2 = U21(X1,X2) a__U22(X) = X >= X = U22(X) a__isNat(X) = X >= X = isNat(X) a__U31(X1,X2) = X1 + X2 >= X1 + X2 = U31(X1,X2) a__U32(X) = X + 4 >= X + 4 = U32(X) a__U41(X1,X2,X3) = X1 + X2 + 5X3 >= X1 + X2 + 5X3 = U41(X1,X2,X3) a__U42(X1,X2) = X1 + 5X2 >= X1 + 5X2 = U42(X1,X2) a__U43(X) = X >= X = U43(X) a__isNatIList(X) = 4X >= 4X = isNatIList(X) a__U51(X1,X2,X3) = X1 + X2 + 2X3 >= X1 + X2 + 2X3 = U51(X1,X2,X3) a__U52(X1,X2) = X1 + X2 >= X1 + X2 = U52(X1,X2) a__U53(X) = X >= X = U53(X) a__U61(X1,X2) = X1 + 6X2 >= X1 + 6X2 = U61(X1,X2) a__length(X) = 3X >= 3X = length(X) a__and(X1,X2) = 2X1 + 2X2 >= 2X1 + 2X2 = and(X1,X2) a__isNatIListKind(X) = 2X >= 2X = isNatIListKind(X) a__isNatKind(X) = X >= X = isNatKind(X) problem: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() 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__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(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(isNatIList(X)) -> a__isNatIList(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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2) -> U61(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dim=1 interpretation: [U61](x0, x1) = x0 + 7x1, [U53](x0) = x0, [U52](x0, x1) = x0 + x1, [U51](x0, x1, x2) = x0 + 4x1 + x2, [isNatIList](x0) = 2x0 + 1, [U43](x0) = x0, [U42](x0, x1) = x0 + 4x1 + 1, [U41](x0, x1, x2) = 3x0 + 2x1 + 4x2 + 1, [U32](x0) = x0 + 1, [U31](x0, x1) = x0 + x1, [U22](x0) = x0, [U21](x0, x1) = 2x0 + 2x1, [isNatList](x0) = x0, [U12](x0) = x0, [U11](x0, x1) = x0 + 4x1, [and](x0, x1) = 2x0 + 2x1, [isNatKind](x0) = x0, [isNat](x0) = 2x0, [isNatIListKind](x0) = x0, [a__isNatKind](x0) = x0, [a__isNatIListKind](x0) = x0, [length](x0) = 3x0, [a__and](x0, x1) = 2x0 + 2x1, [s](x0) = 2x0, [a__length](x0) = 3x0, [mark](x0) = x0, [a__U61](x0, x1) = x0 + 7x1, [a__U53](x0) = x0, [a__U52](x0, x1) = x0 + x1, [a__U51](x0, x1, x2) = x0 + 4x1 + x2, [a__U43](x0) = x0, [a__isNatIList](x0) = 2x0 + 1, [a__U42](x0, x1) = x0 + 4x1 + 1, [a__U41](x0, x1, x2) = 3x0 + 2x1 + 4x2 + 1, [a__U32](x0) = x0 + 1, [a__U31](x0, x1) = x0 + x1, [a__U22](x0) = x0, [a__isNat](x0) = 2x0, [a__U21](x0, x1) = 2x0 + 2x1, [a__U12](x0) = x0, [a__isNatList](x0) = x0, [a__U11](x0, x1) = x0 + 4x1, [tt] = 0, [cons](x0, x1) = 6x0 + 5x1, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),V1) = 4V1 >= V1 = a__U12(a__isNatList(V1)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = 2V1 >= 2V1 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U41(tt(),V1,V2) = 2V1 + 4V2 + 1 >= 2V1 + 4V2 + 1 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = 4V2 + 1 >= 2V2 + 1 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = 4V1 + V2 >= 2V1 + V2 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = V2 >= V2 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),L) = 7L >= 6L = s(a__length(mark(L))) a__and(tt(),X) = 2X >= X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 6V1 >= 5V1 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 4V1 >= 4V1 = a__U21(a__isNatKind(V1),V1) a__isNatIList(V) = 2V + 1 >= 2V = a__U31(a__isNatIListKind(V),V) a__isNatIList(cons(V1,V2)) = 12V1 + 10V2 + 1 >= 8V1 + 10V2 + 1 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 6V1 + 5V2 >= 2V1 + 2V2 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = 3V1 >= V1 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 2V1 >= V1 = a__isNatKind(V1) a__isNatList(cons(V1,V2)) = 6V1 + 5V2 >= 6V1 + 3V2 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(cons(N,L)) = 15L + 18N >= 15L + 12N = a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = X1 + 4X2 >= X1 + 4X2 = a__U11(mark(X1),X2) mark(U12(X)) = X >= X = a__U12(mark(X)) mark(isNatList(X)) = X >= X = a__isNatList(X) mark(U21(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = a__U21(mark(X1),X2) mark(U22(X)) = X >= X = a__U22(mark(X)) mark(isNat(X)) = 2X >= 2X = a__isNat(X) mark(U31(X1,X2)) = X1 + X2 >= X1 + X2 = a__U31(mark(X1),X2) mark(U32(X)) = X + 1 >= X + 1 = a__U32(mark(X)) mark(U41(X1,X2,X3)) = 3X1 + 2X2 + 4X3 + 1 >= 3X1 + 2X2 + 4X3 + 1 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = X1 + 4X2 + 1 >= X1 + 4X2 + 1 = a__U42(mark(X1),X2) mark(U43(X)) = X >= X = a__U43(mark(X)) mark(isNatIList(X)) = 2X + 1 >= 2X + 1 = a__isNatIList(X) mark(U51(X1,X2,X3)) = X1 + 4X2 + X3 >= X1 + 4X2 + X3 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = X1 + X2 >= X1 + X2 = a__U52(mark(X1),X2) mark(U53(X)) = X >= X = a__U53(mark(X)) mark(U61(X1,X2)) = X1 + 7X2 >= X1 + 7X2 = a__U61(mark(X1),X2) mark(length(X)) = 3X >= 3X = a__length(mark(X)) mark(and(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = X >= X = a__isNatIListKind(X) mark(isNatKind(X)) = X >= X = a__isNatKind(X) mark(cons(X1,X2)) = 6X1 + 5X2 >= 6X1 + 5X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 2X >= 2X = s(mark(X)) a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = X1 + 4X2 >= X1 + 4X2 = U11(X1,X2) a__U12(X) = X >= X = U12(X) a__isNatList(X) = X >= X = isNatList(X) a__U21(X1,X2) = 2X1 + 2X2 >= 2X1 + 2X2 = U21(X1,X2) a__U22(X) = X >= X = U22(X) a__isNat(X) = 2X >= 2X = isNat(X) a__U31(X1,X2) = X1 + X2 >= X1 + X2 = U31(X1,X2) a__U32(X) = X + 1 >= X + 1 = U32(X) a__U41(X1,X2,X3) = 3X1 + 2X2 + 4X3 + 1 >= 3X1 + 2X2 + 4X3 + 1 = U41(X1,X2,X3) a__U42(X1,X2) = X1 + 4X2 + 1 >= X1 + 4X2 + 1 = U42(X1,X2) a__U43(X) = X >= X = U43(X) a__isNatIList(X) = 2X + 1 >= 2X + 1 = isNatIList(X) a__U51(X1,X2,X3) = X1 + 4X2 + X3 >= X1 + 4X2 + X3 = U51(X1,X2,X3) a__U52(X1,X2) = X1 + X2 >= X1 + X2 = U52(X1,X2) a__U53(X) = X >= X = U53(X) a__U61(X1,X2) = X1 + 7X2 >= X1 + 7X2 = U61(X1,X2) a__length(X) = 3X >= 3X = length(X) a__and(X1,X2) = 2X1 + 2X2 >= 2X1 + 2X2 = and(X1,X2) a__isNatIListKind(X) = X >= X = isNatIListKind(X) a__isNatKind(X) = X >= X = isNatKind(X) problem: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() 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__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(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(isNatIList(X)) -> a__isNatIList(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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2) -> U61(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dim=1 interpretation: [U61](x0, x1) = x0 + 4x1, [U53](x0) = x0, [U52](x0, x1) = x0 + x1, [U51](x0, x1, x2) = x0 + 4x1 + x2, [isNatIList](x0) = x0, [U43](x0) = x0, [U42](x0, x1) = x0 + x1, [U41](x0, x1, x2) = 2x0 + 3x1 + x2, [U32](x0) = 2x0 + 4, [U31](x0, x1) = x0 + 4x1, [U22](x0) = x0, [U21](x0, x1) = 2x0 + 3x1, [isNatList](x0) = x0, [U12](x0) = x0, [U11](x0, x1) = 2x0 + x1, [and](x0, x1) = x0 + x1, [isNatKind](x0) = x0, [isNat](x0) = 2x0, [isNatIListKind](x0) = x0, [a__isNatKind](x0) = x0, [a__isNatIListKind](x0) = x0, [length](x0) = x0, [a__and](x0, x1) = x0 + 2x1, [s](x0) = 2x0, [a__length](x0) = x0, [mark](x0) = 2x0, [a__U61](x0, x1) = x0 + 4x1, [a__U53](x0) = x0, [a__U52](x0, x1) = x0 + x1, [a__U51](x0, x1, x2) = x0 + 5x1 + x2, [a__U43](x0) = x0, [a__isNatIList](x0) = x0, [a__U42](x0, x1) = x0 + x1, [a__U41](x0, x1, x2) = 2x0 + 3x1 + x2, [a__U32](x0) = 2x0 + 7, [a__U31](x0, x1) = x0 + 6x1, [a__U22](x0) = x0, [a__isNat](x0) = 3x0, [a__U21](x0, x1) = 2x0 + 3x1, [a__U12](x0) = x0, [a__isNatList](x0) = x0, [a__U11](x0, x1) = 2x0 + x1, [tt] = 0, [cons](x0, x1) = 6x0 + 7x1, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),V1) = V1 >= V1 = a__U12(a__isNatList(V1)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = 3V1 >= 3V1 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U41(tt(),V1,V2) = 3V1 + V2 >= 3V1 + V2 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = V2 >= V2 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = 5V1 + V2 >= 3V1 + V2 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = V2 >= V2 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),L) = 4L >= 4L = s(a__length(mark(L))) a__and(tt(),X) = 2X >= 2X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 3V1 >= 3V1 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 6V1 >= 5V1 = a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) = 6V1 + 7V2 >= 5V1 + 5V2 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 6V1 + 7V2 >= V1 + 2V2 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = V1 >= V1 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 2V1 >= V1 = a__isNatKind(V1) a__isNatList(cons(V1,V2)) = 6V1 + 7V2 >= 6V1 + 3V2 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(cons(N,L)) = 7L + 6N >= 7L + 6N = a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = 4X1 + 2X2 >= 4X1 + X2 = a__U11(mark(X1),X2) mark(U12(X)) = 2X >= 2X = a__U12(mark(X)) mark(isNatList(X)) = 2X >= X = a__isNatList(X) mark(U21(X1,X2)) = 4X1 + 6X2 >= 4X1 + 3X2 = a__U21(mark(X1),X2) mark(U22(X)) = 2X >= 2X = a__U22(mark(X)) mark(isNat(X)) = 4X >= 3X = a__isNat(X) mark(U31(X1,X2)) = 2X1 + 8X2 >= 2X1 + 6X2 = a__U31(mark(X1),X2) mark(U32(X)) = 4X + 8 >= 4X + 7 = a__U32(mark(X)) mark(U41(X1,X2,X3)) = 4X1 + 6X2 + 2X3 >= 4X1 + 3X2 + X3 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = 2X1 + 2X2 >= 2X1 + X2 = a__U42(mark(X1),X2) mark(U43(X)) = 2X >= 2X = a__U43(mark(X)) mark(isNatIList(X)) = 2X >= X = a__isNatIList(X) mark(U51(X1,X2,X3)) = 2X1 + 8X2 + 2X3 >= 2X1 + 5X2 + X3 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = 2X1 + 2X2 >= 2X1 + X2 = a__U52(mark(X1),X2) mark(U53(X)) = 2X >= 2X = a__U53(mark(X)) mark(U61(X1,X2)) = 2X1 + 8X2 >= 2X1 + 4X2 = a__U61(mark(X1),X2) mark(length(X)) = 2X >= 2X = a__length(mark(X)) mark(and(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 2X >= X = a__isNatIListKind(X) mark(isNatKind(X)) = 2X >= X = a__isNatKind(X) mark(cons(X1,X2)) = 12X1 + 14X2 >= 12X1 + 7X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 4X >= 4X = s(mark(X)) a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = 2X1 + X2 >= 2X1 + X2 = U11(X1,X2) a__U12(X) = X >= X = U12(X) a__isNatList(X) = X >= X = isNatList(X) a__U21(X1,X2) = 2X1 + 3X2 >= 2X1 + 3X2 = U21(X1,X2) a__U22(X) = X >= X = U22(X) a__isNat(X) = 3X >= 2X = isNat(X) a__U31(X1,X2) = X1 + 6X2 >= X1 + 4X2 = U31(X1,X2) a__U32(X) = 2X + 7 >= 2X + 4 = U32(X) a__U41(X1,X2,X3) = 2X1 + 3X2 + X3 >= 2X1 + 3X2 + X3 = U41(X1,X2,X3) a__U42(X1,X2) = X1 + X2 >= X1 + X2 = U42(X1,X2) a__U43(X) = X >= X = U43(X) a__isNatIList(X) = X >= X = isNatIList(X) a__U51(X1,X2,X3) = X1 + 5X2 + X3 >= X1 + 4X2 + X3 = U51(X1,X2,X3) a__U52(X1,X2) = X1 + X2 >= X1 + X2 = U52(X1,X2) a__U53(X) = X >= X = U53(X) a__U61(X1,X2) = X1 + 4X2 >= X1 + 4X2 = U61(X1,X2) a__length(X) = X >= X = length(X) a__and(X1,X2) = X1 + 2X2 >= X1 + X2 = and(X1,X2) a__isNatIListKind(X) = X >= X = isNatIListKind(X) a__isNatKind(X) = X >= X = isNatKind(X) problem: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() 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__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) 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(isNatIList(X)) -> a__isNatIList(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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2) -> U61(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dim=1 interpretation: [U61](x0, x1) = x0 + 4x1, [U53](x0) = x0, [U52](x0, x1) = 2x0 + x1, [U51](x0, x1, x2) = x0 + 2x1 + 2x2, [isNatIList](x0) = x0 + 4, [U43](x0) = x0, [U42](x0, x1) = x0 + 2x1 + 4, [U41](x0, x1, x2) = x0 + 4x1 + 4x2 + 4, [U31](x0, x1) = x0 + 2x1, [U22](x0) = x0, [U21](x0, x1) = x0 + 2x1, [isNatList](x0) = x0, [U12](x0) = x0, [U11](x0, x1) = x0 + x1, [and](x0, x1) = x0 + x1, [isNatKind](x0) = x0, [isNat](x0) = 2x0, [isNatIListKind](x0) = x0, [a__isNatKind](x0) = x0, [a__isNatIListKind](x0) = x0, [length](x0) = x0, [a__and](x0, x1) = x0 + 2x1, [s](x0) = 2x0, [a__length](x0) = x0, [mark](x0) = 2x0, [a__U61](x0, x1) = x0 + 4x1, [a__U53](x0) = x0, [a__U52](x0, x1) = 2x0 + x1, [a__U51](x0, x1, x2) = x0 + 4x1 + 4x2, [a__U43](x0) = x0, [a__isNatIList](x0) = 2x0 + 4, [a__U42](x0, x1) = x0 + 4x1 + 4, [a__U41](x0, x1, x2) = x0 + 7x1 + 4x2 + 4, [a__U31](x0, x1) = x0 + 4x1, [a__U22](x0) = x0, [a__isNat](x0) = 2x0, [a__U21](x0, x1) = x0 + 3x1, [a__U12](x0) = x0, [a__isNatList](x0) = x0, [a__U11](x0, x1) = x0 + x1, [tt] = 0, [cons](x0, x1) = 6x0 + 7x1, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),V1) = V1 >= V1 = a__U12(a__isNatList(V1)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = 3V1 >= 2V1 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U41(tt(),V1,V2) = 7V1 + 4V2 + 4 >= 2V1 + 4V2 + 4 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = 4V2 + 4 >= 2V2 + 4 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = 4V1 + 4V2 >= 4V1 + V2 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = V2 >= V2 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),L) = 4L >= 4L = s(a__length(mark(L))) a__and(tt(),X) = 2X >= 2X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 2V1 >= 2V1 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 4V1 >= 4V1 = a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) = 12V1 + 14V2 + 4 >= 8V1 + 6V2 + 4 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 6V1 + 7V2 >= V1 + 2V2 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = V1 >= V1 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 2V1 >= V1 = a__isNatKind(V1) a__isNatList(cons(V1,V2)) = 6V1 + 7V2 >= 5V1 + 6V2 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(cons(N,L)) = 7L + 6N >= 7L + 6N = a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = 2X1 + 2X2 >= 2X1 + X2 = a__U11(mark(X1),X2) mark(U12(X)) = 2X >= 2X = a__U12(mark(X)) mark(isNatList(X)) = 2X >= X = a__isNatList(X) mark(U21(X1,X2)) = 2X1 + 4X2 >= 2X1 + 3X2 = a__U21(mark(X1),X2) mark(U22(X)) = 2X >= 2X = a__U22(mark(X)) mark(isNat(X)) = 4X >= 2X = a__isNat(X) mark(U31(X1,X2)) = 2X1 + 4X2 >= 2X1 + 4X2 = a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) = 2X1 + 8X2 + 8X3 + 8 >= 2X1 + 7X2 + 4X3 + 4 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = 2X1 + 4X2 + 8 >= 2X1 + 4X2 + 4 = a__U42(mark(X1),X2) mark(U43(X)) = 2X >= 2X = a__U43(mark(X)) mark(isNatIList(X)) = 2X + 8 >= 2X + 4 = a__isNatIList(X) mark(U51(X1,X2,X3)) = 2X1 + 4X2 + 4X3 >= 2X1 + 4X2 + 4X3 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = 4X1 + 2X2 >= 4X1 + X2 = a__U52(mark(X1),X2) mark(U53(X)) = 2X >= 2X = a__U53(mark(X)) mark(U61(X1,X2)) = 2X1 + 8X2 >= 2X1 + 4X2 = a__U61(mark(X1),X2) mark(length(X)) = 2X >= 2X = a__length(mark(X)) mark(and(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 2X >= X = a__isNatIListKind(X) mark(isNatKind(X)) = 2X >= X = a__isNatKind(X) mark(cons(X1,X2)) = 12X1 + 14X2 >= 12X1 + 7X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 4X >= 4X = s(mark(X)) a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = X1 + X2 >= X1 + X2 = U11(X1,X2) a__U12(X) = X >= X = U12(X) a__isNatList(X) = X >= X = isNatList(X) a__U21(X1,X2) = X1 + 3X2 >= X1 + 2X2 = U21(X1,X2) a__U22(X) = X >= X = U22(X) a__isNat(X) = 2X >= 2X = isNat(X) a__U31(X1,X2) = X1 + 4X2 >= X1 + 2X2 = U31(X1,X2) a__U41(X1,X2,X3) = X1 + 7X2 + 4X3 + 4 >= X1 + 4X2 + 4X3 + 4 = U41(X1,X2,X3) a__U42(X1,X2) = X1 + 4X2 + 4 >= X1 + 2X2 + 4 = U42(X1,X2) a__U43(X) = X >= X = U43(X) a__isNatIList(X) = 2X + 4 >= X + 4 = isNatIList(X) a__U51(X1,X2,X3) = X1 + 4X2 + 4X3 >= X1 + 2X2 + 2X3 = U51(X1,X2,X3) a__U52(X1,X2) = 2X1 + X2 >= 2X1 + X2 = U52(X1,X2) a__U53(X) = X >= X = U53(X) a__U61(X1,X2) = X1 + 4X2 >= X1 + 4X2 = U61(X1,X2) a__length(X) = X >= X = length(X) a__and(X1,X2) = X1 + 2X2 >= X1 + X2 = and(X1,X2) a__isNatIListKind(X) = X >= X = isNatIListKind(X) a__isNatKind(X) = X >= X = isNatKind(X) problem: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() 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__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1,X2)) -> a__U31(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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2) -> U61(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dim=1 interpretation: [U61](x0, x1) = x0 + 4x1, [U53](x0) = x0, [U52](x0, x1) = x0 + x1, [U51](x0, x1, x2) = 2x0 + 3x1 + 2x2, [isNatIList](x0) = x0, [U43](x0) = 2x0, [U42](x0, x1) = x0 + x1, [U41](x0, x1, x2) = x0 + 2x1 + 3x2, [U31](x0, x1) = 4x0 + 2x1 + 4, [U22](x0) = x0, [U21](x0, x1) = x0 + 3x1, [isNatList](x0) = x0, [U12](x0) = x0, [U11](x0, x1) = x0 + x1, [and](x0, x1) = x0 + x1, [isNatKind](x0) = x0, [isNat](x0) = 2x0, [isNatIListKind](x0) = x0, [a__isNatKind](x0) = 2x0, [a__isNatIListKind](x0) = 2x0, [length](x0) = x0, [a__and](x0, x1) = x0 + 2x1, [s](x0) = 2x0, [a__length](x0) = x0, [mark](x0) = 2x0, [a__U61](x0, x1) = x0 + 4x1, [a__U53](x0) = x0, [a__U52](x0, x1) = x0 + x1, [a__U51](x0, x1, x2) = 2x0 + 3x1 + 2x2, [a__U43](x0) = 2x0, [a__isNatIList](x0) = x0, [a__U42](x0, x1) = x0 + 2x1, [a__U41](x0, x1, x2) = 2x0 + 3x1 + 3x2, [a__U31](x0, x1) = 4x0 + 3x1 + 6, [a__U22](x0) = x0, [a__isNat](x0) = 3x0, [a__U21](x0, x1) = x0 + 3x1, [a__U12](x0) = x0, [a__isNatList](x0) = x0, [a__U11](x0, x1) = x0 + x1, [tt] = 0, [cons](x0, x1) = 7x0 + 7x1, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),V1) = V1 >= V1 = a__U12(a__isNatList(V1)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = 3V1 >= 3V1 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U41(tt(),V1,V2) = 3V1 + 3V2 >= 3V1 + 2V2 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = 2V2 >= 2V2 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = 3V1 + 2V2 >= 3V1 + V2 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = V2 >= V2 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),L) = 4L >= 4L = s(a__length(mark(L))) a__and(tt(),X) = 2X >= 2X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 3V1 >= 3V1 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 6V1 >= 5V1 = a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) = 7V1 + 7V2 >= 7V1 + 7V2 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 14V1 + 14V2 >= 2V1 + 2V2 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = 2V1 >= 2V1 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 4V1 >= 2V1 = a__isNatKind(V1) a__isNatList(cons(V1,V2)) = 7V1 + 7V2 >= 7V1 + 6V2 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(cons(N,L)) = 7L + 7N >= 7L + 6N = a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = 2X1 + 2X2 >= 2X1 + X2 = a__U11(mark(X1),X2) mark(U12(X)) = 2X >= 2X = a__U12(mark(X)) mark(isNatList(X)) = 2X >= X = a__isNatList(X) mark(U21(X1,X2)) = 2X1 + 6X2 >= 2X1 + 3X2 = a__U21(mark(X1),X2) mark(U22(X)) = 2X >= 2X = a__U22(mark(X)) mark(isNat(X)) = 4X >= 3X = a__isNat(X) mark(U31(X1,X2)) = 8X1 + 4X2 + 8 >= 8X1 + 3X2 + 6 = a__U31(mark(X1),X2) mark(U43(X)) = 4X >= 4X = a__U43(mark(X)) mark(U51(X1,X2,X3)) = 4X1 + 6X2 + 4X3 >= 4X1 + 3X2 + 2X3 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = 2X1 + 2X2 >= 2X1 + X2 = a__U52(mark(X1),X2) mark(U53(X)) = 2X >= 2X = a__U53(mark(X)) mark(U61(X1,X2)) = 2X1 + 8X2 >= 2X1 + 4X2 = a__U61(mark(X1),X2) mark(length(X)) = 2X >= 2X = a__length(mark(X)) mark(and(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 2X >= 2X = a__isNatIListKind(X) mark(isNatKind(X)) = 2X >= 2X = a__isNatKind(X) mark(cons(X1,X2)) = 14X1 + 14X2 >= 14X1 + 7X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 4X >= 4X = s(mark(X)) a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = X1 + X2 >= X1 + X2 = U11(X1,X2) a__U12(X) = X >= X = U12(X) a__isNatList(X) = X >= X = isNatList(X) a__U21(X1,X2) = X1 + 3X2 >= X1 + 3X2 = U21(X1,X2) a__U22(X) = X >= X = U22(X) a__isNat(X) = 3X >= 2X = isNat(X) a__U31(X1,X2) = 4X1 + 3X2 + 6 >= 4X1 + 2X2 + 4 = U31(X1,X2) a__U41(X1,X2,X3) = 2X1 + 3X2 + 3X3 >= X1 + 2X2 + 3X3 = U41(X1,X2,X3) a__U42(X1,X2) = X1 + 2X2 >= X1 + X2 = U42(X1,X2) a__U43(X) = 2X >= 2X = U43(X) a__isNatIList(X) = X >= X = isNatIList(X) a__U51(X1,X2,X3) = 2X1 + 3X2 + 2X3 >= 2X1 + 3X2 + 2X3 = U51(X1,X2,X3) a__U52(X1,X2) = X1 + X2 >= X1 + X2 = U52(X1,X2) a__U53(X) = X >= X = U53(X) a__U61(X1,X2) = X1 + 4X2 >= X1 + 4X2 = U61(X1,X2) a__length(X) = X >= X = length(X) a__and(X1,X2) = X1 + 2X2 >= X1 + X2 = and(X1,X2) a__isNatIListKind(X) = 2X >= X = isNatIListKind(X) a__isNatKind(X) = 2X >= X = isNatKind(X) problem: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() 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__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2) -> U61(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dim=1 interpretation: [U61](x0, x1) = x0 + 6x1, [U53](x0) = x0, [U52](x0, x1) = x0 + x1, [U51](x0, x1, x2) = x0 + 2x1 + x2, [isNatIList](x0) = 2x0 + 2, [U43](x0) = x0, [U42](x0, x1) = x0 + 2x1 + 2, [U41](x0, x1, x2) = x0 + 2x1 + 4x2, [U22](x0) = x0, [U21](x0, x1) = x0 + 3x1, [isNatList](x0) = x0, [U12](x0) = x0, [U11](x0, x1) = 4x0 + x1, [and](x0, x1) = 2x0 + 2x1, [isNatKind](x0) = x0, [isNat](x0) = 2x0, [isNatIListKind](x0) = x0, [a__isNatKind](x0) = x0, [a__isNatIListKind](x0) = x0, [length](x0) = 3x0, [a__and](x0, x1) = 2x0 + 2x1, [s](x0) = 2x0, [a__length](x0) = 3x0, [mark](x0) = x0, [a__U61](x0, x1) = x0 + 6x1, [a__U53](x0) = x0, [a__U52](x0, x1) = x0 + x1, [a__U51](x0, x1, x2) = x0 + 2x1 + x2, [a__U43](x0) = x0, [a__isNatIList](x0) = 4x0 + 2, [a__U42](x0, x1) = x0 + 4x1 + 2, [a__U41](x0, x1, x2) = 2x0 + 2x1 + 6x2 + 2, [a__U22](x0) = x0, [a__isNat](x0) = 2x0, [a__U21](x0, x1) = x0 + 3x1, [a__U12](x0) = x0, [a__isNatList](x0) = x0, [a__U11](x0, x1) = 4x0 + x1, [tt] = 0, [cons](x0, x1) = 4x0 + 5x1, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),V1) = V1 >= V1 = a__U12(a__isNatList(V1)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = 3V1 >= 2V1 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U41(tt(),V1,V2) = 2V1 + 6V2 + 2 >= 2V1 + 4V2 + 2 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = 4V2 + 2 >= 4V2 + 2 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = 2V1 + V2 >= 2V1 + V2 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = V2 >= V2 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),L) = 6L >= 6L = s(a__length(mark(L))) a__and(tt(),X) = 2X >= X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 6V1 >= 5V1 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 4V1 >= 4V1 = a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) = 16V1 + 20V2 + 2 >= 6V1 + 10V2 + 2 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 4V1 + 5V2 >= 2V1 + 2V2 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = 3V1 >= V1 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 2V1 >= V1 = a__isNatKind(V1) a__isNatList(cons(V1,V2)) = 4V1 + 5V2 >= 4V1 + 3V2 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(cons(N,L)) = 15L + 12N >= 14L + 12N = a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = 4X1 + X2 >= 4X1 + X2 = a__U11(mark(X1),X2) mark(U12(X)) = X >= X = a__U12(mark(X)) mark(isNatList(X)) = X >= X = a__isNatList(X) mark(U21(X1,X2)) = X1 + 3X2 >= X1 + 3X2 = a__U21(mark(X1),X2) mark(U22(X)) = X >= X = a__U22(mark(X)) mark(isNat(X)) = 2X >= 2X = a__isNat(X) mark(U43(X)) = X >= X = a__U43(mark(X)) mark(U51(X1,X2,X3)) = X1 + 2X2 + X3 >= X1 + 2X2 + X3 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = X1 + X2 >= X1 + X2 = a__U52(mark(X1),X2) mark(U53(X)) = X >= X = a__U53(mark(X)) mark(U61(X1,X2)) = X1 + 6X2 >= X1 + 6X2 = a__U61(mark(X1),X2) mark(length(X)) = 3X >= 3X = a__length(mark(X)) mark(and(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = X >= X = a__isNatIListKind(X) mark(isNatKind(X)) = X >= X = a__isNatKind(X) mark(cons(X1,X2)) = 4X1 + 5X2 >= 4X1 + 5X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 2X >= 2X = s(mark(X)) a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = 4X1 + X2 >= 4X1 + X2 = U11(X1,X2) a__U12(X) = X >= X = U12(X) a__isNatList(X) = X >= X = isNatList(X) a__U21(X1,X2) = X1 + 3X2 >= X1 + 3X2 = U21(X1,X2) a__U22(X) = X >= X = U22(X) a__isNat(X) = 2X >= 2X = isNat(X) a__U41(X1,X2,X3) = 2X1 + 2X2 + 6X3 + 2 >= X1 + 2X2 + 4X3 = U41(X1,X2,X3) a__U42(X1,X2) = X1 + 4X2 + 2 >= X1 + 2X2 + 2 = U42(X1,X2) a__U43(X) = X >= X = U43(X) a__isNatIList(X) = 4X + 2 >= 2X + 2 = isNatIList(X) a__U51(X1,X2,X3) = X1 + 2X2 + X3 >= X1 + 2X2 + X3 = U51(X1,X2,X3) a__U52(X1,X2) = X1 + X2 >= X1 + X2 = U52(X1,X2) a__U53(X) = X >= X = U53(X) a__U61(X1,X2) = X1 + 6X2 >= X1 + 6X2 = U61(X1,X2) a__length(X) = 3X >= 3X = length(X) a__and(X1,X2) = 2X1 + 2X2 >= 2X1 + 2X2 = and(X1,X2) a__isNatIListKind(X) = X >= X = isNatIListKind(X) a__isNatKind(X) = X >= X = isNatKind(X) problem: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() 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__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2) -> U61(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dim=1 interpretation: [U61](x0, x1) = x0 + 4x1, [U53](x0) = 4x0, [U52](x0, x1) = x0 + 4x1, [U51](x0, x1, x2) = x0 + x1 + 3x2, [isNatIList](x0) = x0 + 1, [U43](x0) = x0, [U42](x0, x1) = x0 + x1, [U22](x0) = x0, [U21](x0, x1) = x0 + 2x1, [isNatList](x0) = x0, [U12](x0) = x0, [U11](x0, x1) = x0 + x1, [and](x0, x1) = x0 + 2x1, [isNatKind](x0) = x0, [isNat](x0) = x0, [isNatIListKind](x0) = x0, [a__isNatKind](x0) = 2x0, [a__isNatIListKind](x0) = x0, [length](x0) = x0, [a__and](x0, x1) = x0 + 2x1, [s](x0) = 2x0, [a__length](x0) = x0, [mark](x0) = 2x0, [a__U61](x0, x1) = x0 + 4x1, [a__U53](x0) = 4x0, [a__U52](x0, x1) = x0 + 4x1, [a__U51](x0, x1, x2) = x0 + 2x1 + 5x2, [a__U43](x0) = x0, [a__isNatIList](x0) = x0 + 1, [a__U42](x0, x1) = x0 + x1 + 1, [a__U41](x0, x1, x2) = x0 + 2x1 + 4x2 + 1, [a__U22](x0) = x0, [a__isNat](x0) = 2x0, [a__U21](x0, x1) = x0 + 2x1, [a__U12](x0) = x0, [a__isNatList](x0) = x0, [a__U11](x0, x1) = x0 + x1, [tt] = 0, [cons](x0, x1) = 6x0 + 7x1, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),V1) = V1 >= V1 = a__U12(a__isNatList(V1)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = 2V1 >= 2V1 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U41(tt(),V1,V2) = 2V1 + 4V2 + 1 >= 2V1 + V2 + 1 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = V2 + 1 >= V2 + 1 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = 2V1 + 5V2 >= 2V1 + 4V2 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = 4V2 >= 4V2 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),L) = 4L >= 4L = s(a__length(mark(L))) a__and(tt(),X) = 2X >= 2X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 2V1 >= 2V1 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 4V1 >= 4V1 = a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) = 6V1 + 7V2 + 1 >= 4V1 + 6V2 + 1 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 6V1 + 7V2 >= 2V1 + 2V2 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = 2V1 >= V1 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 4V1 >= 2V1 = a__isNatKind(V1) a__isNatList(cons(V1,V2)) = 6V1 + 7V2 >= 4V1 + 7V2 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(cons(N,L)) = 7L + 6N >= 7L + 6N = a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = 2X1 + 2X2 >= 2X1 + X2 = a__U11(mark(X1),X2) mark(U12(X)) = 2X >= 2X = a__U12(mark(X)) mark(isNatList(X)) = 2X >= X = a__isNatList(X) mark(U21(X1,X2)) = 2X1 + 4X2 >= 2X1 + 2X2 = a__U21(mark(X1),X2) mark(U22(X)) = 2X >= 2X = a__U22(mark(X)) mark(isNat(X)) = 2X >= 2X = a__isNat(X) mark(U43(X)) = 2X >= 2X = a__U43(mark(X)) mark(U51(X1,X2,X3)) = 2X1 + 2X2 + 6X3 >= 2X1 + 2X2 + 5X3 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = 2X1 + 8X2 >= 2X1 + 4X2 = a__U52(mark(X1),X2) mark(U53(X)) = 8X >= 8X = a__U53(mark(X)) mark(U61(X1,X2)) = 2X1 + 8X2 >= 2X1 + 4X2 = a__U61(mark(X1),X2) mark(length(X)) = 2X >= 2X = a__length(mark(X)) mark(and(X1,X2)) = 2X1 + 4X2 >= 2X1 + 2X2 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 2X >= X = a__isNatIListKind(X) mark(isNatKind(X)) = 2X >= 2X = a__isNatKind(X) mark(cons(X1,X2)) = 12X1 + 14X2 >= 12X1 + 7X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 4X >= 4X = s(mark(X)) a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = X1 + X2 >= X1 + X2 = U11(X1,X2) a__U12(X) = X >= X = U12(X) a__isNatList(X) = X >= X = isNatList(X) a__U21(X1,X2) = X1 + 2X2 >= X1 + 2X2 = U21(X1,X2) a__U22(X) = X >= X = U22(X) a__isNat(X) = 2X >= X = isNat(X) a__U42(X1,X2) = X1 + X2 + 1 >= X1 + X2 = U42(X1,X2) a__U43(X) = X >= X = U43(X) a__isNatIList(X) = X + 1 >= X + 1 = isNatIList(X) a__U51(X1,X2,X3) = X1 + 2X2 + 5X3 >= X1 + X2 + 3X3 = U51(X1,X2,X3) a__U52(X1,X2) = X1 + 4X2 >= X1 + 4X2 = U52(X1,X2) a__U53(X) = 4X >= 4X = U53(X) a__U61(X1,X2) = X1 + 4X2 >= X1 + 4X2 = U61(X1,X2) a__length(X) = X >= X = length(X) a__and(X1,X2) = X1 + 2X2 >= X1 + 2X2 = and(X1,X2) a__isNatIListKind(X) = X >= X = isNatIListKind(X) a__isNatKind(X) = 2X >= X = isNatKind(X) problem: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() 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__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2) -> U61(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dim=1 interpretation: [U61](x0, x1) = x0 + 4x1, [U53](x0) = x0, [U52](x0, x1) = x0 + x1, [U51](x0, x1, x2) = x0 + 4x1 + 5x2, [isNatIList](x0) = x0, [U43](x0) = x0, [U22](x0) = x0, [U21](x0, x1) = x0 + 4x1, [isNatList](x0) = x0, [U12](x0) = x0, [U11](x0, x1) = 2x0 + x1, [and](x0, x1) = x0 + x1, [isNatKind](x0) = 2x0, [isNat](x0) = 4x0, [isNatIListKind](x0) = x0, [a__isNatKind](x0) = 2x0, [a__isNatIListKind](x0) = x0, [length](x0) = x0, [a__and](x0, x1) = x0 + x1, [s](x0) = 4x0, [a__length](x0) = x0, [mark](x0) = x0, [a__U61](x0, x1) = x0 + 4x1, [a__U53](x0) = x0, [a__U52](x0, x1) = x0 + x1, [a__U51](x0, x1, x2) = x0 + 4x1 + 5x2, [a__U43](x0) = x0, [a__isNatIList](x0) = x0 + 2, [a__U42](x0, x1) = x0 + x1 + 2, [a__U41](x0, x1, x2) = x0 + 5x1 + 5x2 + 2, [a__U22](x0) = x0, [a__isNat](x0) = 4x0, [a__U21](x0, x1) = x0 + 4x1, [a__U12](x0) = x0, [a__isNatList](x0) = x0, [a__U11](x0, x1) = 2x0 + x1, [tt] = 0, [cons](x0, x1) = 7x0 + 6x1, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),V1) = V1 >= V1 = a__U12(a__isNatList(V1)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = 4V1 >= 4V1 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U41(tt(),V1,V2) = 5V1 + 5V2 + 2 >= 4V1 + V2 + 2 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = V2 + 2 >= V2 + 2 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = 4V1 + 5V2 >= 4V1 + V2 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = V2 >= V2 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),L) = 4L >= 4L = s(a__length(mark(L))) a__and(tt(),X) = X >= X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 4V1 >= 3V1 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 16V1 >= 6V1 = a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) = 7V1 + 6V2 + 2 >= 7V1 + 6V2 + 2 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 7V1 + 6V2 >= 2V1 + V2 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = 2V1 >= V1 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 8V1 >= 2V1 = a__isNatKind(V1) a__isNatList(cons(V1,V2)) = 7V1 + 6V2 >= 6V1 + 6V2 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(cons(N,L)) = 6L + 7N >= 6L + 6N = a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = a__U11(mark(X1),X2) mark(U12(X)) = X >= X = a__U12(mark(X)) mark(isNatList(X)) = X >= X = a__isNatList(X) mark(U21(X1,X2)) = X1 + 4X2 >= X1 + 4X2 = a__U21(mark(X1),X2) mark(U22(X)) = X >= X = a__U22(mark(X)) mark(isNat(X)) = 4X >= 4X = a__isNat(X) mark(U43(X)) = X >= X = a__U43(mark(X)) mark(U51(X1,X2,X3)) = X1 + 4X2 + 5X3 >= X1 + 4X2 + 5X3 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = X1 + X2 >= X1 + X2 = a__U52(mark(X1),X2) mark(U53(X)) = X >= X = a__U53(mark(X)) mark(U61(X1,X2)) = X1 + 4X2 >= X1 + 4X2 = a__U61(mark(X1),X2) mark(length(X)) = X >= X = a__length(mark(X)) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = X >= X = a__isNatIListKind(X) mark(isNatKind(X)) = 2X >= 2X = a__isNatKind(X) mark(cons(X1,X2)) = 7X1 + 6X2 >= 7X1 + 6X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 4X >= 4X = s(mark(X)) a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = 2X1 + X2 >= 2X1 + X2 = U11(X1,X2) a__U12(X) = X >= X = U12(X) a__isNatList(X) = X >= X = isNatList(X) a__U21(X1,X2) = X1 + 4X2 >= X1 + 4X2 = U21(X1,X2) a__U22(X) = X >= X = U22(X) a__isNat(X) = 4X >= 4X = isNat(X) a__U43(X) = X >= X = U43(X) a__isNatIList(X) = X + 2 >= X = isNatIList(X) a__U51(X1,X2,X3) = X1 + 4X2 + 5X3 >= X1 + 4X2 + 5X3 = U51(X1,X2,X3) a__U52(X1,X2) = X1 + X2 >= X1 + X2 = U52(X1,X2) a__U53(X) = X >= X = U53(X) a__U61(X1,X2) = X1 + 4X2 >= X1 + 4X2 = U61(X1,X2) a__length(X) = X >= X = length(X) a__and(X1,X2) = X1 + X2 >= X1 + X2 = and(X1,X2) a__isNatIListKind(X) = X >= X = isNatIListKind(X) a__isNatKind(X) = 2X >= 2X = isNatKind(X) problem: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() 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__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U43(X) -> U43(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2) -> U61(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1 1 1] [U61](x0, x1) = [0 0 0]x0 + [1 1 1]x1 [0 0 0] [1 0 1] , [1 0 1] [U53](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [1 0 1] [U52](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [1 1 1] [1 0 1] [U51](x0, x1, x2) = [0 0 0]x0 + [1 0 0]x1 + [0 0 0]x2 [0 0 0] [0 0 0] [0 0 0] , [1 0 1] [U43](x0) = [0 0 0]x0 [1 0 1] , [1 0 0] [0] [U22](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 0] [1 1 1] [0] [U21](x0, x1) = [0 0 1]x0 + [0 0 0]x1 + [0] [0 0 0] [0 1 0] [1], [1 0 1] [isNatList](x0) = [1 0 0]x0 [0 0 0] , [1 0 0] [U12](x0) = [0 0 1]x0 [0 0 1] , [1 0 0] [1 0 1] [0] [U11](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1] [0 0 0] [1 0 0] [0], [1 0 0] [1 0 0] [and](x0, x1) = [0 1 0]x0 + [0 1 0]x1 [0 1 0] [0 1 1] , [1 0 0] [0] [isNatKind](x0) = [0 0 1]x0 + [0] [1 0 0] [1], [1 1 1] [0] [isNat](x0) = [1 0 0]x0 + [1] [0 0 1] [1], [1 0 0] [0] [isNatIListKind](x0) = [1 0 0]x0 + [0] [1 1 0] [1], [1 0 0] [0] [a__isNatKind](x0) = [0 0 1]x0 + [0] [1 0 0] [1], [1 0 0] [0] [a__isNatIListKind](x0) = [1 0 0]x0 + [0] [1 1 0] [1], [1 1 1] [length](x0) = [0 0 1]x0 [1 0 0] , [1 0 0] [1 0 0] [a__and](x0, x1) = [0 1 0]x0 + [0 1 0]x1 [0 1 0] [0 1 1] , [1 0 0] [s](x0) = [1 0 0]x0 [0 1 1] , [1 1 1] [a__length](x0) = [0 0 1]x0 [1 0 0] , [mark](x0) = x0 , [1 0 0] [1 1 1] [a__U61](x0, x1) = [0 0 0]x0 + [1 1 1]x1 [0 0 0] [1 0 1] , [1 0 1] [a__U53](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [1 0 1] [a__U52](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [1 1 1] [1 0 1] [a__U51](x0, x1, x2) = [0 0 0]x0 + [1 0 0]x1 + [0 0 0]x2 [0 0 0] [0 0 0] [0 0 0] , [1 0 1] [a__U43](x0) = [0 0 0]x0 [1 0 1] , [1 1 0] [a__isNatIList](x0) = [0 1 0]x0 [0 0 1] , [1 0 0] [1 1 1] [a__U42](x0, x1) = [0 0 0]x0 + [0 1 0]x1 [0 0 0] [1 1 1] , [1 0 0] [1 1 1] [1 1 1] [a__U41](x0, x1, x2) = [0 0 0]x0 + [1 1 0]x1 + [0 1 0]x2 [0 0 0] [0 0 1] [1 1 1] , [1 0 0] [0] [a__U22](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 1 1] [0] [a__isNat](x0) = [1 0 0]x0 + [1] [0 0 1] [1], [1 0 0] [1 1 1] [0] [a__U21](x0, x1) = [0 0 1]x0 + [0 0 0]x1 + [0] [0 0 0] [0 1 0] [1], [1 0 0] [a__U12](x0) = [0 0 1]x0 [0 0 1] , [1 0 1] [a__isNatList](x0) = [1 0 0]x0 [0 0 0] , [1 0 0] [1 0 1] [0] [a__U11](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1] [0 0 0] [1 0 0] [0], [0] [tt] = [0] [1], [1 0 1] [1 0 1] [cons](x0, x1) = [1 1 0]x0 + [1 1 0]x1 [1 1 1] [1 1 1] , [0] [zeros] = [0] [0], [0] [0] = [0] [0], [0] [a__zeros] = [0] [0] orientation: [0] [0] a__zeros() = [0] >= [0] = cons(0(),zeros()) [0] [0] [1 0 1] [0] [1 0 1] a__U11(tt(),V1) = [0 0 0]V1 + [1] >= [0 0 0]V1 = a__U12(a__isNatList(V1)) [1 0 0] [0] [0 0 0] [0] [0] a__U12(tt()) = [1] >= [0] = tt() [1] [1] [1 1 1] [0] [1 1 1] [0] a__U21(tt(),V1) = [0 0 0]V1 + [1] >= [0 0 0]V1 + [0] = a__U22(a__isNat(V1)) [0 1 0] [1] [0 0 0] [1] [0] [0] a__U22(tt()) = [0] >= [0] = tt() [1] [1] [1 1 1] [1 1 1] [1 1 1] [1 1 1] a__U41(tt(),V1,V2) = [1 1 0]V1 + [0 1 0]V2 >= [0 0 0]V1 + [0 1 0]V2 = a__U42(a__isNat(V1),V2) [0 0 1] [1 1 1] [0 0 0] [1 1 1] [1 1 1] [1 1 1] a__U42(tt(),V2) = [0 1 0]V2 >= [0 0 0]V2 = a__U43(a__isNatIList(V2)) [1 1 1] [1 1 1] [1] [0] a__U43(tt()) = [0] >= [0] = tt() [1] [1] [1 1 1] [1 0 1] [1 1 1] [1 0 1] a__U51(tt(),V1,V2) = [1 0 0]V1 + [0 0 0]V2 >= [0 0 0]V1 + [0 0 0]V2 = a__U52(a__isNat(V1),V2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] a__U52(tt(),V2) = [0 0 0]V2 >= [0 0 0]V2 = a__U53(a__isNatList(V2)) [0 0 0] [0 0 0] [1] [0] a__U53(tt()) = [0] >= [0] = tt() [1] [1] [1 1 1] [1 1 1] a__U61(tt(),L) = [1 1 1]L >= [1 1 1]L = s(a__length(mark(L))) [1 0 1] [1 0 1] [1 0 0] a__and(tt(),X) = [0 1 0]X >= X = mark(X) [0 1 1] [0] [0] a__isNat(0()) = [1] >= [0] = tt() [1] [1] [2 1 2] [0] [2 0 1] [0] a__isNat(length(V1)) = [1 1 1]V1 + [1] >= [0 0 0]V1 + [1] = a__U11(a__isNatIListKind(V1),V1) [1 0 0] [1] [1 0 0] [0] [2 1 1] [0] [2 1 1] [0] a__isNat(s(V1)) = [1 0 0]V1 + [1] >= [1 0 0]V1 + [1] = a__U21(a__isNatKind(V1),V1) [0 1 1] [1] [0 1 0] [1] [2 1 1] [2 1 1] [2 1 1] [2 1 1] a__isNatIList(cons(V1,V2)) = [1 1 0]V1 + [1 1 0]V2 >= [1 1 0]V1 + [0 1 0]V2 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) [1 1 1] [1 1 1] [0 0 1] [1 1 1] [0] [0] a__isNatIListKind(zeros()) = [0] >= [0] = tt() [1] [1] [1 0 1] [1 0 1] [0] [1 0 0] [1 0 0] [0] a__isNatIListKind(cons(V1,V2)) = [1 0 1]V1 + [1 0 1]V2 + [0] >= [0 0 1]V1 + [1 0 0]V2 + [0] = a__and(a__isNatKind(V1),isNatIListKind(V2)) [2 1 1] [2 1 1] [1] [0 0 1] [2 1 0] [1] [0] [0] a__isNatKind(0()) = [0] >= [0] = tt() [1] [1] [1 1 1] [0] [1 0 0] [0] a__isNatKind(length(V1)) = [1 0 0]V1 + [0] >= [1 0 0]V1 + [0] = a__isNatIListKind(V1) [1 1 1] [1] [1 1 0] [1] [1 0 0] [0] [1 0 0] [0] a__isNatKind(s(V1)) = [0 1 1]V1 + [0] >= [0 0 1]V1 + [0] = a__isNatKind(V1) [1 0 0] [1] [1 0 0] [1] [2 1 2] [2 1 2] [2 1 1] [2 0 1] a__isNatList(cons(V1,V2)) = [1 0 1]V1 + [1 0 1]V2 >= [1 0 0]V1 + [0 0 0]V2 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [3 2 2] [3 2 2] [3 1 2] [2 1 1] a__length(cons(N,L)) = [1 1 1]L + [1 1 1]N >= [1 1 1]L + [0 0 0]N = a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) [1 0 1] [1 0 1] [1 0 1] [0 0 0] [0] [0] mark(zeros()) = [0] >= [0] = a__zeros() [0] [0] [1 0 0] [1 0 1] [0] [1 0 0] [1 0 1] [0] mark(U11(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = a__U11(mark(X1),X2) [0 0 0] [1 0 0] [0] [0 0 0] [1 0 0] [0] [1 0 0] [1 0 0] mark(U12(X)) = [0 0 1]X >= [0 0 1]X = a__U12(mark(X)) [0 0 1] [0 0 1] [1 0 1] [1 0 1] mark(isNatList(X)) = [1 0 0]X >= [1 0 0]X = a__isNatList(X) [0 0 0] [0 0 0] [1 0 0] [1 1 1] [0] [1 0 0] [1 1 1] [0] mark(U21(X1,X2)) = [0 0 1]X1 + [0 0 0]X2 + [0] >= [0 0 1]X1 + [0 0 0]X2 + [0] = a__U21(mark(X1),X2) [0 0 0] [0 1 0] [1] [0 0 0] [0 1 0] [1] [1 0 0] [0] [1 0 0] [0] mark(U22(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = a__U22(mark(X)) [0 0 0] [1] [0 0 0] [1] [1 1 1] [0] [1 1 1] [0] mark(isNat(X)) = [1 0 0]X + [1] >= [1 0 0]X + [1] = a__isNat(X) [0 0 1] [1] [0 0 1] [1] [1 0 1] [1 0 1] mark(U43(X)) = [0 0 0]X >= [0 0 0]X = a__U43(mark(X)) [1 0 1] [1 0 1] [1 0 0] [1 1 1] [1 0 1] [1 0 0] [1 1 1] [1 0 1] mark(U51(X1,X2,X3)) = [0 0 0]X1 + [1 0 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [1 0 0]X2 + [0 0 0]X3 = a__U51(mark(X1),X2,X3) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 1] [1 0 0] [1 0 1] mark(U52(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = a__U52(mark(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] mark(U53(X)) = [0 0 0]X >= [0 0 0]X = a__U53(mark(X)) [0 0 1] [0 0 1] [1 0 0] [1 1 1] [1 0 0] [1 1 1] mark(U61(X1,X2)) = [0 0 0]X1 + [1 1 1]X2 >= [0 0 0]X1 + [1 1 1]X2 = a__U61(mark(X1),X2) [0 0 0] [1 0 1] [0 0 0] [1 0 1] [1 1 1] [1 1 1] mark(length(X)) = [0 0 1]X >= [0 0 1]X = a__length(mark(X)) [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] mark(and(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 >= [0 1 0]X1 + [0 1 0]X2 = a__and(mark(X1),X2) [0 1 0] [0 1 1] [0 1 0] [0 1 1] [1 0 0] [0] [1 0 0] [0] mark(isNatIListKind(X)) = [1 0 0]X + [0] >= [1 0 0]X + [0] = a__isNatIListKind(X) [1 1 0] [1] [1 1 0] [1] [1 0 0] [0] [1 0 0] [0] mark(isNatKind(X)) = [0 0 1]X + [0] >= [0 0 1]X + [0] = a__isNatKind(X) [1 0 0] [1] [1 0 0] [1] [1 0 1] [1 0 1] [1 0 1] [1 0 1] mark(cons(X1,X2)) = [1 1 0]X1 + [1 1 0]X2 >= [1 1 0]X1 + [1 1 0]X2 = cons(mark(X1),X2) [1 1 1] [1 1 1] [1 1 1] [1 1 1] [0] [0] mark(0()) = [0] >= [0] = 0() [0] [0] [0] [0] mark(tt()) = [0] >= [0] = tt() [1] [1] [1 0 0] [1 0 0] mark(s(X)) = [1 0 0]X >= [1 0 0]X = s(mark(X)) [0 1 1] [0 1 1] [0] [0] a__zeros() = [0] >= [0] = zeros() [0] [0] [1 0 0] [1 0 1] [0] [1 0 0] [1 0 1] [0] a__U11(X1,X2) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = U11(X1,X2) [0 0 0] [1 0 0] [0] [0 0 0] [1 0 0] [0] [1 0 0] [1 0 0] a__U12(X) = [0 0 1]X >= [0 0 1]X = U12(X) [0 0 1] [0 0 1] [1 0 1] [1 0 1] a__isNatList(X) = [1 0 0]X >= [1 0 0]X = isNatList(X) [0 0 0] [0 0 0] [1 0 0] [1 1 1] [0] [1 0 0] [1 1 1] [0] a__U21(X1,X2) = [0 0 1]X1 + [0 0 0]X2 + [0] >= [0 0 1]X1 + [0 0 0]X2 + [0] = U21(X1,X2) [0 0 0] [0 1 0] [1] [0 0 0] [0 1 0] [1] [1 0 0] [0] [1 0 0] [0] a__U22(X) = [0 0 0]X + [0] >= [0 0 0]X + [0] = U22(X) [0 0 0] [1] [0 0 0] [1] [1 1 1] [0] [1 1 1] [0] a__isNat(X) = [1 0 0]X + [1] >= [1 0 0]X + [1] = isNat(X) [0 0 1] [1] [0 0 1] [1] [1 0 1] [1 0 1] a__U43(X) = [0 0 0]X >= [0 0 0]X = U43(X) [1 0 1] [1 0 1] [1 0 0] [1 1 1] [1 0 1] [1 0 0] [1 1 1] [1 0 1] a__U51(X1,X2,X3) = [0 0 0]X1 + [1 0 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [1 0 0]X2 + [0 0 0]X3 = U51(X1,X2,X3) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 1] [1 0 0] [1 0 1] a__U52(X1,X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = U52(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] a__U53(X) = [0 0 0]X >= [0 0 0]X = U53(X) [0 0 1] [0 0 1] [1 0 0] [1 1 1] [1 0 0] [1 1 1] a__U61(X1,X2) = [0 0 0]X1 + [1 1 1]X2 >= [0 0 0]X1 + [1 1 1]X2 = U61(X1,X2) [0 0 0] [1 0 1] [0 0 0] [1 0 1] [1 1 1] [1 1 1] a__length(X) = [0 0 1]X >= [0 0 1]X = length(X) [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] a__and(X1,X2) = [0 1 0]X1 + [0 1 0]X2 >= [0 1 0]X1 + [0 1 0]X2 = and(X1,X2) [0 1 0] [0 1 1] [0 1 0] [0 1 1] [1 0 0] [0] [1 0 0] [0] a__isNatIListKind(X) = [1 0 0]X + [0] >= [1 0 0]X + [0] = isNatIListKind(X) [1 1 0] [1] [1 1 0] [1] [1 0 0] [0] [1 0 0] [0] a__isNatKind(X) = [0 0 1]X + [0] >= [0 0 1]X + [0] = isNatKind(X) [1 0 0] [1] [1 0 0] [1] problem: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U61(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() 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__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U43(X) -> U43(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2) -> U61(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1 1 1] [0] [U61](x0, x1) = [0 0 0]x0 + [1 0 1]x1 + [1] [0 0 0] [1 1 1] [0], [1 0 0] [U53](x0) = [0 0 1]x0 [0 0 0] , [1 0 0] [1 0 1] [U52](x0, x1) = [0 0 0]x0 + [0 1 0]x1 [0 0 0] [0 1 0] , [1 0 0] [1 1 1] [1 0 1] [U51](x0, x1, x2) = [0 1 0]x0 + [0 0 0]x1 + [1 1 0]x2 [0 1 0] [0 0 0] [0 1 0] , [1 0 0] [U43](x0) = [0 0 0]x0 [0 1 0] , [1 0 0] [0] [U22](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [1 0 0] [1 1 1] [0] [U21](x0, x1) = [1 0 0]x0 + [0 0 0]x1 + [1] [0 0 0] [0 0 0] [0], [1 0 1] [0] [isNatList](x0) = [0 1 0]x0 + [1] [0 1 0] [0], [1 0 0] [1] [U12](x0) = [0 1 0]x0 + [0] [0 0 0] [0], [1 0 0] [1 0 1] [1] [U11](x0, x1) = [1 0 0]x0 + [0 1 1]x1 + [1] [0 0 0] [0 0 0] [1], [1 0 0] [and](x0, x1) = [0 0 1]x0 + x1 [0 1 1] , [1 0 0] [isNatKind](x0) = [0 0 0]x0 [0 0 0] , [1 1 1] [0] [isNat](x0) = [1 0 0]x0 + [1] [0 1 1] [1], [1 0 0] [isNatIListKind](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [a__isNatKind](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [a__isNatIListKind](x0) = [0 0 0]x0 [0 0 0] , [1 1 1] [0] [length](x0) = [0 0 1]x0 + [1] [1 0 0] [0], [1 0 0] [a__and](x0, x1) = [0 0 1]x0 + x1 [0 1 1] , [1 0 0] [s](x0) = [0 1 1]x0 [1 0 0] , [1 1 1] [0] [a__length](x0) = [0 0 1]x0 + [1] [1 0 0] [0], [mark](x0) = x0 , [1 0 0] [1 1 1] [0] [a__U61](x0, x1) = [0 0 0]x0 + [1 0 1]x1 + [1] [0 0 0] [1 1 1] [0], [1 0 0] [a__U53](x0) = [0 0 1]x0 [0 0 0] , [1 0 0] [1 0 1] [a__U52](x0, x1) = [0 0 0]x0 + [0 1 0]x1 [0 0 0] [0 1 0] , [1 0 0] [1 1 1] [1 0 1] [a__U51](x0, x1, x2) = [0 1 0]x0 + [0 0 0]x1 + [1 1 0]x2 [0 1 0] [0 0 0] [0 1 0] , [1 0 0] [a__U43](x0) = [0 0 0]x0 [0 1 0] , [1 1 1] [0] [a__isNatIList](x0) = [1 1 1]x0 + [0] [1 0 0] [1], [1 0 0] [1 1 1] [a__U42](x0, x1) = [0 0 0]x0 + [1 1 0]x1 [0 0 1] [1 1 1] , [1 0 0] [1 1 1] [1 1 1] [0] [a__U41](x0, x1, x2) = [0 0 0]x0 + [1 0 1]x1 + [1 1 0]x2 + [0] [0 0 0] [1 1 1] [1 1 1] [1], [1 0 0] [0] [a__U22](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [1 1 1] [0] [a__isNat](x0) = [1 0 0]x0 + [1] [0 1 1] [1], [1 0 0] [1 1 1] [0] [a__U21](x0, x1) = [1 0 0]x0 + [0 0 0]x1 + [1] [0 0 0] [0 0 0] [0], [1 0 0] [1] [a__U12](x0) = [0 1 0]x0 + [0] [0 0 0] [0], [1 0 1] [0] [a__isNatList](x0) = [0 1 0]x0 + [1] [0 1 0] [0], [1 0 0] [1 0 1] [1] [a__U11](x0, x1) = [1 0 0]x0 + [0 1 1]x1 + [1] [0 0 0] [0 0 0] [1], [0] [tt] = [0] [0], [1 1 1] [1 1 1] [cons](x0, x1) = [0 0 0]x0 + [1 1 0]x1 [1 0 1] [1 0 1] , [0] [zeros] = [0] [0], [0] [0] = [0] [0], [0] [a__zeros] = [0] [0] orientation: [0] [0] a__zeros() = [0] >= [0] = cons(0(),zeros()) [0] [0] [1 0 1] [1] [1 0 1] [1] a__U11(tt(),V1) = [0 1 1]V1 + [1] >= [0 1 0]V1 + [1] = a__U12(a__isNatList(V1)) [0 0 0] [1] [0 0 0] [0] [1] [0] a__U12(tt()) = [0] >= [0] = tt() [0] [0] [1 1 1] [0] [1 1 1] [0] a__U21(tt(),V1) = [0 0 0]V1 + [1] >= [0 0 0]V1 + [1] = a__U22(a__isNat(V1)) [0 0 0] [0] [0 0 0] [0] [0] [0] a__U22(tt()) = [1] >= [0] = tt() [0] [0] [1 1 1] [1 1 1] [0] [1 1 1] [1 1 1] [0] a__U41(tt(),V1,V2) = [1 0 1]V1 + [1 1 0]V2 + [0] >= [0 0 0]V1 + [1 1 0]V2 + [0] = a__U42(a__isNat(V1),V2) [1 1 1] [1 1 1] [1] [0 1 1] [1 1 1] [1] [1 1 1] [1 1 1] a__U42(tt(),V2) = [1 1 0]V2 >= [0 0 0]V2 = a__U43(a__isNatIList(V2)) [1 1 1] [1 1 1] [1 1 1] [1 0 1] [1 1 1] [1 0 1] a__U51(tt(),V1,V2) = [0 0 0]V1 + [1 1 0]V2 >= [0 0 0]V1 + [0 1 0]V2 = a__U52(a__isNat(V1),V2) [0 0 0] [0 1 0] [0 0 0] [0 1 0] [1 0 1] [1 0 1] a__U52(tt(),V2) = [0 1 0]V2 >= [0 1 0]V2 = a__U53(a__isNatList(V2)) [0 1 0] [0 0 0] [1 1 1] [0] [1 1 1] [0] a__U61(tt(),L) = [1 0 1]L + [1] >= [1 0 1]L + [1] = s(a__length(mark(L))) [1 1 1] [0] [1 1 1] [0] a__and(tt(),X) = X >= X = mark(X) [0] [0] a__isNat(0()) = [1] >= [0] = tt() [1] [0] [2 1 2] [1] [2 0 1] [1] a__isNat(length(V1)) = [1 1 1]V1 + [1] >= [1 1 1]V1 + [1] = a__U11(a__isNatIListKind(V1),V1) [1 0 1] [2] [0 0 0] [1] [2 1 1] [0] [2 1 1] [0] a__isNat(s(V1)) = [1 0 0]V1 + [1] >= [1 0 0]V1 + [1] = a__U21(a__isNatKind(V1),V1) [1 1 1] [1] [0 0 0] [0] [2 1 2] [3 2 2] [0] [2 1 1] [2 1 1] [0] a__isNatIList(cons(V1,V2)) = [2 1 2]V1 + [3 2 2]V2 + [0] >= [1 0 1]V1 + [1 1 0]V2 + [0] = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) [1 1 1] [1 1 1] [1] [1 1 1] [1 1 1] [1] [0] [0] a__isNatIListKind(zeros()) = [0] >= [0] = tt() [0] [0] [1 1 1] [1 1 1] [1 0 0] [1 0 0] a__isNatIListKind(cons(V1,V2)) = [0 0 0]V1 + [0 0 0]V2 >= [0 0 0]V1 + [0 0 0]V2 = a__and(a__isNatKind(V1),isNatIListKind(V2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] a__isNatKind(0()) = [0] >= [0] = tt() [0] [0] [1 1 1] [1 0 0] a__isNatKind(length(V1)) = [0 0 0]V1 >= [0 0 0]V1 = a__isNatIListKind(V1) [0 0 0] [0 0 0] [1 0 0] [1 0 0] a__isNatKind(s(V1)) = [0 0 0]V1 >= [0 0 0]V1 = a__isNatKind(V1) [0 0 0] [0 0 0] [2 1 2] [2 1 2] [0] [2 1 1] [2 0 1] a__isNatList(cons(V1,V2)) = [0 0 0]V1 + [1 1 0]V2 + [1] >= [0 0 0]V1 + [1 1 0]V2 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) [0 0 0] [1 1 0] [0] [0 0 0] [0 1 0] [3 2 2] [2 1 2] [0] [3 1 2] [2 1 1] [0] a__length(cons(N,L)) = [1 0 1]L + [1 0 1]N + [1] >= [1 0 1]L + [0 0 0]N + [1] = a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) [1 1 1] [1 1 1] [0] [1 1 1] [0 0 0] [0] [0] [0] mark(zeros()) = [0] >= [0] = a__zeros() [0] [0] [1 0 0] [1 0 1] [1] [1 0 0] [1 0 1] [1] mark(U11(X1,X2)) = [1 0 0]X1 + [0 1 1]X2 + [1] >= [1 0 0]X1 + [0 1 1]X2 + [1] = a__U11(mark(X1),X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [1] [1 0 0] [1] mark(U12(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = a__U12(mark(X)) [0 0 0] [0] [0 0 0] [0] [1 0 1] [0] [1 0 1] [0] mark(isNatList(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = a__isNatList(X) [0 1 0] [0] [0 1 0] [0] [1 0 0] [1 1 1] [0] [1 0 0] [1 1 1] [0] mark(U21(X1,X2)) = [1 0 0]X1 + [0 0 0]X2 + [1] >= [1 0 0]X1 + [0 0 0]X2 + [1] = a__U21(mark(X1),X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] mark(U22(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = a__U22(mark(X)) [0 0 0] [0] [0 0 0] [0] [1 1 1] [0] [1 1 1] [0] mark(isNat(X)) = [1 0 0]X + [1] >= [1 0 0]X + [1] = a__isNat(X) [0 1 1] [1] [0 1 1] [1] [1 0 0] [1 0 0] mark(U43(X)) = [0 0 0]X >= [0 0 0]X = a__U43(mark(X)) [0 1 0] [0 1 0] [1 0 0] [1 1 1] [1 0 1] [1 0 0] [1 1 1] [1 0 1] mark(U51(X1,X2,X3)) = [0 1 0]X1 + [0 0 0]X2 + [1 1 0]X3 >= [0 1 0]X1 + [0 0 0]X2 + [1 1 0]X3 = a__U51(mark(X1),X2,X3) [0 1 0] [0 0 0] [0 1 0] [0 1 0] [0 0 0] [0 1 0] [1 0 0] [1 0 1] [1 0 0] [1 0 1] mark(U52(X1,X2)) = [0 0 0]X1 + [0 1 0]X2 >= [0 0 0]X1 + [0 1 0]X2 = a__U52(mark(X1),X2) [0 0 0] [0 1 0] [0 0 0] [0 1 0] [1 0 0] [1 0 0] mark(U53(X)) = [0 0 1]X >= [0 0 1]X = a__U53(mark(X)) [0 0 0] [0 0 0] [1 0 0] [1 1 1] [0] [1 0 0] [1 1 1] [0] mark(U61(X1,X2)) = [0 0 0]X1 + [1 0 1]X2 + [1] >= [0 0 0]X1 + [1 0 1]X2 + [1] = a__U61(mark(X1),X2) [0 0 0] [1 1 1] [0] [0 0 0] [1 1 1] [0] [1 1 1] [0] [1 1 1] [0] mark(length(X)) = [0 0 1]X + [1] >= [0 0 1]X + [1] = a__length(mark(X)) [1 0 0] [0] [1 0 0] [0] [1 0 0] [1 0 0] mark(and(X1,X2)) = [0 0 1]X1 + X2 >= [0 0 1]X1 + X2 = a__and(mark(X1),X2) [0 1 1] [0 1 1] [1 0 0] [1 0 0] mark(isNatIListKind(X)) = [0 0 0]X >= [0 0 0]X = a__isNatIListKind(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] mark(isNatKind(X)) = [0 0 0]X >= [0 0 0]X = a__isNatKind(X) [0 0 0] [0 0 0] [1 1 1] [1 1 1] [1 1 1] [1 1 1] mark(cons(X1,X2)) = [0 0 0]X1 + [1 1 0]X2 >= [0 0 0]X1 + [1 1 0]X2 = cons(mark(X1),X2) [1 0 1] [1 0 1] [1 0 1] [1 0 1] [0] [0] mark(0()) = [0] >= [0] = 0() [0] [0] [0] [0] mark(tt()) = [0] >= [0] = tt() [0] [0] [1 0 0] [1 0 0] mark(s(X)) = [0 1 1]X >= [0 1 1]X = s(mark(X)) [1 0 0] [1 0 0] [0] [0] a__zeros() = [0] >= [0] = zeros() [0] [0] [1 0 0] [1 0 1] [1] [1 0 0] [1 0 1] [1] a__U11(X1,X2) = [1 0 0]X1 + [0 1 1]X2 + [1] >= [1 0 0]X1 + [0 1 1]X2 + [1] = U11(X1,X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [1] [1 0 0] [1] a__U12(X) = [0 1 0]X + [0] >= [0 1 0]X + [0] = U12(X) [0 0 0] [0] [0 0 0] [0] [1 0 1] [0] [1 0 1] [0] a__isNatList(X) = [0 1 0]X + [1] >= [0 1 0]X + [1] = isNatList(X) [0 1 0] [0] [0 1 0] [0] [1 0 0] [1 1 1] [0] [1 0 0] [1 1 1] [0] a__U21(X1,X2) = [1 0 0]X1 + [0 0 0]X2 + [1] >= [1 0 0]X1 + [0 0 0]X2 + [1] = U21(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] a__U22(X) = [0 0 0]X + [1] >= [0 0 0]X + [1] = U22(X) [0 0 0] [0] [0 0 0] [0] [1 1 1] [0] [1 1 1] [0] a__isNat(X) = [1 0 0]X + [1] >= [1 0 0]X + [1] = isNat(X) [0 1 1] [1] [0 1 1] [1] [1 0 0] [1 0 0] a__U43(X) = [0 0 0]X >= [0 0 0]X = U43(X) [0 1 0] [0 1 0] [1 0 0] [1 1 1] [1 0 1] [1 0 0] [1 1 1] [1 0 1] a__U51(X1,X2,X3) = [0 1 0]X1 + [0 0 0]X2 + [1 1 0]X3 >= [0 1 0]X1 + [0 0 0]X2 + [1 1 0]X3 = U51(X1,X2,X3) [0 1 0] [0 0 0] [0 1 0] [0 1 0] [0 0 0] [0 1 0] [1 0 0] [1 0 1] [1 0 0] [1 0 1] a__U52(X1,X2) = [0 0 0]X1 + [0 1 0]X2 >= [0 0 0]X1 + [0 1 0]X2 = U52(X1,X2) [0 0 0] [0 1 0] [0 0 0] [0 1 0] [1 0 0] [1 0 0] a__U53(X) = [0 0 1]X >= [0 0 1]X = U53(X) [0 0 0] [0 0 0] [1 0 0] [1 1 1] [0] [1 0 0] [1 1 1] [0] a__U61(X1,X2) = [0 0 0]X1 + [1 0 1]X2 + [1] >= [0 0 0]X1 + [1 0 1]X2 + [1] = U61(X1,X2) [0 0 0] [1 1 1] [0] [0 0 0] [1 1 1] [0] [1 1 1] [0] [1 1 1] [0] a__length(X) = [0 0 1]X + [1] >= [0 0 1]X + [1] = length(X) [1 0 0] [0] [1 0 0] [0] [1 0 0] [1 0 0] a__and(X1,X2) = [0 0 1]X1 + X2 >= [0 0 1]X1 + X2 = and(X1,X2) [0 1 1] [0 1 1] [1 0 0] [1 0 0] a__isNatIListKind(X) = [0 0 0]X >= [0 0 0]X = isNatIListKind(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] a__isNatKind(X) = [0 0 0]X >= [0 0 0]X = isNatKind(X) [0 0 0] [0 0 0] problem: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U61(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() 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__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U43(X) -> U43(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2) -> U61(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1 1 1] [0] [U61](x0, x1) = [0 0 0]x0 + [1 1 1]x1 + [1] [0 0 0] [1 1 1] [0], [1 0 0] [0] [U53](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 0] [1 1 1] [0] [U52](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 0] [1], [1 0 0] [1 1 1] [1 1 1] [0] [U51](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 + [0] [0 0 0] [0 0 0] [0 0 0] [1], [1 0 0] [U43](x0) = [0 0 0]x0 [0 1 0] , [1 0 0] [U22](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1 1 1] [U21](x0, x1) = [0 1 0]x0 + [0 0 0]x1 [1 0 1] [0 1 0] , [1 1 0] [0] [isNatList](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 0] [U12](x0) = [0 0 1]x0 [0 0 0] , [1 0 0] [1 1 0] [1] [U11](x0, x1) = [0 1 0]x0 + [0 0 0]x1 + [1] [0 0 0] [0 1 0] [0], [1 0 0] [and](x0, x1) = [1 0 1]x0 + x1 [0 0 1] , [1 0 0] [isNatKind](x0) = [0 1 0]x0 [1 0 0] , [1 1 1] [0] [isNat](x0) = [0 1 0]x0 + [1] [1 1 1] [0], [1 0 0] [isNatIListKind](x0) = [0 1 1]x0 [1 0 0] , [1 0 0] [a__isNatKind](x0) = [0 1 0]x0 [1 0 0] , [1 0 0] [a__isNatIListKind](x0) = [0 1 1]x0 [1 0 0] , [1 1 1] [0] [length](x0) = [0 1 1]x0 + [1] [1 0 0] [0], [1 0 0] [a__and](x0, x1) = [1 0 1]x0 + x1 [0 0 1] , [1 0 0] [s](x0) = [0 1 1]x0 [1 0 0] , [1 1 1] [0] [a__length](x0) = [0 1 1]x0 + [1] [1 0 0] [0], [mark](x0) = x0 , [1 0 0] [1 1 1] [0] [a__U61](x0, x1) = [0 0 0]x0 + [1 1 1]x1 + [1] [0 0 0] [1 1 1] [0], [1 0 0] [0] [a__U53](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 0] [1 1 1] [0] [a__U52](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 0] [1], [1 0 0] [1 1 1] [1 1 1] [0] [a__U51](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 + [0] [0 0 0] [0 0 0] [0 0 0] [1], [1 0 0] [a__U43](x0) = [0 0 0]x0 [0 1 0] , [1 1 0] [a__isNatIList](x0) = [1 0 1]x0 [1 0 1] , [1 0 0] [1 1 1] [a__U42](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [1 0 1] , [1 0 0] [1 1 1] [1 1 1] [a__U41](x0, x1, x2) = [1 0 0]x0 + [1 0 0]x1 + [1 1 0]x2 [1 0 0] [1 0 0] [1 0 1] , [1 0 0] [a__U22](x0) = [0 0 0]x0 [0 0 0] , [1 1 1] [0] [a__isNat](x0) = [0 1 0]x0 + [1] [1 1 1] [0], [1 0 0] [1 1 1] [a__U21](x0, x1) = [0 1 0]x0 + [0 0 0]x1 [1 0 1] [0 1 0] , [1 0 0] [a__U12](x0) = [0 0 1]x0 [0 0 0] , [1 1 0] [0] [a__isNatList](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 0] [1 1 0] [1] [a__U11](x0, x1) = [0 1 0]x0 + [0 0 0]x1 + [1] [0 0 0] [0 1 0] [0], [0] [tt] = [0] [0], [1 0 0] [1 1 1] [cons](x0, x1) = [1 1 1]x0 + [1 0 0]x1 [1 0 0] [1 1 1] , [0] [zeros] = [0] [0], [0] [0] = [0] [0], [0] [a__zeros] = [0] [0] orientation: [0] [0] a__zeros() = [0] >= [0] = cons(0(),zeros()) [0] [0] [1 1 0] [1] [1 1 0] [0] a__U11(tt(),V1) = [0 0 0]V1 + [1] >= [0 0 0]V1 + [1] = a__U12(a__isNatList(V1)) [0 1 0] [0] [0 0 0] [0] [1 1 1] [1 1 1] a__U21(tt(),V1) = [0 0 0]V1 >= [0 0 0]V1 = a__U22(a__isNat(V1)) [0 1 0] [0 0 0] [0] [0] a__U22(tt()) = [0] >= [0] = tt() [0] [0] [1 1 1] [1 1 1] [1 1 1] [1 1 1] a__U41(tt(),V1,V2) = [1 0 0]V1 + [1 1 0]V2 >= [0 0 0]V1 + [0 0 0]V2 = a__U42(a__isNat(V1),V2) [1 0 0] [1 0 1] [0 0 0] [1 0 1] [1 1 1] [1 1 0] a__U42(tt(),V2) = [0 0 0]V2 >= [0 0 0]V2 = a__U43(a__isNatIList(V2)) [1 0 1] [1 0 1] [1 1 1] [1 1 1] [0] [1 1 1] [1 1 1] [0] a__U51(tt(),V1,V2) = [0 0 0]V1 + [0 0 0]V2 + [0] >= [0 0 0]V1 + [0 0 0]V2 + [0] = a__U52(a__isNat(V1),V2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 1 1] [0] [1 1 0] [0] a__U52(tt(),V2) = [0 0 0]V2 + [0] >= [0 0 0]V2 + [0] = a__U53(a__isNatList(V2)) [0 0 0] [1] [0 0 0] [1] [1 1 1] [0] [1 1 1] [0] a__U61(tt(),L) = [1 1 1]L + [1] >= [1 1 1]L + [1] = s(a__length(mark(L))) [1 1 1] [0] [1 1 1] [0] a__and(tt(),X) = X >= X = mark(X) [0] [0] a__isNat(0()) = [1] >= [0] = tt() [0] [0] [2 2 2] [1] [2 1 0] [1] a__isNat(length(V1)) = [0 1 1]V1 + [2] >= [0 1 1]V1 + [1] = a__U11(a__isNatIListKind(V1),V1) [2 2 2] [1] [0 1 0] [0] [2 1 1] [0] [2 1 1] a__isNat(s(V1)) = [0 1 1]V1 + [1] >= [0 1 0]V1 = a__U21(a__isNatKind(V1),V1) [2 1 1] [0] [2 1 0] [2 1 1] [2 1 1] [2 1 1] [2 1 1] a__isNatIList(cons(V1,V2)) = [2 0 0]V1 + [2 2 2]V2 >= [2 0 0]V1 + [2 1 0]V2 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) [2 0 0] [2 2 2] [2 0 0] [2 0 1] [0] [0] a__isNatIListKind(zeros()) = [0] >= [0] = tt() [0] [0] [1 0 0] [1 1 1] [1 0 0] [1 0 0] a__isNatIListKind(cons(V1,V2)) = [2 1 1]V1 + [2 1 1]V2 >= [2 0 0]V1 + [0 1 1]V2 = a__and(a__isNatKind(V1),isNatIListKind(V2)) [1 0 0] [1 1 1] [1 0 0] [1 0 0] [0] [0] a__isNatKind(0()) = [0] >= [0] = tt() [0] [0] [1 1 1] [0] [1 0 0] a__isNatKind(length(V1)) = [0 1 1]V1 + [1] >= [0 1 1]V1 = a__isNatIListKind(V1) [1 1 1] [0] [1 0 0] [1 0 0] [1 0 0] a__isNatKind(s(V1)) = [0 1 1]V1 >= [0 1 0]V1 = a__isNatKind(V1) [1 0 0] [1 0 0] [2 1 1] [2 1 1] [0] [2 1 1] [2 1 1] [0] a__isNatList(cons(V1,V2)) = [0 0 0]V1 + [0 0 0]V2 + [0] >= [0 0 0]V1 + [0 0 0]V2 + [0] = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [3 2 2] [3 1 1] [0] [3 2 1] [2 1 1] [0] a__length(cons(N,L)) = [2 1 1]L + [2 1 1]N + [1] >= [1 1 1]L + [0 0 0]N + [1] = a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) [1 1 1] [1 0 0] [0] [1 1 1] [0 0 0] [0] [0] [0] mark(zeros()) = [0] >= [0] = a__zeros() [0] [0] [1 0 0] [1 1 0] [1] [1 0 0] [1 1 0] [1] mark(U11(X1,X2)) = [0 1 0]X1 + [0 0 0]X2 + [1] >= [0 1 0]X1 + [0 0 0]X2 + [1] = a__U11(mark(X1),X2) [0 0 0] [0 1 0] [0] [0 0 0] [0 1 0] [0] [1 0 0] [1 0 0] mark(U12(X)) = [0 0 1]X >= [0 0 1]X = a__U12(mark(X)) [0 0 0] [0 0 0] [1 1 0] [0] [1 1 0] [0] mark(isNatList(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = a__isNatList(X) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 1 1] [1 0 0] [1 1 1] mark(U21(X1,X2)) = [0 1 0]X1 + [0 0 0]X2 >= [0 1 0]X1 + [0 0 0]X2 = a__U21(mark(X1),X2) [1 0 1] [0 1 0] [1 0 1] [0 1 0] [1 0 0] [1 0 0] mark(U22(X)) = [0 0 0]X >= [0 0 0]X = a__U22(mark(X)) [0 0 0] [0 0 0] [1 1 1] [0] [1 1 1] [0] mark(isNat(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = a__isNat(X) [1 1 1] [0] [1 1 1] [0] [1 0 0] [1 0 0] mark(U43(X)) = [0 0 0]X >= [0 0 0]X = a__U43(mark(X)) [0 1 0] [0 1 0] [1 0 0] [1 1 1] [1 1 1] [0] [1 0 0] [1 1 1] [1 1 1] [0] mark(U51(X1,X2,X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] = a__U51(mark(X1),X2,X3) [0 0 0] [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0 0 0] [1] [1 0 0] [1 1 1] [0] [1 0 0] [1 1 1] [0] mark(U52(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = a__U52(mark(X1),X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] mark(U53(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = a__U53(mark(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 1 1] [0] [1 0 0] [1 1 1] [0] mark(U61(X1,X2)) = [0 0 0]X1 + [1 1 1]X2 + [1] >= [0 0 0]X1 + [1 1 1]X2 + [1] = a__U61(mark(X1),X2) [0 0 0] [1 1 1] [0] [0 0 0] [1 1 1] [0] [1 1 1] [0] [1 1 1] [0] mark(length(X)) = [0 1 1]X + [1] >= [0 1 1]X + [1] = a__length(mark(X)) [1 0 0] [0] [1 0 0] [0] [1 0 0] [1 0 0] mark(and(X1,X2)) = [1 0 1]X1 + X2 >= [1 0 1]X1 + X2 = a__and(mark(X1),X2) [0 0 1] [0 0 1] [1 0 0] [1 0 0] mark(isNatIListKind(X)) = [0 1 1]X >= [0 1 1]X = a__isNatIListKind(X) [1 0 0] [1 0 0] [1 0 0] [1 0 0] mark(isNatKind(X)) = [0 1 0]X >= [0 1 0]X = a__isNatKind(X) [1 0 0] [1 0 0] [1 0 0] [1 1 1] [1 0 0] [1 1 1] mark(cons(X1,X2)) = [1 1 1]X1 + [1 0 0]X2 >= [1 1 1]X1 + [1 0 0]X2 = cons(mark(X1),X2) [1 0 0] [1 1 1] [1 0 0] [1 1 1] [0] [0] mark(0()) = [0] >= [0] = 0() [0] [0] [0] [0] mark(tt()) = [0] >= [0] = tt() [0] [0] [1 0 0] [1 0 0] mark(s(X)) = [0 1 1]X >= [0 1 1]X = s(mark(X)) [1 0 0] [1 0 0] [0] [0] a__zeros() = [0] >= [0] = zeros() [0] [0] [1 0 0] [1 1 0] [1] [1 0 0] [1 1 0] [1] a__U11(X1,X2) = [0 1 0]X1 + [0 0 0]X2 + [1] >= [0 1 0]X1 + [0 0 0]X2 + [1] = U11(X1,X2) [0 0 0] [0 1 0] [0] [0 0 0] [0 1 0] [0] [1 0 0] [1 0 0] a__U12(X) = [0 0 1]X >= [0 0 1]X = U12(X) [0 0 0] [0 0 0] [1 1 0] [0] [1 1 0] [0] a__isNatList(X) = [0 0 0]X + [0] >= [0 0 0]X + [0] = isNatList(X) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 1 1] [1 0 0] [1 1 1] a__U21(X1,X2) = [0 1 0]X1 + [0 0 0]X2 >= [0 1 0]X1 + [0 0 0]X2 = U21(X1,X2) [1 0 1] [0 1 0] [1 0 1] [0 1 0] [1 0 0] [1 0 0] a__U22(X) = [0 0 0]X >= [0 0 0]X = U22(X) [0 0 0] [0 0 0] [1 1 1] [0] [1 1 1] [0] a__isNat(X) = [0 1 0]X + [1] >= [0 1 0]X + [1] = isNat(X) [1 1 1] [0] [1 1 1] [0] [1 0 0] [1 0 0] a__U43(X) = [0 0 0]X >= [0 0 0]X = U43(X) [0 1 0] [0 1 0] [1 0 0] [1 1 1] [1 1 1] [0] [1 0 0] [1 1 1] [1 1 1] [0] a__U51(X1,X2,X3) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] = U51(X1,X2,X3) [0 0 0] [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0 0 0] [1] [1 0 0] [1 1 1] [0] [1 0 0] [1 1 1] [0] a__U52(X1,X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = U52(X1,X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] a__U53(X) = [0 0 0]X + [0] >= [0 0 0]X + [0] = U53(X) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 1 1] [0] [1 0 0] [1 1 1] [0] a__U61(X1,X2) = [0 0 0]X1 + [1 1 1]X2 + [1] >= [0 0 0]X1 + [1 1 1]X2 + [1] = U61(X1,X2) [0 0 0] [1 1 1] [0] [0 0 0] [1 1 1] [0] [1 1 1] [0] [1 1 1] [0] a__length(X) = [0 1 1]X + [1] >= [0 1 1]X + [1] = length(X) [1 0 0] [0] [1 0 0] [0] [1 0 0] [1 0 0] a__and(X1,X2) = [1 0 1]X1 + X2 >= [1 0 1]X1 + X2 = and(X1,X2) [0 0 1] [0 0 1] [1 0 0] [1 0 0] a__isNatIListKind(X) = [0 1 1]X >= [0 1 1]X = isNatIListKind(X) [1 0 0] [1 0 0] [1 0 0] [1 0 0] a__isNatKind(X) = [0 1 0]X >= [0 1 0]X = isNatKind(X) [1 0 0] [1 0 0] problem: a__zeros() -> cons(0(),zeros()) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U61(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() 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__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U43(X) -> U43(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2) -> U61(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dim=1 interpretation: [U61](x0, x1) = x0 + 3x1, [U53](x0) = x0, [U52](x0, x1) = x0 + x1, [U51](x0, x1, x2) = 2x0 + x1 + x2, [U43](x0) = x0, [U22](x0) = x0, [U21](x0, x1) = 2x0 + 2x1, [isNatList](x0) = x0, [U12](x0) = 4x0 + 4, [U11](x0, x1) = x0 + x1, [and](x0, x1) = x0 + x1, [isNatKind](x0) = x0, [isNat](x0) = x0, [isNatIListKind](x0) = x0, [a__isNatKind](x0) = x0, [a__isNatIListKind](x0) = x0, [length](x0) = x0, [a__and](x0, x1) = x0 + 2x1, [s](x0) = 2x0, [a__length](x0) = x0, [mark](x0) = 2x0, [a__U61](x0, x1) = x0 + 4x1, [a__U53](x0) = x0, [a__U52](x0, x1) = x0 + x1, [a__U51](x0, x1, x2) = 2x0 + 2x1 + 2x2, [a__U43](x0) = x0, [a__isNatIList](x0) = 4x0 + 4, [a__U42](x0, x1) = 2x0 + 4x1 + 4, [a__U41](x0, x1, x2) = x0 + 4x1 + 4x2 + 4, [a__U22](x0) = x0, [a__isNat](x0) = 2x0, [a__U21](x0, x1) = 2x0 + 2x1, [a__U12](x0) = 4x0 + 4, [a__isNatList](x0) = x0, [a__U11](x0, x1) = x0 + x1, [tt] = 0, [cons](x0, x1) = 4x0 + 7x1, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U21(tt(),V1) = 2V1 >= 2V1 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U41(tt(),V1,V2) = 4V1 + 4V2 + 4 >= 4V1 + 4V2 + 4 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = 4V2 + 4 >= 4V2 + 4 = a__U43(a__isNatIList(V2)) a__U51(tt(),V1,V2) = 2V1 + 2V2 >= 2V1 + V2 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = V2 >= V2 = a__U53(a__isNatList(V2)) a__U61(tt(),L) = 4L >= 4L = s(a__length(mark(L))) a__and(tt(),X) = 2X >= 2X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 2V1 >= 2V1 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 4V1 >= 4V1 = a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) = 16V1 + 28V2 + 4 >= 5V1 + 6V2 + 4 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 4V1 + 7V2 >= V1 + 2V2 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = V1 >= V1 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 2V1 >= V1 = a__isNatKind(V1) a__isNatList(cons(V1,V2)) = 4V1 + 7V2 >= 4V1 + 6V2 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(cons(N,L)) = 7L + 4N >= 7L + 4N = a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = 2X1 + 2X2 >= 2X1 + X2 = a__U11(mark(X1),X2) mark(U12(X)) = 8X + 8 >= 8X + 4 = a__U12(mark(X)) mark(isNatList(X)) = 2X >= X = a__isNatList(X) mark(U21(X1,X2)) = 4X1 + 4X2 >= 4X1 + 2X2 = a__U21(mark(X1),X2) mark(U22(X)) = 2X >= 2X = a__U22(mark(X)) mark(isNat(X)) = 2X >= 2X = a__isNat(X) mark(U43(X)) = 2X >= 2X = a__U43(mark(X)) mark(U51(X1,X2,X3)) = 4X1 + 2X2 + 2X3 >= 4X1 + 2X2 + 2X3 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = 2X1 + 2X2 >= 2X1 + X2 = a__U52(mark(X1),X2) mark(U53(X)) = 2X >= 2X = a__U53(mark(X)) mark(U61(X1,X2)) = 2X1 + 6X2 >= 2X1 + 4X2 = a__U61(mark(X1),X2) mark(length(X)) = 2X >= 2X = a__length(mark(X)) mark(and(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 2X >= X = a__isNatIListKind(X) mark(isNatKind(X)) = 2X >= X = a__isNatKind(X) mark(cons(X1,X2)) = 8X1 + 14X2 >= 8X1 + 7X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 4X >= 4X = s(mark(X)) a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = X1 + X2 >= X1 + X2 = U11(X1,X2) a__U12(X) = 4X + 4 >= 4X + 4 = U12(X) a__isNatList(X) = X >= X = isNatList(X) a__U21(X1,X2) = 2X1 + 2X2 >= 2X1 + 2X2 = U21(X1,X2) a__U22(X) = X >= X = U22(X) a__isNat(X) = 2X >= X = isNat(X) a__U43(X) = X >= X = U43(X) a__U51(X1,X2,X3) = 2X1 + 2X2 + 2X3 >= 2X1 + X2 + X3 = U51(X1,X2,X3) a__U52(X1,X2) = X1 + X2 >= X1 + X2 = U52(X1,X2) a__U53(X) = X >= X = U53(X) a__U61(X1,X2) = X1 + 4X2 >= X1 + 3X2 = U61(X1,X2) a__length(X) = X >= X = length(X) a__and(X1,X2) = X1 + 2X2 >= X1 + X2 = and(X1,X2) a__isNatIListKind(X) = X >= X = isNatIListKind(X) a__isNatKind(X) = X >= X = isNatKind(X) problem: a__zeros() -> cons(0(),zeros()) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U61(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() 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__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U43(X) -> U43(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2) -> U61(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dim=1 interpretation: [U61](x0, x1) = x0 + 4x1, [U53](x0) = x0, [U52](x0, x1) = x0 + x1, [U51](x0, x1, x2) = 2x0 + x1 + 2x2, [U43](x0) = x0, [U22](x0) = x0, [U21](x0, x1) = x0 + x1, [isNatList](x0) = x0, [U12](x0) = x0, [U11](x0, x1) = x0 + x1, [and](x0, x1) = 2x0 + 2x1, [isNatKind](x0) = x0, [isNat](x0) = x0, [isNatIListKind](x0) = x0, [a__isNatKind](x0) = x0, [a__isNatIListKind](x0) = x0, [length](x0) = 2x0, [a__and](x0, x1) = 2x0 + 2x1, [s](x0) = 2x0, [a__length](x0) = 2x0, [mark](x0) = x0, [a__U61](x0, x1) = x0 + 4x1, [a__U53](x0) = x0, [a__U52](x0, x1) = x0 + x1, [a__U51](x0, x1, x2) = 2x0 + x1 + 2x2, [a__U43](x0) = x0, [a__isNatIList](x0) = x0 + 3, [a__U42](x0, x1) = x0 + x1 + 3, [a__U41](x0, x1, x2) = 2x0 + x1 + x2 + 3, [a__U22](x0) = x0, [a__isNat](x0) = x0, [a__U21](x0, x1) = x0 + x1, [a__U12](x0) = x0 + 1, [a__isNatList](x0) = x0, [a__U11](x0, x1) = x0 + x1, [tt] = 0, [cons](x0, x1) = 5x0 + 6x1, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U21(tt(),V1) = V1 >= V1 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U41(tt(),V1,V2) = V1 + V2 + 3 >= V1 + V2 + 3 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = V2 + 3 >= V2 + 3 = a__U43(a__isNatIList(V2)) a__U51(tt(),V1,V2) = V1 + 2V2 >= V1 + V2 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = V2 >= V2 = a__U53(a__isNatList(V2)) a__U61(tt(),L) = 4L >= 4L = s(a__length(mark(L))) a__and(tt(),X) = 2X >= X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 2V1 >= 2V1 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 2V1 >= 2V1 = a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) = 5V1 + 6V2 + 3 >= 5V1 + 5V2 + 3 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 5V1 + 6V2 >= 2V1 + 2V2 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = 2V1 >= V1 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 2V1 >= V1 = a__isNatKind(V1) a__isNatList(cons(V1,V2)) = 5V1 + 6V2 >= 5V1 + 6V2 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(cons(N,L)) = 12L + 10N >= 12L + 8N = a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = X1 + X2 >= X1 + X2 = a__U11(mark(X1),X2) mark(isNatList(X)) = X >= X = a__isNatList(X) mark(U21(X1,X2)) = X1 + X2 >= X1 + X2 = a__U21(mark(X1),X2) mark(U22(X)) = X >= X = a__U22(mark(X)) mark(isNat(X)) = X >= X = a__isNat(X) mark(U43(X)) = X >= X = a__U43(mark(X)) mark(U51(X1,X2,X3)) = 2X1 + X2 + 2X3 >= 2X1 + X2 + 2X3 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = X1 + X2 >= X1 + X2 = a__U52(mark(X1),X2) mark(U53(X)) = X >= X = a__U53(mark(X)) mark(U61(X1,X2)) = X1 + 4X2 >= X1 + 4X2 = a__U61(mark(X1),X2) mark(length(X)) = 2X >= 2X = a__length(mark(X)) mark(and(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = X >= X = a__isNatIListKind(X) mark(isNatKind(X)) = X >= X = a__isNatKind(X) mark(cons(X1,X2)) = 5X1 + 6X2 >= 5X1 + 6X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 2X >= 2X = s(mark(X)) a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = X1 + X2 >= X1 + X2 = U11(X1,X2) a__U12(X) = X + 1 >= X = U12(X) a__isNatList(X) = X >= X = isNatList(X) a__U21(X1,X2) = X1 + X2 >= X1 + X2 = U21(X1,X2) a__U22(X) = X >= X = U22(X) a__isNat(X) = X >= X = isNat(X) a__U43(X) = X >= X = U43(X) a__U51(X1,X2,X3) = 2X1 + X2 + 2X3 >= 2X1 + X2 + 2X3 = U51(X1,X2,X3) a__U52(X1,X2) = X1 + X2 >= X1 + X2 = U52(X1,X2) a__U53(X) = X >= X = U53(X) a__U61(X1,X2) = X1 + 4X2 >= X1 + 4X2 = U61(X1,X2) a__length(X) = 2X >= 2X = length(X) a__and(X1,X2) = 2X1 + 2X2 >= 2X1 + 2X2 = and(X1,X2) a__isNatIListKind(X) = X >= X = isNatIListKind(X) a__isNatKind(X) = X >= X = isNatKind(X) problem: a__zeros() -> cons(0(),zeros()) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U61(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() 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__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U43(X) -> U43(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2) -> U61(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1 1 1] [0] [U61](x0, x1) = [0 0 0]x0 + [1 1 0]x1 + [1] [0 0 0] [1 1 1] [0], [1 0 0] [U53](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1 1 0] [U52](x0, x1) = [0 0 0]x0 + [0 1 0]x1 [0 0 0] [0 0 0] , [1 0 0] [1 1 1] [1 1 1] [U51](x0, x1, x2) = [0 0 0]x0 + [1 0 1]x1 + [1 1 0]x2 [1 0 0] [0 0 0] [0 0 0] , [1 0 0] [U43](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [U22](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [1 1 1] [U21](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 1 0] [isNatList](x0) = [0 1 1]x0 [0 1 0] , [1 0 0] [1 0 0] [U11](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [1 0 1] [0 0 0] , [1 0 0] [and](x0, x1) = [0 0 0]x0 + x1 [1 0 0] , [1 0 0] [isNatKind](x0) = [0 0 0]x0 [0 1 0] , [1 1 1] [isNat](x0) = [0 0 0]x0 [1 1 0] , [1 0 0] [isNatIListKind](x0) = [0 0 0]x0 [1 0 0] , [1 0 0] [a__isNatKind](x0) = [0 0 0]x0 [0 1 0] , [1 0 0] [a__isNatIListKind](x0) = [0 0 0]x0 [1 0 0] , [1 1 1] [0] [length](x0) = [1 0 0]x0 + [1] [0 1 0] [0], [1 0 0] [a__and](x0, x1) = [0 0 0]x0 + x1 [1 0 0] , [1 0 0] [s](x0) = [0 1 1]x0 [1 0 0] , [1 1 1] [0] [a__length](x0) = [1 0 0]x0 + [1] [0 1 0] [0], [mark](x0) = x0 , [1 0 0] [1 1 1] [0] [a__U61](x0, x1) = [0 0 0]x0 + [1 1 0]x1 + [1] [0 0 0] [1 1 1] [0], [1 0 0] [a__U53](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1 1 0] [a__U52](x0, x1) = [0 0 0]x0 + [0 1 0]x1 [0 0 0] [0 0 0] , [1 0 0] [1 1 1] [1 1 1] [a__U51](x0, x1, x2) = [0 0 0]x0 + [1 0 1]x1 + [1 1 0]x2 [1 0 0] [0 0 0] [0 0 0] , [1 0 0] [a__U43](x0) = [0 0 0]x0 [0 0 1] , [1 1 0] [0] [a__isNatIList](x0) = [0 0 0]x0 + [1] [1 1 0] [1], [1 0 0] [1 1 0] [0] [a__U42](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1] [0 0 0] [1 1 1] [1], [1 0 0] [1 1 1] [1 1 0] [0] [a__U41](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 + [1] [1 0 0] [1 0 0] [1 1 1] [1], [1 0 0] [a__U22](x0) = [0 1 0]x0 [0 0 0] , [1 1 1] [a__isNat](x0) = [0 0 0]x0 [1 1 0] , [1 0 0] [1 1 1] [a__U21](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 1 0] [a__isNatList](x0) = [0 1 1]x0 [0 1 0] , [1 0 0] [1 0 0] [a__U11](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [1 0 1] [0 0 0] , [0] [tt] = [0] [0], [1 0 1] [1 1 0] [cons](x0, x1) = [1 1 1]x0 + [1 1 1]x1 [0 1 1] [1 0 0] , [0] [zeros] = [0] [0], [0] [0] = [0] [0], [0] [a__zeros] = [0] [0] orientation: [0] [0] a__zeros() = [0] >= [0] = cons(0(),zeros()) [0] [0] [1 1 1] [1 1 1] a__U21(tt(),V1) = [0 0 0]V1 >= [0 0 0]V1 = a__U22(a__isNat(V1)) [0 0 0] [0 0 0] [0] [0] a__U22(tt()) = [0] >= [0] = tt() [0] [0] [1 1 1] [1 1 0] [0] [1 1 1] [1 1 0] [0] a__U41(tt(),V1,V2) = [0 0 0]V1 + [0 0 0]V2 + [1] >= [0 0 0]V1 + [0 0 0]V2 + [1] = a__U42(a__isNat(V1),V2) [1 0 0] [1 1 1] [1] [0 0 0] [1 1 1] [1] [1 1 0] [0] [1 1 0] [0] a__U42(tt(),V2) = [0 0 0]V2 + [1] >= [0 0 0]V2 + [0] = a__U43(a__isNatIList(V2)) [1 1 1] [1] [1 1 0] [1] [1 1 1] [1 1 1] [1 1 1] [1 1 0] a__U51(tt(),V1,V2) = [1 0 1]V1 + [1 1 0]V2 >= [0 0 0]V1 + [0 1 0]V2 = a__U52(a__isNat(V1),V2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] a__U52(tt(),V2) = [0 1 0]V2 >= [0 0 0]V2 = a__U53(a__isNatList(V2)) [0 0 0] [0 0 0] [1 1 1] [0] [1 1 1] [0] a__U61(tt(),L) = [1 1 0]L + [1] >= [1 1 0]L + [1] = s(a__length(mark(L))) [1 1 1] [0] [1 1 1] [0] a__and(tt(),X) = X >= X = mark(X) [0] [0] a__isNat(0()) = [0] >= [0] = tt() [0] [0] [2 2 1] [1] [2 0 0] a__isNat(length(V1)) = [0 0 0]V1 + [0] >= [0 0 0]V1 = a__U11(a__isNatIListKind(V1),V1) [2 1 1] [1] [2 0 0] [2 1 1] [2 1 1] a__isNat(s(V1)) = [0 0 0]V1 >= [0 0 0]V1 = a__U21(a__isNatKind(V1),V1) [1 1 1] [0 0 0] [2 1 2] [2 2 1] [0] [2 1 1] [2 1 0] [0] a__isNatIList(cons(V1,V2)) = [0 0 0]V1 + [0 0 0]V2 + [1] >= [0 0 0]V1 + [0 0 0]V2 + [1] = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) [2 1 2] [2 2 1] [1] [2 0 0] [2 1 1] [1] [0] [0] a__isNatIListKind(zeros()) = [0] >= [0] = tt() [0] [0] [1 0 1] [1 1 0] [1 0 0] [1 0 0] a__isNatIListKind(cons(V1,V2)) = [0 0 0]V1 + [0 0 0]V2 >= [0 0 0]V1 + [0 0 0]V2 = a__and(a__isNatKind(V1),isNatIListKind(V2)) [1 0 1] [1 1 0] [1 0 0] [1 0 0] [0] [0] a__isNatKind(0()) = [0] >= [0] = tt() [0] [0] [1 1 1] [0] [1 0 0] a__isNatKind(length(V1)) = [0 0 0]V1 + [0] >= [0 0 0]V1 = a__isNatIListKind(V1) [1 0 0] [1] [1 0 0] [1 0 0] [1 0 0] a__isNatKind(s(V1)) = [0 0 0]V1 >= [0 0 0]V1 = a__isNatKind(V1) [0 1 1] [0 1 0] [2 1 2] [2 2 1] [2 1 1] [2 1 1] a__isNatList(cons(V1,V2)) = [1 2 2]V1 + [2 1 1]V2 >= [1 0 1]V1 + [1 1 0]V2 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) [1 1 1] [1 1 1] [1 0 0] [1 0 0] [3 2 1] [2 2 3] [0] [3 2 1] [2 1 1] [0] a__length(cons(N,L)) = [1 1 0]L + [1 0 1]N + [1] >= [1 1 0]L + [0 0 0]N + [1] = a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) [1 1 1] [1 1 1] [0] [1 1 1] [0 0 0] [0] [0] [0] mark(zeros()) = [0] >= [0] = a__zeros() [0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] mark(U11(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = a__U11(mark(X1),X2) [1 0 1] [0 0 0] [1 0 1] [0 0 0] [1 1 0] [1 1 0] mark(isNatList(X)) = [0 1 1]X >= [0 1 1]X = a__isNatList(X) [0 1 0] [0 1 0] [1 0 0] [1 1 1] [1 0 0] [1 1 1] mark(U21(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = a__U21(mark(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] mark(U22(X)) = [0 1 0]X >= [0 1 0]X = a__U22(mark(X)) [0 0 0] [0 0 0] [1 1 1] [1 1 1] mark(isNat(X)) = [0 0 0]X >= [0 0 0]X = a__isNat(X) [1 1 0] [1 1 0] [1 0 0] [1 0 0] mark(U43(X)) = [0 0 0]X >= [0 0 0]X = a__U43(mark(X)) [0 0 1] [0 0 1] [1 0 0] [1 1 1] [1 1 1] [1 0 0] [1 1 1] [1 1 1] mark(U51(X1,X2,X3)) = [0 0 0]X1 + [1 0 1]X2 + [1 1 0]X3 >= [0 0 0]X1 + [1 0 1]X2 + [1 1 0]X3 = a__U51(mark(X1),X2,X3) [1 0 0] [0 0 0] [0 0 0] [1 0 0] [0 0 0] [0 0 0] [1 0 0] [1 1 0] [1 0 0] [1 1 0] mark(U52(X1,X2)) = [0 0 0]X1 + [0 1 0]X2 >= [0 0 0]X1 + [0 1 0]X2 = a__U52(mark(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] mark(U53(X)) = [0 0 0]X >= [0 0 0]X = a__U53(mark(X)) [0 0 0] [0 0 0] [1 0 0] [1 1 1] [0] [1 0 0] [1 1 1] [0] mark(U61(X1,X2)) = [0 0 0]X1 + [1 1 0]X2 + [1] >= [0 0 0]X1 + [1 1 0]X2 + [1] = a__U61(mark(X1),X2) [0 0 0] [1 1 1] [0] [0 0 0] [1 1 1] [0] [1 1 1] [0] [1 1 1] [0] mark(length(X)) = [1 0 0]X + [1] >= [1 0 0]X + [1] = a__length(mark(X)) [0 1 0] [0] [0 1 0] [0] [1 0 0] [1 0 0] mark(and(X1,X2)) = [0 0 0]X1 + X2 >= [0 0 0]X1 + X2 = a__and(mark(X1),X2) [1 0 0] [1 0 0] [1 0 0] [1 0 0] mark(isNatIListKind(X)) = [0 0 0]X >= [0 0 0]X = a__isNatIListKind(X) [1 0 0] [1 0 0] [1 0 0] [1 0 0] mark(isNatKind(X)) = [0 0 0]X >= [0 0 0]X = a__isNatKind(X) [0 1 0] [0 1 0] [1 0 1] [1 1 0] [1 0 1] [1 1 0] mark(cons(X1,X2)) = [1 1 1]X1 + [1 1 1]X2 >= [1 1 1]X1 + [1 1 1]X2 = cons(mark(X1),X2) [0 1 1] [1 0 0] [0 1 1] [1 0 0] [0] [0] mark(0()) = [0] >= [0] = 0() [0] [0] [0] [0] mark(tt()) = [0] >= [0] = tt() [0] [0] [1 0 0] [1 0 0] mark(s(X)) = [0 1 1]X >= [0 1 1]X = s(mark(X)) [1 0 0] [1 0 0] [0] [0] a__zeros() = [0] >= [0] = zeros() [0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] a__U11(X1,X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = U11(X1,X2) [1 0 1] [0 0 0] [1 0 1] [0 0 0] [1 1 0] [1 1 0] a__isNatList(X) = [0 1 1]X >= [0 1 1]X = isNatList(X) [0 1 0] [0 1 0] [1 0 0] [1 1 1] [1 0 0] [1 1 1] a__U21(X1,X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = U21(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] a__U22(X) = [0 1 0]X >= [0 1 0]X = U22(X) [0 0 0] [0 0 0] [1 1 1] [1 1 1] a__isNat(X) = [0 0 0]X >= [0 0 0]X = isNat(X) [1 1 0] [1 1 0] [1 0 0] [1 0 0] a__U43(X) = [0 0 0]X >= [0 0 0]X = U43(X) [0 0 1] [0 0 1] [1 0 0] [1 1 1] [1 1 1] [1 0 0] [1 1 1] [1 1 1] a__U51(X1,X2,X3) = [0 0 0]X1 + [1 0 1]X2 + [1 1 0]X3 >= [0 0 0]X1 + [1 0 1]X2 + [1 1 0]X3 = U51(X1,X2,X3) [1 0 0] [0 0 0] [0 0 0] [1 0 0] [0 0 0] [0 0 0] [1 0 0] [1 1 0] [1 0 0] [1 1 0] a__U52(X1,X2) = [0 0 0]X1 + [0 1 0]X2 >= [0 0 0]X1 + [0 1 0]X2 = U52(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] a__U53(X) = [0 0 0]X >= [0 0 0]X = U53(X) [0 0 0] [0 0 0] [1 0 0] [1 1 1] [0] [1 0 0] [1 1 1] [0] a__U61(X1,X2) = [0 0 0]X1 + [1 1 0]X2 + [1] >= [0 0 0]X1 + [1 1 0]X2 + [1] = U61(X1,X2) [0 0 0] [1 1 1] [0] [0 0 0] [1 1 1] [0] [1 1 1] [0] [1 1 1] [0] a__length(X) = [1 0 0]X + [1] >= [1 0 0]X + [1] = length(X) [0 1 0] [0] [0 1 0] [0] [1 0 0] [1 0 0] a__and(X1,X2) = [0 0 0]X1 + X2 >= [0 0 0]X1 + X2 = and(X1,X2) [1 0 0] [1 0 0] [1 0 0] [1 0 0] a__isNatIListKind(X) = [0 0 0]X >= [0 0 0]X = isNatIListKind(X) [1 0 0] [1 0 0] [1 0 0] [1 0 0] a__isNatKind(X) = [0 0 0]X >= [0 0 0]X = isNatKind(X) [0 1 0] [0 1 0] problem: a__zeros() -> cons(0(),zeros()) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U61(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() 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__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U43(X) -> U43(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2) -> U61(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dim=1 interpretation: [U61](x0, x1) = x0 + 2x1, [U53](x0) = x0, [U52](x0, x1) = x0 + x1, [U51](x0, x1, x2) = 2x0 + x1 + x2, [U43](x0) = 2x0, [U22](x0) = x0, [U21](x0, x1) = x0 + x1, [isNatList](x0) = x0, [U11](x0, x1) = x0 + x1 + 4, [and](x0, x1) = x0 + 2x1, [isNatKind](x0) = x0, [isNat](x0) = x0, [isNatIListKind](x0) = x0, [a__isNatKind](x0) = x0, [a__isNatIListKind](x0) = x0, [length](x0) = x0, [a__and](x0, x1) = x0 + 2x1, [s](x0) = 2x0, [a__length](x0) = x0, [mark](x0) = 2x0, [a__U61](x0, x1) = x0 + 4x1, [a__U53](x0) = x0, [a__U52](x0, x1) = x0 + x1, [a__U51](x0, x1, x2) = 2x0 + 2x1 + x2, [a__U43](x0) = 2x0, [a__isNatIList](x0) = x0, [a__U42](x0, x1) = 2x0 + 2x1, [a__U41](x0, x1, x2) = 2x0 + 2x1 + 2x2, [a__U22](x0) = x0, [a__isNat](x0) = x0, [a__U21](x0, x1) = x0 + x1, [a__isNatList](x0) = x0, [a__U11](x0, x1) = x0 + x1 + 5, [tt] = 0, [cons](x0, x1) = 6x0 + 7x1, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U21(tt(),V1) = V1 >= V1 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U41(tt(),V1,V2) = 2V1 + 2V2 >= 2V1 + 2V2 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = 2V2 >= 2V2 = a__U43(a__isNatIList(V2)) a__U51(tt(),V1,V2) = 2V1 + V2 >= V1 + V2 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = V2 >= V2 = a__U53(a__isNatList(V2)) a__U61(tt(),L) = 4L >= 4L = s(a__length(mark(L))) a__and(tt(),X) = 2X >= 2X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(s(V1)) = 2V1 >= 2V1 = a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) = 6V1 + 7V2 >= 4V1 + 6V2 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 6V1 + 7V2 >= V1 + 2V2 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = V1 >= V1 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 2V1 >= V1 = a__isNatKind(V1) a__isNatList(cons(V1,V2)) = 6V1 + 7V2 >= 4V1 + 5V2 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(cons(N,L)) = 7L + 6N >= 7L + 6N = a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = 2X1 + 2X2 + 8 >= 2X1 + X2 + 5 = a__U11(mark(X1),X2) mark(isNatList(X)) = 2X >= X = a__isNatList(X) mark(U21(X1,X2)) = 2X1 + 2X2 >= 2X1 + X2 = a__U21(mark(X1),X2) mark(U22(X)) = 2X >= 2X = a__U22(mark(X)) mark(isNat(X)) = 2X >= X = a__isNat(X) mark(U43(X)) = 4X >= 4X = a__U43(mark(X)) mark(U51(X1,X2,X3)) = 4X1 + 2X2 + 2X3 >= 4X1 + 2X2 + X3 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = 2X1 + 2X2 >= 2X1 + X2 = a__U52(mark(X1),X2) mark(U53(X)) = 2X >= 2X = a__U53(mark(X)) mark(U61(X1,X2)) = 2X1 + 4X2 >= 2X1 + 4X2 = a__U61(mark(X1),X2) mark(length(X)) = 2X >= 2X = a__length(mark(X)) mark(and(X1,X2)) = 2X1 + 4X2 >= 2X1 + 2X2 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 2X >= X = a__isNatIListKind(X) mark(isNatKind(X)) = 2X >= X = a__isNatKind(X) mark(cons(X1,X2)) = 12X1 + 14X2 >= 12X1 + 7X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 4X >= 4X = s(mark(X)) a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = X1 + X2 + 5 >= X1 + X2 + 4 = U11(X1,X2) a__isNatList(X) = X >= X = isNatList(X) a__U21(X1,X2) = X1 + X2 >= X1 + X2 = U21(X1,X2) a__U22(X) = X >= X = U22(X) a__isNat(X) = X >= X = isNat(X) a__U43(X) = 2X >= 2X = U43(X) a__U51(X1,X2,X3) = 2X1 + 2X2 + X3 >= 2X1 + X2 + X3 = U51(X1,X2,X3) a__U52(X1,X2) = X1 + X2 >= X1 + X2 = U52(X1,X2) a__U53(X) = X >= X = U53(X) a__U61(X1,X2) = X1 + 4X2 >= X1 + 2X2 = U61(X1,X2) a__length(X) = X >= X = length(X) a__and(X1,X2) = X1 + 2X2 >= X1 + 2X2 = and(X1,X2) a__isNatIListKind(X) = X >= X = isNatIListKind(X) a__isNatKind(X) = X >= X = isNatKind(X) problem: a__zeros() -> cons(0(),zeros()) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U61(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() 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__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) mark(zeros()) -> a__zeros() mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U43(X) -> U43(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2) -> U61(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dim=2 interpretation: [2 0] [2 2] [U61](x0, x1) = [0 0]x0 + [0 0]x1, [1 0] [U53](x0) = [0 0]x0, [1 0] [1 0] [U52](x0, x1) = [0 0]x0 + [1 0]x1, [1 0] [2 0] [1 0] [U51](x0, x1, x2) = [0 0]x0 + [3 0]x1 + [2 1]x2, [1 0] [U43](x0) = [0 0]x0, [1 2] [U22](x0) = [0 1]x0, [1 0] [3 2] [U21](x0, x1) = [0 0]x0 + [0 2]x1, [isNatList](x0) = x0, [1 0] [and](x0, x1) = [0 0]x0 + x1, [1 0] [0] [isNatKind](x0) = [0 0]x0 + [2], [2 0] [isNat](x0) = [0 1]x0, [1 0] [0] [isNatIListKind](x0) = [0 0]x0 + [2], [1 0] [0] [a__isNatKind](x0) = [0 0]x0 + [2], [1 0] [0] [a__isNatIListKind](x0) = [0 0]x0 + [2], [1 1] [length](x0) = [0 0]x0, [1 0] [a__and](x0, x1) = [0 0]x0 + x1, [2 1] [s](x0) = [0 2]x0, [1 1] [a__length](x0) = [0 0]x0, [mark](x0) = x0, [2 0] [2 2] [a__U61](x0, x1) = [0 0]x0 + [0 0]x1, [1 0] [a__U53](x0) = [0 0]x0, [1 0] [1 0] [a__U52](x0, x1) = [0 0]x0 + [1 0]x1, [1 0] [2 0] [1 0] [a__U51](x0, x1, x2) = [0 0]x0 + [3 0]x1 + [2 1]x2, [1 0] [a__U43](x0) = [0 0]x0, [2 0] [0] [a__isNatIList](x0) = [0 0]x0 + [1], [1 0] [2 0] [a__U42](x0, x1) = [0 0]x0 + [0 0]x1, [1 0] [2 0] [2 0] [a__U41](x0, x1, x2) = [0 0]x0 + [0 0]x1 + [0 0]x2, [1 2] [a__U22](x0) = [0 1]x0, [2 0] [a__isNat](x0) = [0 1]x0, [1 0] [3 2] [a__U21](x0, x1) = [0 0]x0 + [0 2]x1, [a__isNatList](x0) = x0, [0] [tt] = [2], [3 0] [3 0] [cons](x0, x1) = [3 0]x0 + [3 2]x1, [0] [zeros] = [0], [0] [0] = [2], [0] [a__zeros] = [0] orientation: [0] [0] a__zeros() = [0] >= [0] = cons(0(),zeros()) [3 2] [2 2] a__U21(tt(),V1) = [0 2]V1 >= [0 1]V1 = a__U22(a__isNat(V1)) [4] [0] a__U22(tt()) = [2] >= [2] = tt() [2 0] [2 0] [2 0] [2 0] a__U41(tt(),V1,V2) = [0 0]V1 + [0 0]V2 >= [0 0]V1 + [0 0]V2 = a__U42(a__isNat(V1),V2) [2 0] [2 0] a__U42(tt(),V2) = [0 0]V2 >= [0 0]V2 = a__U43(a__isNatIList(V2)) [2 0] [1 0] [2 0] [1 0] a__U51(tt(),V1,V2) = [3 0]V1 + [2 1]V2 >= [0 0]V1 + [1 0]V2 = a__U52(a__isNat(V1),V2) [1 0] [1 0] a__U52(tt(),V2) = [1 0]V2 >= [0 0]V2 = a__U53(a__isNatList(V2)) [2 2] [2 2] a__U61(tt(),L) = [0 0]L >= [0 0]L = s(a__length(mark(L))) a__and(tt(),X) = X >= X = mark(X) [0] [0] a__isNat(0()) = [2] >= [2] = tt() [4 2] [4 2] a__isNat(s(V1)) = [0 2]V1 >= [0 2]V1 = a__U21(a__isNatKind(V1),V1) [6 0] [6 0] [0] [3 0] [3 0] a__isNatIList(cons(V1,V2)) = [0 0]V1 + [0 0]V2 + [1] >= [0 0]V1 + [0 0]V2 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) [0] [0] a__isNatIListKind(zeros()) = [2] >= [2] = tt() [3 0] [3 0] [0] [1 0] [1 0] [0] a__isNatIListKind(cons(V1,V2)) = [0 0]V1 + [0 0]V2 + [2] >= [0 0]V1 + [0 0]V2 + [2] = a__and(a__isNatKind(V1),isNatIListKind(V2)) [0] [0] a__isNatKind(0()) = [2] >= [2] = tt() [1 1] [0] [1 0] [0] a__isNatKind(length(V1)) = [0 0]V1 + [2] >= [0 0]V1 + [2] = a__isNatIListKind(V1) [2 1] [0] [1 0] [0] a__isNatKind(s(V1)) = [0 0]V1 + [2] >= [0 0]V1 + [2] = a__isNatKind(V1) [3 0] [3 0] [3 0] [2 0] a__isNatList(cons(V1,V2)) = [3 0]V1 + [3 2]V2 >= [3 0]V1 + [2 1]V2 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) [6 2] [6 0] [6 2] [6 0] a__length(cons(N,L)) = [0 0]L + [0 0]N >= [0 0]L + [0 0]N = a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) [0] [0] mark(zeros()) = [0] >= [0] = a__zeros() mark(isNatList(X)) = X >= X = a__isNatList(X) [1 0] [3 2] [1 0] [3 2] mark(U21(X1,X2)) = [0 0]X1 + [0 2]X2 >= [0 0]X1 + [0 2]X2 = a__U21(mark(X1),X2) [1 2] [1 2] mark(U22(X)) = [0 1]X >= [0 1]X = a__U22(mark(X)) [2 0] [2 0] mark(isNat(X)) = [0 1]X >= [0 1]X = a__isNat(X) [1 0] [1 0] mark(U43(X)) = [0 0]X >= [0 0]X = a__U43(mark(X)) [1 0] [2 0] [1 0] [1 0] [2 0] [1 0] mark(U51(X1,X2,X3)) = [0 0]X1 + [3 0]X2 + [2 1]X3 >= [0 0]X1 + [3 0]X2 + [2 1]X3 = a__U51(mark(X1),X2,X3) [1 0] [1 0] [1 0] [1 0] mark(U52(X1,X2)) = [0 0]X1 + [1 0]X2 >= [0 0]X1 + [1 0]X2 = a__U52(mark(X1),X2) [1 0] [1 0] mark(U53(X)) = [0 0]X >= [0 0]X = a__U53(mark(X)) [2 0] [2 2] [2 0] [2 2] mark(U61(X1,X2)) = [0 0]X1 + [0 0]X2 >= [0 0]X1 + [0 0]X2 = a__U61(mark(X1),X2) [1 1] [1 1] mark(length(X)) = [0 0]X >= [0 0]X = a__length(mark(X)) [1 0] [1 0] mark(and(X1,X2)) = [0 0]X1 + X2 >= [0 0]X1 + X2 = a__and(mark(X1),X2) [1 0] [0] [1 0] [0] mark(isNatIListKind(X)) = [0 0]X + [2] >= [0 0]X + [2] = a__isNatIListKind(X) [1 0] [0] [1 0] [0] mark(isNatKind(X)) = [0 0]X + [2] >= [0 0]X + [2] = a__isNatKind(X) [3 0] [3 0] [3 0] [3 0] mark(cons(X1,X2)) = [3 0]X1 + [3 2]X2 >= [3 0]X1 + [3 2]X2 = cons(mark(X1),X2) [0] [0] mark(0()) = [2] >= [2] = 0() [0] [0] mark(tt()) = [2] >= [2] = tt() [2 1] [2 1] mark(s(X)) = [0 2]X >= [0 2]X = s(mark(X)) [0] [0] a__zeros() = [0] >= [0] = zeros() a__isNatList(X) = X >= X = isNatList(X) [1 0] [3 2] [1 0] [3 2] a__U21(X1,X2) = [0 0]X1 + [0 2]X2 >= [0 0]X1 + [0 2]X2 = U21(X1,X2) [1 2] [1 2] a__U22(X) = [0 1]X >= [0 1]X = U22(X) [2 0] [2 0] a__isNat(X) = [0 1]X >= [0 1]X = isNat(X) [1 0] [1 0] a__U43(X) = [0 0]X >= [0 0]X = U43(X) [1 0] [2 0] [1 0] [1 0] [2 0] [1 0] a__U51(X1,X2,X3) = [0 0]X1 + [3 0]X2 + [2 1]X3 >= [0 0]X1 + [3 0]X2 + [2 1]X3 = U51(X1,X2,X3) [1 0] [1 0] [1 0] [1 0] a__U52(X1,X2) = [0 0]X1 + [1 0]X2 >= [0 0]X1 + [1 0]X2 = U52(X1,X2) [1 0] [1 0] a__U53(X) = [0 0]X >= [0 0]X = U53(X) [2 0] [2 2] [2 0] [2 2] a__U61(X1,X2) = [0 0]X1 + [0 0]X2 >= [0 0]X1 + [0 0]X2 = U61(X1,X2) [1 1] [1 1] a__length(X) = [0 0]X >= [0 0]X = length(X) [1 0] [1 0] a__and(X1,X2) = [0 0]X1 + X2 >= [0 0]X1 + X2 = and(X1,X2) [1 0] [0] [1 0] [0] a__isNatIListKind(X) = [0 0]X + [2] >= [0 0]X + [2] = isNatIListKind(X) [1 0] [0] [1 0] [0] a__isNatKind(X) = [0 0]X + [2] >= [0 0]X + [2] = isNatKind(X) problem: a__zeros() -> cons(0(),zeros()) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U61(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() 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__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) mark(zeros()) -> a__zeros() mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U43(X) -> U43(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2) -> U61(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) DP Processor: DPs: a__U21#(tt(),V1) -> a__isNat#(V1) a__U21#(tt(),V1) -> a__U22#(a__isNat(V1)) a__U41#(tt(),V1,V2) -> a__isNat#(V1) a__U41#(tt(),V1,V2) -> a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) -> a__isNatIList#(V2) a__U42#(tt(),V2) -> a__U43#(a__isNatIList(V2)) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__U52#(tt(),V2) -> a__U53#(a__isNatList(V2)) a__U61#(tt(),L) -> mark#(L) a__U61#(tt(),L) -> a__length#(mark(L)) a__and#(tt(),X) -> mark#(X) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__isNatIList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatIList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIList#(cons(V1,V2)) -> a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) 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) a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length#(cons(N,L)) -> a__isNatList#(L) a__length#(cons(N,L)) -> a__and#(a__isNatList(L),isNatIListKind(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__U61#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) mark#(zeros()) -> a__zeros#() mark#(isNatList(X)) -> a__isNatList#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> a__U22#(mark(X)) mark#(isNat(X)) -> a__isNat#(X) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> a__U43#(mark(X)) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> a__U53#(mark(X)) mark#(U61(X1,X2)) -> mark#(X1) mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(length(X)) -> mark#(X) mark#(length(X)) -> a__length#(mark(X)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U61(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() 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__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) mark(zeros()) -> a__zeros() mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U43(X) -> U43(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2) -> U61(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) TDG Processor: DPs: a__U21#(tt(),V1) -> a__isNat#(V1) a__U21#(tt(),V1) -> a__U22#(a__isNat(V1)) a__U41#(tt(),V1,V2) -> a__isNat#(V1) a__U41#(tt(),V1,V2) -> a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) -> a__isNatIList#(V2) a__U42#(tt(),V2) -> a__U43#(a__isNatIList(V2)) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__U52#(tt(),V2) -> a__U53#(a__isNatList(V2)) a__U61#(tt(),L) -> mark#(L) a__U61#(tt(),L) -> a__length#(mark(L)) a__and#(tt(),X) -> mark#(X) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__isNatIList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatIList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIList#(cons(V1,V2)) -> a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) 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) a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length#(cons(N,L)) -> a__isNatList#(L) a__length#(cons(N,L)) -> a__and#(a__isNatList(L),isNatIListKind(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__U61#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) mark#(zeros()) -> a__zeros#() mark#(isNatList(X)) -> a__isNatList#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> a__U22#(mark(X)) mark#(isNat(X)) -> a__isNat#(X) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> a__U43#(mark(X)) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> a__U53#(mark(X)) mark#(U61(X1,X2)) -> mark#(X1) mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(length(X)) -> mark#(X) mark#(length(X)) -> a__length#(mark(X)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U61(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() 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__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) mark(zeros()) -> a__zeros() mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U43(X) -> U43(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2) -> U61(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) graph: a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) -> a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) -> a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) -> a__and#(tt(),X) -> mark#(X) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) -> a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) -> a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) -> a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) -> a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__and#(tt(),X) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__and#(tt(),X) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__and#(tt(),X) -> mark#(X) -> mark#(isNatKind(X)) -> a__isNatKind#(X) a__and#(tt(),X) -> mark#(X) -> mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) a__and#(tt(),X) -> mark#(X) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__and#(tt(),X) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) a__and#(tt(),X) -> mark#(X) -> mark#(length(X)) -> a__length#(mark(X)) a__and#(tt(),X) -> mark#(X) -> mark#(length(X)) -> mark#(X) a__and#(tt(),X) -> mark#(X) -> mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) a__and#(tt(),X) -> mark#(X) -> mark#(U61(X1,X2)) -> mark#(X1) a__and#(tt(),X) -> mark#(X) -> mark#(U53(X)) -> a__U53#(mark(X)) a__and#(tt(),X) -> mark#(X) -> mark#(U53(X)) -> mark#(X) a__and#(tt(),X) -> mark#(X) -> mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) a__and#(tt(),X) -> mark#(X) -> mark#(U52(X1,X2)) -> mark#(X1) a__and#(tt(),X) -> mark#(X) -> mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) a__and#(tt(),X) -> mark#(X) -> mark#(U51(X1,X2,X3)) -> mark#(X1) a__and#(tt(),X) -> mark#(X) -> mark#(U43(X)) -> a__U43#(mark(X)) a__and#(tt(),X) -> mark#(X) -> mark#(U43(X)) -> mark#(X) a__and#(tt(),X) -> mark#(X) -> mark#(isNat(X)) -> a__isNat#(X) a__and#(tt(),X) -> mark#(X) -> mark#(U22(X)) -> a__U22#(mark(X)) a__and#(tt(),X) -> mark#(X) -> mark#(U22(X)) -> mark#(X) a__and#(tt(),X) -> mark#(X) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) a__and#(tt(),X) -> mark#(X) -> mark#(U21(X1,X2)) -> mark#(X1) a__and#(tt(),X) -> mark#(X) -> mark#(isNatList(X)) -> a__isNatList#(X) a__and#(tt(),X) -> mark#(X) -> mark#(zeros()) -> a__zeros#() a__length#(cons(N,L)) -> a__and#(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) -> a__and#(tt(),X) -> mark#(X) a__length#(cons(N,L)) -> a__and#(a__isNatList(L),isNatIListKind(L)) -> a__and#(tt(),X) -> mark#(X) a__length#(cons(N,L)) -> a__U61#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) -> a__U61#(tt(),L) -> a__length#(mark(L)) a__length#(cons(N,L)) -> a__U61#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) -> a__U61#(tt(),L) -> mark#(L) a__length#(cons(N,L)) -> a__isNatList#(L) -> a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length#(cons(N,L)) -> a__isNatList#(L) -> a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__length#(cons(N,L)) -> a__isNatList#(L) -> a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> a__length#(mark(X)) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> mark#(X) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(U61(X1,X2)) -> mark#(X1) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(U53(X)) -> a__U53#(mark(X)) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(U53(X)) -> mark#(X) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(U52(X1,X2)) -> mark#(X1) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(U43(X)) -> a__U43#(mark(X)) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(U43(X)) -> mark#(X) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(U22(X)) -> a__U22#(mark(X)) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(U22(X)) -> mark#(X) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2)) -> mark#(X1) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(zeros()) -> a__zeros#() mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) -> a__U61#(tt(),L) -> a__length#(mark(L)) mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) -> a__U61#(tt(),L) -> mark#(L) mark#(U53(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(U53(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(U53(X)) -> mark#(X) -> mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(U53(X)) -> mark#(X) -> mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(U53(X)) -> mark#(X) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(U53(X)) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) mark#(U53(X)) -> mark#(X) -> mark#(length(X)) -> a__length#(mark(X)) mark#(U53(X)) -> mark#(X) -> mark#(length(X)) -> mark#(X) mark#(U53(X)) -> mark#(X) -> mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(U53(X)) -> mark#(X) -> mark#(U61(X1,X2)) -> mark#(X1) mark#(U53(X)) -> mark#(X) -> mark#(U53(X)) -> a__U53#(mark(X)) mark#(U53(X)) -> mark#(X) -> mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> mark#(X) -> mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U53(X)) -> mark#(X) -> mark#(U52(X1,X2)) -> mark#(X1) mark#(U53(X)) -> mark#(X) -> mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U53(X)) -> mark#(X) -> mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U53(X)) -> mark#(X) -> mark#(U43(X)) -> a__U43#(mark(X)) mark#(U53(X)) -> mark#(X) -> mark#(U43(X)) -> mark#(X) mark#(U53(X)) -> mark#(X) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U53(X)) -> mark#(X) -> mark#(U22(X)) -> a__U22#(mark(X)) mark#(U53(X)) -> mark#(X) -> mark#(U22(X)) -> mark#(X) mark#(U53(X)) -> mark#(X) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U53(X)) -> mark#(X) -> mark#(U21(X1,X2)) -> mark#(X1) mark#(U53(X)) -> mark#(X) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(U53(X)) -> mark#(X) -> mark#(zeros()) -> a__zeros#() mark#(U52(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> a__length#(mark(X)) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> mark#(X) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(U61(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(U53(X)) -> a__U53#(mark(X)) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(U53(X)) -> mark#(X) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(U43(X)) -> a__U43#(mark(X)) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(U43(X)) -> mark#(X) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(U22(X)) -> a__U22#(mark(X)) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(U22(X)) -> mark#(X) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(zeros()) -> a__zeros#() mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) -> a__U52#(tt(),V2) -> a__U53#(a__isNatList(V2)) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) -> a__U52#(tt(),V2) -> a__isNatList#(V2) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(length(X)) -> a__length#(mark(X)) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(length(X)) -> mark#(X) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(U61(X1,X2)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(U53(X)) -> a__U53#(mark(X)) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(U53(X)) -> mark#(X) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(U52(X1,X2)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(U43(X)) -> a__U43#(mark(X)) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(U43(X)) -> mark#(X) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(U22(X)) -> a__U22#(mark(X)) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(U22(X)) -> mark#(X) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(U21(X1,X2)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(zeros()) -> a__zeros#() mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) -> a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) -> a__U51#(tt(),V1,V2) -> a__isNat#(V1) mark#(U43(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(U43(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(U43(X)) -> mark#(X) -> mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(U43(X)) -> mark#(X) -> mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(U43(X)) -> mark#(X) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(U43(X)) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) mark#(U43(X)) -> mark#(X) -> mark#(length(X)) -> a__length#(mark(X)) mark#(U43(X)) -> mark#(X) -> mark#(length(X)) -> mark#(X) mark#(U43(X)) -> mark#(X) -> mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(U43(X)) -> mark#(X) -> mark#(U61(X1,X2)) -> mark#(X1) mark#(U43(X)) -> mark#(X) -> mark#(U53(X)) -> a__U53#(mark(X)) mark#(U43(X)) -> mark#(X) -> mark#(U53(X)) -> mark#(X) mark#(U43(X)) -> mark#(X) -> mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U43(X)) -> mark#(X) -> mark#(U52(X1,X2)) -> mark#(X1) mark#(U43(X)) -> mark#(X) -> mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U43(X)) -> mark#(X) -> mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U43(X)) -> mark#(X) -> mark#(U43(X)) -> a__U43#(mark(X)) mark#(U43(X)) -> mark#(X) -> mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> mark#(X) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U43(X)) -> mark#(X) -> mark#(U22(X)) -> a__U22#(mark(X)) mark#(U43(X)) -> mark#(X) -> mark#(U22(X)) -> mark#(X) mark#(U43(X)) -> mark#(X) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U43(X)) -> mark#(X) -> mark#(U21(X1,X2)) -> mark#(X1) mark#(U43(X)) -> mark#(X) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(U43(X)) -> mark#(X) -> mark#(zeros()) -> a__zeros#() mark#(U22(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(U22(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(U22(X)) -> mark#(X) -> mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(U22(X)) -> mark#(X) -> mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(U22(X)) -> mark#(X) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(U22(X)) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) mark#(U22(X)) -> mark#(X) -> mark#(length(X)) -> a__length#(mark(X)) mark#(U22(X)) -> mark#(X) -> mark#(length(X)) -> mark#(X) mark#(U22(X)) -> mark#(X) -> mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(U22(X)) -> mark#(X) -> mark#(U61(X1,X2)) -> mark#(X1) mark#(U22(X)) -> mark#(X) -> mark#(U53(X)) -> a__U53#(mark(X)) mark#(U22(X)) -> mark#(X) -> mark#(U53(X)) -> mark#(X) mark#(U22(X)) -> mark#(X) -> mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U22(X)) -> mark#(X) -> mark#(U52(X1,X2)) -> mark#(X1) mark#(U22(X)) -> mark#(X) -> mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U22(X)) -> mark#(X) -> mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U22(X)) -> mark#(X) -> mark#(U43(X)) -> a__U43#(mark(X)) mark#(U22(X)) -> mark#(X) -> mark#(U43(X)) -> mark#(X) mark#(U22(X)) -> mark#(X) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U22(X)) -> mark#(X) -> mark#(U22(X)) -> a__U22#(mark(X)) mark#(U22(X)) -> mark#(X) -> mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> mark#(X) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X)) -> mark#(X) -> mark#(U21(X1,X2)) -> mark#(X1) mark#(U22(X)) -> mark#(X) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(U22(X)) -> mark#(X) -> mark#(zeros()) -> a__zeros#() mark#(U21(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> a__length#(mark(X)) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> mark#(X) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(U61(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(U53(X)) -> a__U53#(mark(X)) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(U53(X)) -> mark#(X) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(U52(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(U43(X)) -> a__U43#(mark(X)) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(U43(X)) -> mark#(X) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(U22(X)) -> a__U22#(mark(X)) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(U22(X)) -> mark#(X) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(zeros()) -> a__zeros#() mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) -> a__U21#(tt(),V1) -> a__U22#(a__isNat(V1)) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) -> a__U21#(tt(),V1) -> a__isNat#(V1) mark#(isNatList(X)) -> a__isNatList#(X) -> a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) mark#(isNatList(X)) -> a__isNatList#(X) -> a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) mark#(isNatList(X)) -> a__isNatList#(X) -> a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) -> a__and#(tt(),X) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> a__length#(mark(X)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U61(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U53(X)) -> a__U53#(mark(X)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U53(X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U52(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U43(X)) -> a__U43#(mark(X)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U43(X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U22(X)) -> a__U22#(mark(X)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U22(X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(zeros()) -> a__zeros#() mark#(isNatKind(X)) -> a__isNatKind#(X) -> a__isNatKind#(s(V1)) -> a__isNatKind#(V1) mark#(isNatKind(X)) -> a__isNatKind#(X) -> a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) mark#(isNat(X)) -> a__isNat#(X) -> a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) mark#(isNat(X)) -> a__isNat#(X) -> a__isNat#(s(V1)) -> a__isNatKind#(V1) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) -> a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) -> a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) mark#(length(X)) -> a__length#(mark(X)) -> a__length#(cons(N,L)) -> a__U61#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) mark#(length(X)) -> a__length#(mark(X)) -> a__length#(cons(N,L)) -> a__and#(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) mark#(length(X)) -> a__length#(mark(X)) -> a__length#(cons(N,L)) -> a__and#(a__isNatList(L),isNatIListKind(L)) mark#(length(X)) -> a__length#(mark(X)) -> a__length#(cons(N,L)) -> a__isNatList#(L) mark#(length(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(length(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(length(X)) -> mark#(X) -> mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(length(X)) -> mark#(X) -> mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(length(X)) -> mark#(X) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(length(X)) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) mark#(length(X)) -> mark#(X) -> mark#(length(X)) -> a__length#(mark(X)) mark#(length(X)) -> mark#(X) -> mark#(length(X)) -> mark#(X) mark#(length(X)) -> mark#(X) -> mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(length(X)) -> mark#(X) -> mark#(U61(X1,X2)) -> mark#(X1) mark#(length(X)) -> mark#(X) -> mark#(U53(X)) -> a__U53#(mark(X)) mark#(length(X)) -> mark#(X) -> mark#(U53(X)) -> mark#(X) mark#(length(X)) -> mark#(X) -> mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(length(X)) -> mark#(X) -> mark#(U52(X1,X2)) -> mark#(X1) mark#(length(X)) -> mark#(X) -> mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(length(X)) -> mark#(X) -> mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(length(X)) -> mark#(X) -> mark#(U43(X)) -> a__U43#(mark(X)) mark#(length(X)) -> mark#(X) -> mark#(U43(X)) -> mark#(X) mark#(length(X)) -> mark#(X) -> mark#(isNat(X)) -> a__isNat#(X) mark#(length(X)) -> mark#(X) -> mark#(U22(X)) -> a__U22#(mark(X)) mark#(length(X)) -> mark#(X) -> mark#(U22(X)) -> mark#(X) mark#(length(X)) -> mark#(X) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(length(X)) -> mark#(X) -> mark#(U21(X1,X2)) -> mark#(X1) mark#(length(X)) -> mark#(X) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(length(X)) -> mark#(X) -> mark#(zeros()) -> a__zeros#() mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(s(X)) -> mark#(X) -> mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(s(X)) -> mark#(X) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(s(X)) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(length(X)) -> a__length#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(length(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(s(X)) -> mark#(X) -> mark#(U61(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(U53(X)) -> a__U53#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(U53(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(s(X)) -> mark#(X) -> mark#(U52(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(s(X)) -> mark#(X) -> mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(U43(X)) -> a__U43#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(U43(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(isNat(X)) -> a__isNat#(X) mark#(s(X)) -> mark#(X) -> mark#(U22(X)) -> a__U22#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(U22(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(s(X)) -> mark#(X) -> mark#(U21(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(s(X)) -> mark#(X) -> mark#(zeros()) -> a__zeros#() mark#(cons(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> a__length#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U61(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U53(X)) -> a__U53#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U53(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U52(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U43(X)) -> a__U43#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U43(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U22(X)) -> a__U22#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U22(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(zeros()) -> a__zeros#() a__U61#(tt(),L) -> a__length#(mark(L)) -> a__length#(cons(N,L)) -> a__U61#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) a__U61#(tt(),L) -> a__length#(mark(L)) -> a__length#(cons(N,L)) -> a__and#(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) a__U61#(tt(),L) -> a__length#(mark(L)) -> a__length#(cons(N,L)) -> a__and#(a__isNatList(L),isNatIListKind(L)) a__U61#(tt(),L) -> a__length#(mark(L)) -> a__length#(cons(N,L)) -> a__isNatList#(L) a__U61#(tt(),L) -> mark#(L) -> mark#(s(X)) -> mark#(X) a__U61#(tt(),L) -> mark#(L) -> mark#(cons(X1,X2)) -> mark#(X1) a__U61#(tt(),L) -> mark#(L) -> mark#(isNatKind(X)) -> a__isNatKind#(X) a__U61#(tt(),L) -> mark#(L) -> mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) a__U61#(tt(),L) -> mark#(L) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__U61#(tt(),L) -> mark#(L) -> mark#(and(X1,X2)) -> mark#(X1) a__U61#(tt(),L) -> mark#(L) -> mark#(length(X)) -> a__length#(mark(X)) a__U61#(tt(),L) -> mark#(L) -> mark#(length(X)) -> mark#(X) a__U61#(tt(),L) -> mark#(L) -> mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) a__U61#(tt(),L) -> mark#(L) -> mark#(U61(X1,X2)) -> mark#(X1) a__U61#(tt(),L) -> mark#(L) -> mark#(U53(X)) -> a__U53#(mark(X)) a__U61#(tt(),L) -> mark#(L) -> mark#(U53(X)) -> mark#(X) a__U61#(tt(),L) -> mark#(L) -> mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) a__U61#(tt(),L) -> mark#(L) -> mark#(U52(X1,X2)) -> mark#(X1) a__U61#(tt(),L) -> mark#(L) -> mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) a__U61#(tt(),L) -> mark#(L) -> mark#(U51(X1,X2,X3)) -> mark#(X1) a__U61#(tt(),L) -> mark#(L) -> mark#(U43(X)) -> a__U43#(mark(X)) a__U61#(tt(),L) -> mark#(L) -> mark#(U43(X)) -> mark#(X) a__U61#(tt(),L) -> mark#(L) -> mark#(isNat(X)) -> a__isNat#(X) a__U61#(tt(),L) -> mark#(L) -> mark#(U22(X)) -> a__U22#(mark(X)) a__U61#(tt(),L) -> mark#(L) -> mark#(U22(X)) -> mark#(X) a__U61#(tt(),L) -> mark#(L) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) a__U61#(tt(),L) -> mark#(L) -> mark#(U21(X1,X2)) -> mark#(X1) a__U61#(tt(),L) -> mark#(L) -> mark#(isNatList(X)) -> a__isNatList#(X) a__U61#(tt(),L) -> mark#(L) -> mark#(zeros()) -> a__zeros#() a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) -> a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) -> a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) -> a__and#(tt(),X) -> mark#(X) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) -> a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) -> a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__U52#(tt(),V2) -> a__isNatList#(V2) -> a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U52#(tt(),V2) -> a__isNatList#(V2) -> a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__U52#(tt(),V2) -> a__isNatList#(V2) -> a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) -> a__U52#(tt(),V2) -> a__U53#(a__isNatList(V2)) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) -> a__U52#(tt(),V2) -> a__isNatList#(V2) a__U51#(tt(),V1,V2) -> a__isNat#(V1) -> a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U51#(tt(),V1,V2) -> a__isNat#(V1) -> a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNatIList#(cons(V1,V2)) -> a__isNatKind#(V1) -> a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatIList#(cons(V1,V2)) -> a__isNatKind#(V1) -> a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) -> a__and#(tt(),X) -> mark#(X) a__isNatIList#(cons(V1,V2)) -> a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) -> a__U41#(tt(),V1,V2) -> a__U42#(a__isNat(V1),V2) a__isNatIList#(cons(V1,V2)) -> a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) -> a__U41#(tt(),V1,V2) -> a__isNat#(V1) a__U42#(tt(),V2) -> a__isNatIList#(V2) -> a__isNatIList#(cons(V1,V2)) -> a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U42#(tt(),V2) -> a__isNatIList#(V2) -> a__isNatIList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__U42#(tt(),V2) -> a__isNatIList#(V2) -> a__isNatIList#(cons(V1,V2)) -> a__isNatKind#(V1) a__U41#(tt(),V1,V2) -> a__U42#(a__isNat(V1),V2) -> a__U42#(tt(),V2) -> a__U43#(a__isNatIList(V2)) a__U41#(tt(),V1,V2) -> a__U42#(a__isNat(V1),V2) -> a__U42#(tt(),V2) -> a__isNatIList#(V2) a__U41#(tt(),V1,V2) -> a__isNat#(V1) -> a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U41#(tt(),V1,V2) -> a__isNat#(V1) -> a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) -> a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) -> a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) -> a__U21#(tt(),V1) -> a__U22#(a__isNat(V1)) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) -> a__U21#(tt(),V1) -> a__isNat#(V1) a__U21#(tt(),V1) -> a__isNat#(V1) -> a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) -> a__isNat#(s(V1)) -> a__isNatKind#(V1) SCC Processor: #sccs: 2 #rules: 44 #arcs: 395/2916 DPs: a__isNatIList#(cons(V1,V2)) -> a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U41#(tt(),V1,V2) -> a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) -> a__isNatIList#(V2) TRS: a__zeros() -> cons(0(),zeros()) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U61(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() 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__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) mark(zeros()) -> a__zeros() mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U43(X) -> U43(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2) -> U61(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Subterm Criterion Processor: simple projection: pi(a__U41#) = 2 pi(a__U42#) = 1 pi(a__isNatIList#) = 0 problem: DPs: a__U41#(tt(),V1,V2) -> a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) -> a__isNatIList#(V2) TRS: a__zeros() -> cons(0(),zeros()) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U61(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() 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__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) mark(zeros()) -> a__zeros() mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U43(X) -> U43(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2) -> U61(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) SCC Processor: #sccs: 0 #rules: 0 #arcs: 3/4 DPs: a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__and#(tt(),X) -> mark#(X) mark#(isNatList(X)) -> a__isNatList#(X) a__isNatList#(cons(V1,V2)) -> a__isNatKind#(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__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X)) -> mark#(X) mark#(isNat(X)) -> a__isNat#(X) mark#(U43(X)) -> mark#(X) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U53(X)) -> mark#(X) mark#(U61(X1,X2)) -> mark#(X1) mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) a__U61#(tt(),L) -> mark#(L) mark#(length(X)) -> mark#(X) mark#(length(X)) -> a__length#(mark(X)) a__length#(cons(N,L)) -> a__isNatList#(L) a__length#(cons(N,L)) -> a__and#(a__isNatList(L),isNatIListKind(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__U61#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) a__U61#(tt(),L) -> a__length#(mark(L)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U61(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() 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__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) mark(zeros()) -> a__zeros() mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U43(X) -> U43(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2) -> U61(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) EDG Processor: DPs: a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__and#(tt(),X) -> mark#(X) mark#(isNatList(X)) -> a__isNatList#(X) a__isNatList#(cons(V1,V2)) -> a__isNatKind#(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__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X)) -> mark#(X) mark#(isNat(X)) -> a__isNat#(X) mark#(U43(X)) -> mark#(X) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U53(X)) -> mark#(X) mark#(U61(X1,X2)) -> mark#(X1) mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) a__U61#(tt(),L) -> mark#(L) mark#(length(X)) -> mark#(X) mark#(length(X)) -> a__length#(mark(X)) a__length#(cons(N,L)) -> a__isNatList#(L) a__length#(cons(N,L)) -> a__and#(a__isNatList(L),isNatIListKind(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__U61#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) a__U61#(tt(),L) -> a__length#(mark(L)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U61(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() 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__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) mark(zeros()) -> a__zeros() mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U43(X) -> U43(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2) -> U61(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) graph: a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) -> a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) -> a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) -> a__and#(tt(),X) -> mark#(X) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) -> a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) -> a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) -> a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) -> a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__and#(tt(),X) -> mark#(X) -> mark#(isNatList(X)) -> a__isNatList#(X) a__and#(tt(),X) -> mark#(X) -> mark#(U21(X1,X2)) -> mark#(X1) a__and#(tt(),X) -> mark#(X) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) a__and#(tt(),X) -> mark#(X) -> mark#(U22(X)) -> mark#(X) a__and#(tt(),X) -> mark#(X) -> mark#(isNat(X)) -> a__isNat#(X) a__and#(tt(),X) -> mark#(X) -> mark#(U43(X)) -> mark#(X) a__and#(tt(),X) -> mark#(X) -> mark#(U51(X1,X2,X3)) -> mark#(X1) a__and#(tt(),X) -> mark#(X) -> mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) a__and#(tt(),X) -> mark#(X) -> mark#(U52(X1,X2)) -> mark#(X1) a__and#(tt(),X) -> mark#(X) -> mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) a__and#(tt(),X) -> mark#(X) -> mark#(U53(X)) -> mark#(X) a__and#(tt(),X) -> mark#(X) -> mark#(U61(X1,X2)) -> mark#(X1) a__and#(tt(),X) -> mark#(X) -> mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) a__and#(tt(),X) -> mark#(X) -> mark#(length(X)) -> mark#(X) a__and#(tt(),X) -> mark#(X) -> mark#(length(X)) -> a__length#(mark(X)) a__and#(tt(),X) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) a__and#(tt(),X) -> mark#(X) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__and#(tt(),X) -> mark#(X) -> mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) a__and#(tt(),X) -> mark#(X) -> mark#(isNatKind(X)) -> a__isNatKind#(X) a__and#(tt(),X) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__and#(tt(),X) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__length#(cons(N,L)) -> a__and#(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))) -> a__and#(tt(),X) -> mark#(X) a__length#(cons(N,L)) -> a__U61#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) -> a__U61#(tt(),L) -> mark#(L) a__length#(cons(N,L)) -> a__U61#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) -> a__U61#(tt(),L) -> a__length#(mark(L)) a__length#(cons(N,L)) -> a__isNatList#(L) -> a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) a__length#(cons(N,L)) -> a__isNatList#(L) -> a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__length#(cons(N,L)) -> a__isNatList#(L) -> a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2)) -> mark#(X1) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(U22(X)) -> mark#(X) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(U43(X)) -> mark#(X) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(U52(X1,X2)) -> mark#(X1) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(U53(X)) -> mark#(X) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(U61(X1,X2)) -> mark#(X1) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> mark#(X) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> a__length#(mark(X)) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(U61(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) -> a__U61#(tt(),L) -> mark#(L) mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) -> a__U61#(tt(),L) -> a__length#(mark(L)) mark#(U53(X)) -> mark#(X) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(U53(X)) -> mark#(X) -> mark#(U21(X1,X2)) -> mark#(X1) mark#(U53(X)) -> mark#(X) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U53(X)) -> mark#(X) -> mark#(U22(X)) -> mark#(X) mark#(U53(X)) -> mark#(X) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U53(X)) -> mark#(X) -> mark#(U43(X)) -> mark#(X) mark#(U53(X)) -> mark#(X) -> mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U53(X)) -> mark#(X) -> mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U53(X)) -> mark#(X) -> mark#(U52(X1,X2)) -> mark#(X1) mark#(U53(X)) -> mark#(X) -> mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U53(X)) -> mark#(X) -> mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> mark#(X) -> mark#(U61(X1,X2)) -> mark#(X1) mark#(U53(X)) -> mark#(X) -> mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(U53(X)) -> mark#(X) -> mark#(length(X)) -> mark#(X) mark#(U53(X)) -> mark#(X) -> mark#(length(X)) -> a__length#(mark(X)) mark#(U53(X)) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) mark#(U53(X)) -> mark#(X) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(U53(X)) -> mark#(X) -> mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(U53(X)) -> mark#(X) -> mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(U53(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(U53(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(U22(X)) -> mark#(X) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(U43(X)) -> mark#(X) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(U53(X)) -> mark#(X) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(U61(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> mark#(X) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> a__length#(mark(X)) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) -> a__U52#(tt(),V2) -> a__isNatList#(V2) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(U21(X1,X2)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(U22(X)) -> mark#(X) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(U43(X)) -> mark#(X) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(U52(X1,X2)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(U53(X)) -> mark#(X) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(U61(X1,X2)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(length(X)) -> mark#(X) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(length(X)) -> a__length#(mark(X)) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) -> a__U51#(tt(),V1,V2) -> a__isNat#(V1) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) -> a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) mark#(U43(X)) -> mark#(X) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(U43(X)) -> mark#(X) -> mark#(U21(X1,X2)) -> mark#(X1) mark#(U43(X)) -> mark#(X) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U43(X)) -> mark#(X) -> mark#(U22(X)) -> mark#(X) mark#(U43(X)) -> mark#(X) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U43(X)) -> mark#(X) -> mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> mark#(X) -> mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U43(X)) -> mark#(X) -> mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U43(X)) -> mark#(X) -> mark#(U52(X1,X2)) -> mark#(X1) mark#(U43(X)) -> mark#(X) -> mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U43(X)) -> mark#(X) -> mark#(U53(X)) -> mark#(X) mark#(U43(X)) -> mark#(X) -> mark#(U61(X1,X2)) -> mark#(X1) mark#(U43(X)) -> mark#(X) -> mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(U43(X)) -> mark#(X) -> mark#(length(X)) -> mark#(X) mark#(U43(X)) -> mark#(X) -> mark#(length(X)) -> a__length#(mark(X)) mark#(U43(X)) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) mark#(U43(X)) -> mark#(X) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(U43(X)) -> mark#(X) -> mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(U43(X)) -> mark#(X) -> mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(U43(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(U43(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(U22(X)) -> mark#(X) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(U22(X)) -> mark#(X) -> mark#(U21(X1,X2)) -> mark#(X1) mark#(U22(X)) -> mark#(X) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X)) -> mark#(X) -> mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> mark#(X) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U22(X)) -> mark#(X) -> mark#(U43(X)) -> mark#(X) mark#(U22(X)) -> mark#(X) -> mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U22(X)) -> mark#(X) -> mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U22(X)) -> mark#(X) -> mark#(U52(X1,X2)) -> mark#(X1) mark#(U22(X)) -> mark#(X) -> mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U22(X)) -> mark#(X) -> mark#(U53(X)) -> mark#(X) mark#(U22(X)) -> mark#(X) -> mark#(U61(X1,X2)) -> mark#(X1) mark#(U22(X)) -> mark#(X) -> mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(U22(X)) -> mark#(X) -> mark#(length(X)) -> mark#(X) mark#(U22(X)) -> mark#(X) -> mark#(length(X)) -> a__length#(mark(X)) mark#(U22(X)) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) mark#(U22(X)) -> mark#(X) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(U22(X)) -> mark#(X) -> mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(U22(X)) -> mark#(X) -> mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(U22(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(U22(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(U22(X)) -> mark#(X) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(U43(X)) -> mark#(X) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(U52(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(U53(X)) -> mark#(X) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(U61(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> mark#(X) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> a__length#(mark(X)) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) -> a__U21#(tt(),V1) -> a__isNat#(V1) mark#(isNatList(X)) -> a__isNatList#(X) -> a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) mark#(isNatList(X)) -> a__isNatList#(X) -> a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) mark#(isNatList(X)) -> a__isNatList#(X) -> a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) -> a__and#(tt(),X) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U22(X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U43(X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U52(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U53(X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U61(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> a__length#(mark(X)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) -> mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(isNatKind(X)) -> a__isNatKind#(X) -> a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) mark#(isNatKind(X)) -> a__isNatKind#(X) -> a__isNatKind#(s(V1)) -> a__isNatKind#(V1) mark#(isNat(X)) -> a__isNat#(X) -> a__isNat#(s(V1)) -> a__isNatKind#(V1) mark#(isNat(X)) -> a__isNat#(X) -> a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) -> a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) -> a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) mark#(length(X)) -> a__length#(mark(X)) -> a__length#(cons(N,L)) -> a__isNatList#(L) mark#(length(X)) -> a__length#(mark(X)) -> a__length#(cons(N,L)) -> a__and#(a__isNatList(L),isNatIListKind(L)) mark#(length(X)) -> a__length#(mark(X)) -> a__length#(cons(N,L)) -> a__and#(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))) mark#(length(X)) -> a__length#(mark(X)) -> a__length#(cons(N,L)) -> a__U61#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) mark#(length(X)) -> mark#(X) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(length(X)) -> mark#(X) -> mark#(U21(X1,X2)) -> mark#(X1) mark#(length(X)) -> mark#(X) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(length(X)) -> mark#(X) -> mark#(U22(X)) -> mark#(X) mark#(length(X)) -> mark#(X) -> mark#(isNat(X)) -> a__isNat#(X) mark#(length(X)) -> mark#(X) -> mark#(U43(X)) -> mark#(X) mark#(length(X)) -> mark#(X) -> mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(length(X)) -> mark#(X) -> mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(length(X)) -> mark#(X) -> mark#(U52(X1,X2)) -> mark#(X1) mark#(length(X)) -> mark#(X) -> mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(length(X)) -> mark#(X) -> mark#(U53(X)) -> mark#(X) mark#(length(X)) -> mark#(X) -> mark#(U61(X1,X2)) -> mark#(X1) mark#(length(X)) -> mark#(X) -> mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(length(X)) -> mark#(X) -> mark#(length(X)) -> mark#(X) mark#(length(X)) -> mark#(X) -> mark#(length(X)) -> a__length#(mark(X)) mark#(length(X)) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) mark#(length(X)) -> mark#(X) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(length(X)) -> mark#(X) -> mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(length(X)) -> mark#(X) -> mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(length(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(length(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(s(X)) -> mark#(X) -> mark#(U21(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(s(X)) -> mark#(X) -> mark#(U22(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(isNat(X)) -> a__isNat#(X) mark#(s(X)) -> mark#(X) -> mark#(U43(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(s(X)) -> mark#(X) -> mark#(U52(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(s(X)) -> mark#(X) -> mark#(U53(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(U61(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(s(X)) -> mark#(X) -> mark#(length(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(length(X)) -> a__length#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(s(X)) -> mark#(X) -> mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(s(X)) -> mark#(X) -> mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(s(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U22(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U43(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U52(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U53(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U61(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> a__length#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) a__U61#(tt(),L) -> a__length#(mark(L)) -> a__length#(cons(N,L)) -> a__isNatList#(L) a__U61#(tt(),L) -> a__length#(mark(L)) -> a__length#(cons(N,L)) -> a__and#(a__isNatList(L),isNatIListKind(L)) a__U61#(tt(),L) -> a__length#(mark(L)) -> a__length#(cons(N,L)) -> a__and#(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))) a__U61#(tt(),L) -> a__length#(mark(L)) -> a__length#(cons(N,L)) -> a__U61#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) a__U61#(tt(),L) -> mark#(L) -> mark#(isNatList(X)) -> a__isNatList#(X) a__U61#(tt(),L) -> mark#(L) -> mark#(U21(X1,X2)) -> mark#(X1) a__U61#(tt(),L) -> mark#(L) -> mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) a__U61#(tt(),L) -> mark#(L) -> mark#(U22(X)) -> mark#(X) a__U61#(tt(),L) -> mark#(L) -> mark#(isNat(X)) -> a__isNat#(X) a__U61#(tt(),L) -> mark#(L) -> mark#(U43(X)) -> mark#(X) a__U61#(tt(),L) -> mark#(L) -> mark#(U51(X1,X2,X3)) -> mark#(X1) a__U61#(tt(),L) -> mark#(L) -> mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) a__U61#(tt(),L) -> mark#(L) -> mark#(U52(X1,X2)) -> mark#(X1) a__U61#(tt(),L) -> mark#(L) -> mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) a__U61#(tt(),L) -> mark#(L) -> mark#(U53(X)) -> mark#(X) a__U61#(tt(),L) -> mark#(L) -> mark#(U61(X1,X2)) -> mark#(X1) a__U61#(tt(),L) -> mark#(L) -> mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) a__U61#(tt(),L) -> mark#(L) -> mark#(length(X)) -> mark#(X) a__U61#(tt(),L) -> mark#(L) -> mark#(length(X)) -> a__length#(mark(X)) a__U61#(tt(),L) -> mark#(L) -> mark#(and(X1,X2)) -> mark#(X1) a__U61#(tt(),L) -> mark#(L) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__U61#(tt(),L) -> mark#(L) -> mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) a__U61#(tt(),L) -> mark#(L) -> mark#(isNatKind(X)) -> a__isNatKind#(X) a__U61#(tt(),L) -> mark#(L) -> mark#(cons(X1,X2)) -> mark#(X1) a__U61#(tt(),L) -> mark#(L) -> mark#(s(X)) -> mark#(X) a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) -> a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) -> a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) -> a__and#(tt(),X) -> mark#(X) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) -> a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) -> a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) -> a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) a__U52#(tt(),V2) -> a__isNatList#(V2) -> a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__U52#(tt(),V2) -> a__isNatList#(V2) -> a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) -> a__U52#(tt(),V2) -> a__isNatList#(V2) a__U51#(tt(),V1,V2) -> a__isNat#(V1) -> a__isNat#(s(V1)) -> a__isNatKind#(V1) a__U51#(tt(),V1,V2) -> a__isNat#(V1) -> a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) -> a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) -> a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) -> a__U21#(tt(),V1) -> a__isNat#(V1) a__U21#(tt(),V1) -> a__isNat#(V1) -> a__isNat#(s(V1)) -> a__isNatKind#(V1) a__U21#(tt(),V1) -> a__isNat#(V1) -> a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) SCC Processor: #sccs: 1 #rules: 40 #arcs: 326/1681 DPs: a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__and#(tt(),X) -> mark#(X) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) mark#(length(X)) -> a__length#(mark(X)) a__length#(cons(N,L)) -> a__U61#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__U61#(tt(),L) -> a__length#(mark(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__isNatList#(L) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__U61#(tt(),L) -> mark#(L) mark#(length(X)) -> mark#(X) mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(U61(X1,X2)) -> mark#(X1) mark#(U53(X)) -> mark#(X) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U52(X1,X2)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U43(X)) -> mark#(X) mark#(isNat(X)) -> a__isNat#(X) mark#(U22(X)) -> mark#(X) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U21(X1,X2)) -> mark#(X1) mark#(isNatList(X)) -> a__isNatList#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U61(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() 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__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) mark(zeros()) -> a__zeros() mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U43(X) -> U43(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2) -> U61(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Usable Rule Processor: DPs: a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__and#(tt(),X) -> mark#(X) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) mark#(length(X)) -> a__length#(mark(X)) a__length#(cons(N,L)) -> a__U61#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__U61#(tt(),L) -> a__length#(mark(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__isNatList#(L) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__U61#(tt(),L) -> mark#(L) mark#(length(X)) -> mark#(X) mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(U61(X1,X2)) -> mark#(X1) mark#(U53(X)) -> mark#(X) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U52(X1,X2)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U43(X)) -> mark#(X) mark#(isNat(X)) -> a__isNat#(X) mark#(U22(X)) -> mark#(X) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U21(X1,X2)) -> mark#(X1) mark#(isNatList(X)) -> a__isNatList#(X) TRS: a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(X) -> isNatIListKind(X) a__and(tt(),X) -> mark(X) a__and(X1,X2) -> and(X1,X2) mark(zeros()) -> a__zeros() mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(X) -> isNatList(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__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__U53(X) -> U53(X) a__U43(X) -> U43(X) a__U61(tt(),L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__length(X) -> length(X) Matrix Interpretation Processor: dim=1 usable rules: a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(X) -> isNatIListKind(X) a__and(tt(),X) -> mark(X) a__and(X1,X2) -> and(X1,X2) mark(zeros()) -> a__zeros() mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(X) -> isNatList(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__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__U53(X) -> U53(X) a__U43(X) -> U43(X) a__U61(tt(),L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__length(X) -> length(X) interpretation: [a__isNatIListKind#](x0) = 1, [a__isNatKind#](x0) = 1, [a__and#](x0, x1) = x0, [a__length#](x0) = 1, [mark#](x0) = 1, [a__U61#](x0, x1) = 1, [a__isNatList#](x0) = 1, [a__U52#](x0, x1) = 1, [a__U51#](x0, x1, x2) = 1, [a__isNat#](x0) = 1, [a__U21#](x0, x1) = 1, [U61](x0, x1) = x0, [U53](x0) = 0, [U52](x0, x1) = 0, [U51](x0, x1, x2) = 0, [U43](x0) = 0, [U22](x0) = 0, [U21](x0, x1) = 0, [isNatList](x0) = 0, [and](x0, x1) = 0, [isNatKind](x0) = 0, [isNat](x0) = 1, [isNatIListKind](x0) = 0, [a__isNatKind](x0) = 1, [a__isNatIListKind](x0) = 1, [length](x0) = 0, [a__and](x0, x1) = x0, [s](x0) = 1, [a__length](x0) = 0, [mark](x0) = 1, [a__U61](x0, x1) = x0, [a__U53](x0) = 0, [a__U52](x0, x1) = 0, [a__U51](x0, x1, x2) = 0, [a__U43](x0) = 1, [a__U22](x0) = 0, [a__isNat](x0) = 1, [a__U21](x0, x1) = 1, [a__isNatList](x0) = 0, [tt] = 1, [cons](x0, x1) = 0, [zeros] = 1, [0] = 0, [a__zeros] = 1 orientation: a__isNatIListKind#(cons(V1,V2)) = 1 >= 1 = a__isNatKind#(V1) a__isNatKind#(s(V1)) = 1 >= 1 = a__isNatKind#(V1) a__isNatKind#(length(V1)) = 1 >= 1 = a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 1 >= 1 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__and#(tt(),X) = 1 >= 1 = mark#(X) mark#(s(X)) = 1 >= 1 = mark#(X) mark#(cons(X1,X2)) = 1 >= 1 = mark#(X1) mark#(isNatKind(X)) = 1 >= 1 = a__isNatKind#(X) mark#(isNatIListKind(X)) = 1 >= 1 = a__isNatIListKind#(X) mark#(and(X1,X2)) = 1 >= 1 = a__and#(mark(X1),X2) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(length(X)) = 1 >= 1 = a__length#(mark(X)) a__length#(cons(N,L)) = 1 >= 1 = a__U61#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__U61#(tt(),L) = 1 >= 1 = a__length#(mark(L)) a__length#(cons(N,L)) = 1 >= 0 = a__and#(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) a__length#(cons(N,L)) = 1 >= 1 = a__isNatList#(L) a__isNatList#(cons(V1,V2)) = 1 >= 1 = a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) = 1 >= 1 = a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) = 1 >= 1 = a__isNatList#(V2) a__isNatList#(cons(V1,V2)) = 1 >= 1 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 1 >= 1 = a__isNatKind#(V1) a__U51#(tt(),V1,V2) = 1 >= 1 = a__isNat#(V1) a__isNat#(s(V1)) = 1 >= 1 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = 1 >= 1 = a__isNat#(V1) a__isNat#(s(V1)) = 1 >= 1 = a__isNatKind#(V1) a__U61#(tt(),L) = 1 >= 1 = mark#(L) mark#(length(X)) = 1 >= 1 = mark#(X) mark#(U61(X1,X2)) = 1 >= 1 = a__U61#(mark(X1),X2) mark#(U61(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U53(X)) = 1 >= 1 = mark#(X) mark#(U52(X1,X2)) = 1 >= 1 = a__U52#(mark(X1),X2) mark#(U52(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U51(X1,X2,X3)) = 1 >= 1 = a__U51#(mark(X1),X2,X3) mark#(U51(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(U43(X)) = 1 >= 1 = mark#(X) mark#(isNat(X)) = 1 >= 1 = a__isNat#(X) mark#(U22(X)) = 1 >= 1 = mark#(X) mark#(U21(X1,X2)) = 1 >= 1 = a__U21#(mark(X1),X2) mark#(U21(X1,X2)) = 1 >= 1 = mark#(X1) mark#(isNatList(X)) = 1 >= 1 = a__isNatList#(X) a__isNatKind(0()) = 1 >= 1 = tt() a__isNatKind(length(V1)) = 1 >= 1 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 1 >= 1 = a__isNatKind(V1) a__isNatKind(X) = 1 >= 0 = isNatKind(X) a__isNatIListKind(zeros()) = 1 >= 1 = tt() a__isNatIListKind(cons(V1,V2)) = 1 >= 1 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(X) = 1 >= 0 = isNatIListKind(X) a__and(tt(),X) = 1 >= 1 = mark(X) a__and(X1,X2) = X1 >= 0 = and(X1,X2) mark(zeros()) = 1 >= 1 = a__zeros() mark(isNatList(X)) = 1 >= 0 = a__isNatList(X) mark(U21(X1,X2)) = 1 >= 1 = a__U21(mark(X1),X2) mark(U22(X)) = 1 >= 0 = a__U22(mark(X)) mark(isNat(X)) = 1 >= 1 = a__isNat(X) mark(U43(X)) = 1 >= 1 = a__U43(mark(X)) mark(U51(X1,X2,X3)) = 1 >= 0 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = 1 >= 0 = a__U52(mark(X1),X2) mark(U53(X)) = 1 >= 0 = a__U53(mark(X)) mark(U61(X1,X2)) = 1 >= 1 = a__U61(mark(X1),X2) mark(length(X)) = 1 >= 0 = a__length(mark(X)) mark(and(X1,X2)) = 1 >= 1 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 1 >= 1 = a__isNatIListKind(X) mark(isNatKind(X)) = 1 >= 1 = a__isNatKind(X) mark(cons(X1,X2)) = 1 >= 0 = cons(mark(X1),X2) mark(0()) = 1 >= 0 = 0() mark(tt()) = 1 >= 1 = tt() mark(s(X)) = 1 >= 1 = s(mark(X)) a__zeros() = 1 >= 0 = cons(0(),zeros()) a__zeros() = 1 >= 1 = zeros() a__isNatList(cons(V1,V2)) = 0 >= 0 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(X) = 0 >= 0 = isNatList(X) a__U51(tt(),V1,V2) = 0 >= 0 = a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) = 0 >= 0 = U51(X1,X2,X3) a__U52(tt(),V2) = 0 >= 0 = a__U53(a__isNatList(V2)) a__U52(X1,X2) = 0 >= 0 = U52(X1,X2) a__isNat(0()) = 1 >= 1 = tt() a__isNat(s(V1)) = 1 >= 1 = a__U21(a__isNatKind(V1),V1) a__isNat(X) = 1 >= 1 = isNat(X) a__U21(tt(),V1) = 1 >= 0 = a__U22(a__isNat(V1)) a__U21(X1,X2) = 1 >= 0 = U21(X1,X2) a__U22(X) = 0 >= 0 = U22(X) a__U53(X) = 0 >= 0 = U53(X) a__U43(X) = 1 >= 0 = U43(X) a__U61(tt(),L) = 1 >= 1 = s(a__length(mark(L))) a__U61(X1,X2) = X1 >= X1 = U61(X1,X2) a__length(cons(N,L)) = 0 >= 0 = a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__length(X) = 0 >= 0 = length(X) problem: DPs: a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__and#(tt(),X) -> mark#(X) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) mark#(length(X)) -> a__length#(mark(X)) a__length#(cons(N,L)) -> a__U61#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__U61#(tt(),L) -> a__length#(mark(L)) a__length#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__U61#(tt(),L) -> mark#(L) mark#(length(X)) -> mark#(X) mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(U61(X1,X2)) -> mark#(X1) mark#(U53(X)) -> mark#(X) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U52(X1,X2)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U43(X)) -> mark#(X) mark#(isNat(X)) -> a__isNat#(X) mark#(U22(X)) -> mark#(X) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U21(X1,X2)) -> mark#(X1) mark#(isNatList(X)) -> a__isNatList#(X) TRS: a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(X) -> isNatIListKind(X) a__and(tt(),X) -> mark(X) a__and(X1,X2) -> and(X1,X2) mark(zeros()) -> a__zeros() mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(X) -> isNatList(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__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__U53(X) -> U53(X) a__U43(X) -> U43(X) a__U61(tt(),L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__length(X) -> length(X) Restore Modifier: DPs: a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__and#(tt(),X) -> mark#(X) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) mark#(length(X)) -> a__length#(mark(X)) a__length#(cons(N,L)) -> a__U61#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__U61#(tt(),L) -> a__length#(mark(L)) a__length#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__U61#(tt(),L) -> mark#(L) mark#(length(X)) -> mark#(X) mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(U61(X1,X2)) -> mark#(X1) mark#(U53(X)) -> mark#(X) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U52(X1,X2)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U43(X)) -> mark#(X) mark#(isNat(X)) -> a__isNat#(X) mark#(U22(X)) -> mark#(X) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U21(X1,X2)) -> mark#(X1) mark#(isNatList(X)) -> a__isNatList#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U61(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() 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__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) mark(zeros()) -> a__zeros() mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U43(X) -> U43(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2) -> U61(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Usable Rule Processor: DPs: a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__and#(tt(),X) -> mark#(X) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) mark#(length(X)) -> a__length#(mark(X)) a__length#(cons(N,L)) -> a__U61#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__U61#(tt(),L) -> a__length#(mark(L)) a__length#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__U61#(tt(),L) -> mark#(L) mark#(length(X)) -> mark#(X) mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(U61(X1,X2)) -> mark#(X1) mark#(U53(X)) -> mark#(X) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U52(X1,X2)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U43(X)) -> mark#(X) mark#(isNat(X)) -> a__isNat#(X) mark#(U22(X)) -> mark#(X) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U21(X1,X2)) -> mark#(X1) mark#(isNatList(X)) -> a__isNatList#(X) TRS: a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(X) -> isNatIListKind(X) a__and(tt(),X) -> mark(X) a__and(X1,X2) -> and(X1,X2) mark(zeros()) -> a__zeros() mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(X) -> isNatList(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__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__U53(X) -> U53(X) a__U43(X) -> U43(X) a__U61(tt(),L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__length(X) -> length(X) Arctic Interpretation Processor: dimension: 1 usable rules: a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(X) -> isNatIListKind(X) a__and(tt(),X) -> mark(X) a__and(X1,X2) -> and(X1,X2) mark(zeros()) -> a__zeros() mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(X) -> isNatList(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__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__U53(X) -> U53(X) a__U43(X) -> U43(X) a__U61(tt(),L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__length(X) -> length(X) interpretation: [a__isNatIListKind#](x0) = x0, [a__isNatKind#](x0) = x0, [a__and#](x0, x1) = x0 + x1 + 0, [a__length#](x0) = x0 + 0, [mark#](x0) = x0 + 0, [a__U61#](x0, x1) = x0 + x1 + 0, [a__isNatList#](x0) = x0 + 0, [a__U52#](x0, x1) = x0 + x1, [a__U51#](x0, x1, x2) = x0 + x1 + x2, [a__isNat#](x0) = x0, [a__U21#](x0, x1) = x0 + x1, [U61](x0, x1) = x0 + x1, [U53](x0) = x0, [U52](x0, x1) = x0 + x1 + 0, [U51](x0, x1, x2) = x0 + x1 + x2 + 0, [U43](x0) = 6x0 + 1, [U22](x0) = x0 + 0, [U21](x0, x1) = x0 + x1, [isNatList](x0) = x0, [and](x0, x1) = x0 + x1, [isNatKind](x0) = x0, [isNat](x0) = x0, [isNatIListKind](x0) = x0, [a__isNatKind](x0) = x0, [a__isNatIListKind](x0) = x0, [length](x0) = x0 + 0, [a__and](x0, x1) = x0 + x1, [s](x0) = x0, [a__length](x0) = x0 + 0, [mark](x0) = x0, [a__U61](x0, x1) = x0 + x1, [a__U53](x0) = x0, [a__U52](x0, x1) = x0 + x1 + 0, [a__U51](x0, x1, x2) = x0 + x1 + x2 + 0, [a__U43](x0) = 6x0 + 1, [a__U22](x0) = x0 + 0, [a__isNat](x0) = x0, [a__U21](x0, x1) = x0 + x1, [a__isNatList](x0) = x0, [tt] = 0, [cons](x0, x1) = x0 + x1 + 6, [zeros] = 6, [0] = 4, [a__zeros] = 6 orientation: a__isNatIListKind#(cons(V1,V2)) = V1 + V2 + 6 >= V1 = a__isNatKind#(V1) a__isNatKind#(s(V1)) = V1 >= V1 = a__isNatKind#(V1) a__isNatKind#(length(V1)) = V1 + 0 >= V1 = a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) = V1 + V2 + 6 >= V1 + V2 + 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__and#(tt(),X) = X + 0 >= X + 0 = mark#(X) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) mark#(cons(X1,X2)) = X1 + X2 + 6 >= X1 + 0 = mark#(X1) mark#(isNatKind(X)) = X + 0 >= X = a__isNatKind#(X) mark#(isNatIListKind(X)) = X + 0 >= X = a__isNatIListKind#(X) mark#(and(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__and#(mark(X1),X2) mark#(and(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(length(X)) = X + 0 >= X + 0 = a__length#(mark(X)) a__length#(cons(N,L)) = L + N + 6 >= L + N + 0 = a__U61#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__U61#(tt(),L) = L + 0 >= L + 0 = a__length#(mark(L)) a__length#(cons(N,L)) = L + N + 6 >= L + 0 = a__isNatList#(L) a__isNatList#(cons(V1,V2)) = V1 + V2 + 6 >= V1 + V2 = a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) = V1 + V2 + 0 >= V1 + V2 = a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) = V2 + 0 >= V2 + 0 = a__isNatList#(V2) a__isNatList#(cons(V1,V2)) = V1 + V2 + 6 >= V1 + V2 + 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = V1 + V2 + 6 >= V1 = a__isNatKind#(V1) a__U51#(tt(),V1,V2) = V1 + V2 + 0 >= V1 = a__isNat#(V1) a__isNat#(s(V1)) = V1 >= V1 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = V1 + 0 >= V1 = a__isNat#(V1) a__isNat#(s(V1)) = V1 >= V1 = a__isNatKind#(V1) a__U61#(tt(),L) = L + 0 >= L + 0 = mark#(L) mark#(length(X)) = X + 0 >= X + 0 = mark#(X) mark#(U61(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__U61#(mark(X1),X2) mark#(U61(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(U53(X)) = X + 0 >= X + 0 = mark#(X) mark#(U52(X1,X2)) = X1 + X2 + 0 >= X1 + X2 = a__U52#(mark(X1),X2) mark#(U52(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(U51(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 = a__U51#(mark(X1),X2,X3) mark#(U51(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + 0 = mark#(X1) mark#(U43(X)) = 6X + 1 >= X + 0 = mark#(X) mark#(isNat(X)) = X + 0 >= X = a__isNat#(X) mark#(U22(X)) = X + 0 >= X + 0 = mark#(X) mark#(U21(X1,X2)) = X1 + X2 + 0 >= X1 + X2 = a__U21#(mark(X1),X2) mark#(U21(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(isNatList(X)) = X + 0 >= X + 0 = a__isNatList#(X) a__isNatKind(0()) = 4 >= 0 = tt() a__isNatKind(length(V1)) = V1 + 0 >= V1 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = V1 >= V1 = a__isNatKind(V1) a__isNatKind(X) = X >= X = isNatKind(X) a__isNatIListKind(zeros()) = 6 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = V1 + V2 + 6 >= V1 + V2 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(X) = X >= X = isNatIListKind(X) a__and(tt(),X) = X + 0 >= X = mark(X) a__and(X1,X2) = X1 + X2 >= X1 + X2 = and(X1,X2) mark(zeros()) = 6 >= 6 = a__zeros() mark(isNatList(X)) = X >= X = a__isNatList(X) mark(U21(X1,X2)) = X1 + X2 >= X1 + X2 = a__U21(mark(X1),X2) mark(U22(X)) = X + 0 >= X + 0 = a__U22(mark(X)) mark(isNat(X)) = X >= X = a__isNat(X) mark(U43(X)) = 6X + 1 >= 6X + 1 = a__U43(mark(X)) mark(U51(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__U52(mark(X1),X2) mark(U53(X)) = X >= X = a__U53(mark(X)) mark(U61(X1,X2)) = X1 + X2 >= X1 + X2 = a__U61(mark(X1),X2) mark(length(X)) = X + 0 >= X + 0 = a__length(mark(X)) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = X >= X = a__isNatIListKind(X) mark(isNatKind(X)) = X >= X = a__isNatKind(X) mark(cons(X1,X2)) = X1 + X2 + 6 >= X1 + X2 + 6 = cons(mark(X1),X2) mark(0()) = 4 >= 4 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X >= X = s(mark(X)) a__zeros() = 6 >= 6 = cons(0(),zeros()) a__zeros() = 6 >= 6 = zeros() a__isNatList(cons(V1,V2)) = V1 + V2 + 6 >= V1 + V2 + 0 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(X) = X >= X = isNatList(X) a__U51(tt(),V1,V2) = V1 + V2 + 0 >= V1 + V2 + 0 = a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = U51(X1,X2,X3) a__U52(tt(),V2) = V2 + 0 >= V2 = a__U53(a__isNatList(V2)) a__U52(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = U52(X1,X2) a__isNat(0()) = 4 >= 0 = tt() a__isNat(s(V1)) = V1 >= V1 = a__U21(a__isNatKind(V1),V1) a__isNat(X) = X >= X = isNat(X) a__U21(tt(),V1) = V1 + 0 >= V1 + 0 = a__U22(a__isNat(V1)) a__U21(X1,X2) = X1 + X2 >= X1 + X2 = U21(X1,X2) a__U22(X) = X + 0 >= X + 0 = U22(X) a__U53(X) = X >= X = U53(X) a__U43(X) = 6X + 1 >= 6X + 1 = U43(X) a__U61(tt(),L) = L + 0 >= L + 0 = s(a__length(mark(L))) a__U61(X1,X2) = X1 + X2 >= X1 + X2 = U61(X1,X2) a__length(cons(N,L)) = L + N + 6 >= L + N = a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__length(X) = X + 0 >= X + 0 = length(X) problem: DPs: a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__and#(tt(),X) -> mark#(X) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) mark#(length(X)) -> a__length#(mark(X)) a__length#(cons(N,L)) -> a__U61#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__U61#(tt(),L) -> a__length#(mark(L)) a__length#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__U61#(tt(),L) -> mark#(L) mark#(length(X)) -> mark#(X) mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(U61(X1,X2)) -> mark#(X1) mark#(U53(X)) -> mark#(X) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U52(X1,X2)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(isNat(X)) -> a__isNat#(X) mark#(U22(X)) -> mark#(X) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U21(X1,X2)) -> mark#(X1) mark#(isNatList(X)) -> a__isNatList#(X) TRS: a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(X) -> isNatIListKind(X) a__and(tt(),X) -> mark(X) a__and(X1,X2) -> and(X1,X2) mark(zeros()) -> a__zeros() mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(X) -> isNatList(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__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__U53(X) -> U53(X) a__U43(X) -> U43(X) a__U61(tt(),L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__length(X) -> length(X) Restore Modifier: DPs: a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__and#(tt(),X) -> mark#(X) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) mark#(length(X)) -> a__length#(mark(X)) a__length#(cons(N,L)) -> a__U61#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__U61#(tt(),L) -> a__length#(mark(L)) a__length#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__U61#(tt(),L) -> mark#(L) mark#(length(X)) -> mark#(X) mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(U61(X1,X2)) -> mark#(X1) mark#(U53(X)) -> mark#(X) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U52(X1,X2)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(isNat(X)) -> a__isNat#(X) mark#(U22(X)) -> mark#(X) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U21(X1,X2)) -> mark#(X1) mark#(isNatList(X)) -> a__isNatList#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U61(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() 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__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) mark(zeros()) -> a__zeros() mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U43(X) -> U43(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2) -> U61(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Usable Rule Processor: DPs: a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__and#(tt(),X) -> mark#(X) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) mark#(length(X)) -> a__length#(mark(X)) a__length#(cons(N,L)) -> a__U61#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__U61#(tt(),L) -> a__length#(mark(L)) a__length#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__U61#(tt(),L) -> mark#(L) mark#(length(X)) -> mark#(X) mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(U61(X1,X2)) -> mark#(X1) mark#(U53(X)) -> mark#(X) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U52(X1,X2)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(isNat(X)) -> a__isNat#(X) mark#(U22(X)) -> mark#(X) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U21(X1,X2)) -> mark#(X1) mark#(isNatList(X)) -> a__isNatList#(X) TRS: a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(X) -> isNatIListKind(X) a__and(tt(),X) -> mark(X) a__and(X1,X2) -> and(X1,X2) mark(zeros()) -> a__zeros() mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(X) -> isNatList(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__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__U53(X) -> U53(X) a__U43(X) -> U43(X) a__U61(tt(),L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__length(X) -> length(X) Arctic Interpretation Processor: dimension: 1 usable rules: a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(X) -> isNatIListKind(X) a__and(tt(),X) -> mark(X) a__and(X1,X2) -> and(X1,X2) mark(zeros()) -> a__zeros() mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(X) -> isNatList(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__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__U53(X) -> U53(X) a__U43(X) -> U43(X) a__U61(tt(),L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__length(X) -> length(X) interpretation: [a__isNatIListKind#](x0) = x0 + 0, [a__isNatKind#](x0) = x0, [a__and#](x0, x1) = x1, [a__length#](x0) = x0 + 0, [mark#](x0) = x0, [a__U61#](x0, x1) = x0 + x1 + 0, [a__isNatList#](x0) = x0, [a__U52#](x0, x1) = x1, [a__U51#](x0, x1, x2) = x1 + x2 + 0, [a__isNat#](x0) = x0, [a__U21#](x0, x1) = x0 + x1, [U61](x0, x1) = x0 + x1 + 0, [U53](x0) = x0 + 0, [U52](x0, x1) = 2x0 + x1, [U51](x0, x1, x2) = x0 + 2x1 + x2 + 0, [U43](x0) = 5x0, [U22](x0) = x0, [U21](x0, x1) = x0 + x1 + 0, [isNatList](x0) = x0 + 0, [and](x0, x1) = x0 + x1, [isNatKind](x0) = x0, [isNat](x0) = x0, [isNatIListKind](x0) = x0 + 0, [a__isNatKind](x0) = x0, [a__isNatIListKind](x0) = x0 + 0, [length](x0) = x0 + 0, [a__and](x0, x1) = x0 + x1, [s](x0) = x0 + 0, [a__length](x0) = x0 + 0, [mark](x0) = x0, [a__U61](x0, x1) = x0 + x1 + 0, [a__U53](x0) = x0 + 0, [a__U52](x0, x1) = 2x0 + x1, [a__U51](x0, x1, x2) = x0 + 2x1 + x2 + 0, [a__U43](x0) = 5x0, [a__U22](x0) = x0, [a__isNat](x0) = x0, [a__U21](x0, x1) = x0 + x1 + 0, [a__isNatList](x0) = x0 + 0, [tt] = 0, [cons](x0, x1) = 2x0 + x1 + 0, [zeros] = 4, [0] = 2, [a__zeros] = 4 orientation: a__isNatIListKind#(cons(V1,V2)) = 2V1 + V2 + 0 >= V1 = a__isNatKind#(V1) a__isNatKind#(s(V1)) = V1 + 0 >= V1 = a__isNatKind#(V1) a__isNatKind#(length(V1)) = V1 + 0 >= V1 + 0 = a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 2V1 + V2 + 0 >= V2 + 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__and#(tt(),X) = X >= X = mark#(X) mark#(s(X)) = X + 0 >= X = mark#(X) mark#(cons(X1,X2)) = 2X1 + X2 + 0 >= X1 = mark#(X1) mark#(isNatKind(X)) = X >= X = a__isNatKind#(X) mark#(isNatIListKind(X)) = X + 0 >= X + 0 = a__isNatIListKind#(X) mark#(and(X1,X2)) = X1 + X2 >= X2 = a__and#(mark(X1),X2) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(length(X)) = X + 0 >= X + 0 = a__length#(mark(X)) a__length#(cons(N,L)) = L + 2N + 0 >= L + N + 0 = a__U61#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__U61#(tt(),L) = L + 0 >= L + 0 = a__length#(mark(L)) a__length#(cons(N,L)) = L + 2N + 0 >= L = a__isNatList#(L) a__isNatList#(cons(V1,V2)) = 2V1 + V2 + 0 >= V1 + V2 + 0 = a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) = V1 + V2 + 0 >= V2 = a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) = V2 >= V2 = a__isNatList#(V2) a__isNatList#(cons(V1,V2)) = 2V1 + V2 + 0 >= V2 + 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 2V1 + V2 + 0 >= V1 = a__isNatKind#(V1) a__U51#(tt(),V1,V2) = V1 + V2 + 0 >= V1 = a__isNat#(V1) a__isNat#(s(V1)) = V1 + 0 >= V1 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = V1 + 0 >= V1 = a__isNat#(V1) a__isNat#(s(V1)) = V1 + 0 >= V1 = a__isNatKind#(V1) a__U61#(tt(),L) = L + 0 >= L = mark#(L) mark#(length(X)) = X + 0 >= X = mark#(X) mark#(U61(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__U61#(mark(X1),X2) mark#(U61(X1,X2)) = X1 + X2 + 0 >= X1 = mark#(X1) mark#(U53(X)) = X + 0 >= X = mark#(X) mark#(U52(X1,X2)) = 2X1 + X2 >= X2 = a__U52#(mark(X1),X2) mark#(U52(X1,X2)) = 2X1 + X2 >= X1 = mark#(X1) mark#(U51(X1,X2,X3)) = X1 + 2X2 + X3 + 0 >= X2 + X3 + 0 = a__U51#(mark(X1),X2,X3) mark#(U51(X1,X2,X3)) = X1 + 2X2 + X3 + 0 >= X1 = mark#(X1) mark#(isNat(X)) = X >= X = a__isNat#(X) mark#(U22(X)) = X >= X = mark#(X) mark#(U21(X1,X2)) = X1 + X2 + 0 >= X1 + X2 = a__U21#(mark(X1),X2) mark#(U21(X1,X2)) = X1 + X2 + 0 >= X1 = mark#(X1) mark#(isNatList(X)) = X + 0 >= X = a__isNatList#(X) a__isNatKind(0()) = 2 >= 0 = tt() a__isNatKind(length(V1)) = V1 + 0 >= V1 + 0 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = V1 + 0 >= V1 = a__isNatKind(V1) a__isNatKind(X) = X >= X = isNatKind(X) a__isNatIListKind(zeros()) = 4 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 2V1 + V2 + 0 >= V1 + V2 + 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(X) = X + 0 >= X + 0 = isNatIListKind(X) a__and(tt(),X) = X + 0 >= X = mark(X) a__and(X1,X2) = X1 + X2 >= X1 + X2 = and(X1,X2) mark(zeros()) = 4 >= 4 = a__zeros() mark(isNatList(X)) = X + 0 >= X + 0 = a__isNatList(X) mark(U21(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__U21(mark(X1),X2) mark(U22(X)) = X >= X = a__U22(mark(X)) mark(isNat(X)) = X >= X = a__isNat(X) mark(U43(X)) = 5X >= 5X = a__U43(mark(X)) mark(U51(X1,X2,X3)) = X1 + 2X2 + X3 + 0 >= X1 + 2X2 + X3 + 0 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = a__U52(mark(X1),X2) mark(U53(X)) = X + 0 >= X + 0 = a__U53(mark(X)) mark(U61(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__U61(mark(X1),X2) mark(length(X)) = X + 0 >= X + 0 = a__length(mark(X)) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = X + 0 >= X + 0 = a__isNatIListKind(X) mark(isNatKind(X)) = X >= X = a__isNatKind(X) mark(cons(X1,X2)) = 2X1 + X2 + 0 >= 2X1 + X2 + 0 = cons(mark(X1),X2) mark(0()) = 2 >= 2 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) a__zeros() = 4 >= 4 = cons(0(),zeros()) a__zeros() = 4 >= 4 = zeros() a__isNatList(cons(V1,V2)) = 2V1 + V2 + 0 >= 2V1 + V2 + 0 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(X) = X + 0 >= X + 0 = isNatList(X) a__U51(tt(),V1,V2) = 2V1 + V2 + 0 >= 2V1 + V2 = a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) = X1 + 2X2 + X3 + 0 >= X1 + 2X2 + X3 + 0 = U51(X1,X2,X3) a__U52(tt(),V2) = V2 + 2 >= V2 + 0 = a__U53(a__isNatList(V2)) a__U52(X1,X2) = 2X1 + X2 >= 2X1 + X2 = U52(X1,X2) a__isNat(0()) = 2 >= 0 = tt() a__isNat(s(V1)) = V1 + 0 >= V1 + 0 = a__U21(a__isNatKind(V1),V1) a__isNat(X) = X >= X = isNat(X) a__U21(tt(),V1) = V1 + 0 >= V1 = a__U22(a__isNat(V1)) a__U21(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = U21(X1,X2) a__U22(X) = X >= X = U22(X) a__U53(X) = X + 0 >= X + 0 = U53(X) a__U43(X) = 5X >= 5X = U43(X) a__U61(tt(),L) = L + 0 >= L + 0 = s(a__length(mark(L))) a__U61(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = U61(X1,X2) a__length(cons(N,L)) = L + 2N + 0 >= L + N + 0 = a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__length(X) = X + 0 >= X + 0 = length(X) problem: DPs: a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__and#(tt(),X) -> mark#(X) mark#(s(X)) -> mark#(X) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) mark#(length(X)) -> a__length#(mark(X)) a__length#(cons(N,L)) -> a__U61#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__U61#(tt(),L) -> a__length#(mark(L)) a__length#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__U61#(tt(),L) -> mark#(L) mark#(length(X)) -> mark#(X) mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(U61(X1,X2)) -> mark#(X1) mark#(U53(X)) -> mark#(X) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(isNat(X)) -> a__isNat#(X) mark#(U22(X)) -> mark#(X) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U21(X1,X2)) -> mark#(X1) mark#(isNatList(X)) -> a__isNatList#(X) TRS: a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(X) -> isNatIListKind(X) a__and(tt(),X) -> mark(X) a__and(X1,X2) -> and(X1,X2) mark(zeros()) -> a__zeros() mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(X) -> isNatList(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__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__U53(X) -> U53(X) a__U43(X) -> U43(X) a__U61(tt(),L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__length(X) -> length(X) Restore Modifier: DPs: a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__and#(tt(),X) -> mark#(X) mark#(s(X)) -> mark#(X) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) mark#(length(X)) -> a__length#(mark(X)) a__length#(cons(N,L)) -> a__U61#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__U61#(tt(),L) -> a__length#(mark(L)) a__length#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__U61#(tt(),L) -> mark#(L) mark#(length(X)) -> mark#(X) mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(U61(X1,X2)) -> mark#(X1) mark#(U53(X)) -> mark#(X) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(isNat(X)) -> a__isNat#(X) mark#(U22(X)) -> mark#(X) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U21(X1,X2)) -> mark#(X1) mark#(isNatList(X)) -> a__isNatList#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U61(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() 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__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) mark(zeros()) -> a__zeros() mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U43(X) -> U43(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2) -> U61(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Usable Rule Processor: DPs: a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__and#(tt(),X) -> mark#(X) mark#(s(X)) -> mark#(X) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) mark#(length(X)) -> a__length#(mark(X)) a__length#(cons(N,L)) -> a__U61#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__U61#(tt(),L) -> a__length#(mark(L)) a__length#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__U61#(tt(),L) -> mark#(L) mark#(length(X)) -> mark#(X) mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(U61(X1,X2)) -> mark#(X1) mark#(U53(X)) -> mark#(X) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(isNat(X)) -> a__isNat#(X) mark#(U22(X)) -> mark#(X) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U21(X1,X2)) -> mark#(X1) mark#(isNatList(X)) -> a__isNatList#(X) TRS: a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(X) -> isNatIListKind(X) a__and(tt(),X) -> mark(X) a__and(X1,X2) -> and(X1,X2) mark(zeros()) -> a__zeros() mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(X) -> isNatList(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__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__U53(X) -> U53(X) a__U43(X) -> U43(X) a__U61(tt(),L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__length(X) -> length(X) Matrix Interpretation Processor: dim=1 usable rules: a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(X) -> isNatIListKind(X) a__and(tt(),X) -> mark(X) a__and(X1,X2) -> and(X1,X2) mark(zeros()) -> a__zeros() mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(X) -> isNatList(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__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__U53(X) -> U53(X) a__U43(X) -> U43(X) a__U61(tt(),L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__length(X) -> length(X) interpretation: [a__isNatIListKind#](x0) = 1, [a__isNatKind#](x0) = 1, [a__and#](x0, x1) = 1, [a__length#](x0) = 1, [mark#](x0) = 1, [a__U61#](x0, x1) = x0, [a__isNatList#](x0) = 1, [a__U52#](x0, x1) = 1, [a__U51#](x0, x1, x2) = x0, [a__isNat#](x0) = 1, [a__U21#](x0, x1) = 1, [U61](x0, x1) = 0, [U53](x0) = 0, [U52](x0, x1) = 0, [U51](x0, x1, x2) = 0, [U43](x0) = 0, [U22](x0) = 0, [U21](x0, x1) = 0, [isNatList](x0) = 0, [and](x0, x1) = x0, [isNatKind](x0) = 0, [isNat](x0) = 0, [isNatIListKind](x0) = 1, [a__isNatKind](x0) = 1, [a__isNatIListKind](x0) = 1, [length](x0) = 0, [a__and](x0, x1) = x0, [s](x0) = 0, [a__length](x0) = 0, [mark](x0) = 1, [a__U61](x0, x1) = 1/2x0, [a__U53](x0) = 0, [a__U52](x0, x1) = 0, [a__U51](x0, x1, x2) = 0, [a__U43](x0) = 1/2, [a__U22](x0) = 0, [a__isNat](x0) = 1, [a__U21](x0, x1) = x0, [a__isNatList](x0) = 0, [tt] = 1, [cons](x0, x1) = x0, [zeros] = 0, [0] = 1, [a__zeros] = 1 orientation: a__isNatKind#(s(V1)) = 1 >= 1 = a__isNatKind#(V1) a__isNatKind#(length(V1)) = 1 >= 1 = a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 1 >= 1 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__and#(tt(),X) = 1 >= 1 = mark#(X) mark#(s(X)) = 1 >= 1 = mark#(X) mark#(isNatKind(X)) = 1 >= 1 = a__isNatKind#(X) mark#(isNatIListKind(X)) = 1 >= 1 = a__isNatIListKind#(X) mark#(and(X1,X2)) = 1 >= 1 = a__and#(mark(X1),X2) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(length(X)) = 1 >= 1 = a__length#(mark(X)) a__length#(cons(N,L)) = 1 >= 0 = a__U61#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__U61#(tt(),L) = 1 >= 1 = a__length#(mark(L)) a__length#(cons(N,L)) = 1 >= 1 = a__isNatList#(L) a__isNatList#(cons(V1,V2)) = 1 >= 1 = a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) = 1 >= 1 = a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) = 1 >= 1 = a__isNatList#(V2) a__isNatList#(cons(V1,V2)) = 1 >= 1 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__U51#(tt(),V1,V2) = 1 >= 1 = a__isNat#(V1) a__isNat#(s(V1)) = 1 >= 1 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = 1 >= 1 = a__isNat#(V1) a__isNat#(s(V1)) = 1 >= 1 = a__isNatKind#(V1) a__U61#(tt(),L) = 1 >= 1 = mark#(L) mark#(length(X)) = 1 >= 1 = mark#(X) mark#(U61(X1,X2)) = 1 >= 1 = a__U61#(mark(X1),X2) mark#(U61(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U53(X)) = 1 >= 1 = mark#(X) mark#(U52(X1,X2)) = 1 >= 1 = a__U52#(mark(X1),X2) mark#(U51(X1,X2,X3)) = 1 >= 1 = a__U51#(mark(X1),X2,X3) mark#(U51(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(isNat(X)) = 1 >= 1 = a__isNat#(X) mark#(U22(X)) = 1 >= 1 = mark#(X) mark#(U21(X1,X2)) = 1 >= 1 = a__U21#(mark(X1),X2) mark#(U21(X1,X2)) = 1 >= 1 = mark#(X1) mark#(isNatList(X)) = 1 >= 1 = a__isNatList#(X) a__isNatKind(0()) = 1 >= 1 = tt() a__isNatKind(length(V1)) = 1 >= 1 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 1 >= 1 = a__isNatKind(V1) a__isNatKind(X) = 1 >= 0 = isNatKind(X) a__isNatIListKind(zeros()) = 1 >= 1 = tt() a__isNatIListKind(cons(V1,V2)) = 1 >= 1 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(X) = 1 >= 1 = isNatIListKind(X) a__and(tt(),X) = 1 >= 1 = mark(X) a__and(X1,X2) = X1 >= X1 = and(X1,X2) mark(zeros()) = 1 >= 1 = a__zeros() mark(isNatList(X)) = 1 >= 0 = a__isNatList(X) mark(U21(X1,X2)) = 1 >= 1 = a__U21(mark(X1),X2) mark(U22(X)) = 1 >= 0 = a__U22(mark(X)) mark(isNat(X)) = 1 >= 1 = a__isNat(X) mark(U43(X)) = 1 >= 1/2 = a__U43(mark(X)) mark(U51(X1,X2,X3)) = 1 >= 0 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = 1 >= 0 = a__U52(mark(X1),X2) mark(U53(X)) = 1 >= 0 = a__U53(mark(X)) mark(U61(X1,X2)) = 1 >= 1/2 = a__U61(mark(X1),X2) mark(length(X)) = 1 >= 0 = a__length(mark(X)) mark(and(X1,X2)) = 1 >= 1 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 1 >= 1 = a__isNatIListKind(X) mark(isNatKind(X)) = 1 >= 1 = a__isNatKind(X) mark(cons(X1,X2)) = 1 >= 1 = cons(mark(X1),X2) mark(0()) = 1 >= 1 = 0() mark(tt()) = 1 >= 1 = tt() mark(s(X)) = 1 >= 0 = s(mark(X)) a__zeros() = 1 >= 1 = cons(0(),zeros()) a__zeros() = 1 >= 0 = zeros() a__isNatList(cons(V1,V2)) = 0 >= 0 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(X) = 0 >= 0 = isNatList(X) a__U51(tt(),V1,V2) = 0 >= 0 = a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) = 0 >= 0 = U51(X1,X2,X3) a__U52(tt(),V2) = 0 >= 0 = a__U53(a__isNatList(V2)) a__U52(X1,X2) = 0 >= 0 = U52(X1,X2) a__isNat(0()) = 1 >= 1 = tt() a__isNat(s(V1)) = 1 >= 1 = a__U21(a__isNatKind(V1),V1) a__isNat(X) = 1 >= 0 = isNat(X) a__U21(tt(),V1) = 1 >= 0 = a__U22(a__isNat(V1)) a__U21(X1,X2) = X1 >= 0 = U21(X1,X2) a__U22(X) = 0 >= 0 = U22(X) a__U53(X) = 0 >= 0 = U53(X) a__U43(X) = 1/2 >= 0 = U43(X) a__U61(tt(),L) = 1/2 >= 0 = s(a__length(mark(L))) a__U61(X1,X2) = 1/2X1 >= 0 = U61(X1,X2) a__length(cons(N,L)) = 0 >= 0 = a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__length(X) = 0 >= 0 = length(X) problem: DPs: a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__and#(tt(),X) -> mark#(X) mark#(s(X)) -> mark#(X) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) mark#(length(X)) -> a__length#(mark(X)) a__U61#(tt(),L) -> a__length#(mark(L)) a__length#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__U61#(tt(),L) -> mark#(L) mark#(length(X)) -> mark#(X) mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(U61(X1,X2)) -> mark#(X1) mark#(U53(X)) -> mark#(X) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(isNat(X)) -> a__isNat#(X) mark#(U22(X)) -> mark#(X) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U21(X1,X2)) -> mark#(X1) mark#(isNatList(X)) -> a__isNatList#(X) TRS: a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(X) -> isNatIListKind(X) a__and(tt(),X) -> mark(X) a__and(X1,X2) -> and(X1,X2) mark(zeros()) -> a__zeros() mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(X) -> isNatList(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__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__U53(X) -> U53(X) a__U43(X) -> U43(X) a__U61(tt(),L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__length(X) -> length(X) Restore Modifier: DPs: a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__and#(tt(),X) -> mark#(X) mark#(s(X)) -> mark#(X) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) mark#(length(X)) -> a__length#(mark(X)) a__U61#(tt(),L) -> a__length#(mark(L)) a__length#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__U61#(tt(),L) -> mark#(L) mark#(length(X)) -> mark#(X) mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(U61(X1,X2)) -> mark#(X1) mark#(U53(X)) -> mark#(X) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(isNat(X)) -> a__isNat#(X) mark#(U22(X)) -> mark#(X) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U21(X1,X2)) -> mark#(X1) mark#(isNatList(X)) -> a__isNatList#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U61(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() 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__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) mark(zeros()) -> a__zeros() mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U43(X) -> U43(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2) -> U61(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Usable Rule Processor: DPs: a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__and#(tt(),X) -> mark#(X) mark#(s(X)) -> mark#(X) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) mark#(length(X)) -> a__length#(mark(X)) a__U61#(tt(),L) -> a__length#(mark(L)) a__length#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__U61#(tt(),L) -> mark#(L) mark#(length(X)) -> mark#(X) mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(U61(X1,X2)) -> mark#(X1) mark#(U53(X)) -> mark#(X) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(isNat(X)) -> a__isNat#(X) mark#(U22(X)) -> mark#(X) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U21(X1,X2)) -> mark#(X1) mark#(isNatList(X)) -> a__isNatList#(X) TRS: a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(X) -> isNatIListKind(X) a__and(tt(),X) -> mark(X) a__and(X1,X2) -> and(X1,X2) mark(zeros()) -> a__zeros() mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(X) -> isNatList(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__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__U53(X) -> U53(X) a__U43(X) -> U43(X) a__U61(tt(),L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__length(X) -> length(X) Arctic Interpretation Processor: dimension: 1 usable rules: interpretation: [a__isNatIListKind#](x0) = 0, [a__isNatKind#](x0) = 0, [a__and#](x0, x1) = x1 + 0, [a__length#](x0) = 0, [mark#](x0) = x0 + 0, [a__U61#](x0, x1) = x1 + 0, [a__isNatList#](x0) = 0, [a__U52#](x0, x1) = 0, [a__U51#](x0, x1, x2) = 0, [a__isNat#](x0) = 0, [a__U21#](x0, x1) = 0, [U61](x0, x1) = x0 + x1, [U53](x0) = x0 + 0, [U52](x0, x1) = x0 + x1 + 0, [U51](x0, x1, x2) = x0 + x1 + x2 + 0, [U43](x0) = x0 + 4, [U22](x0) = 4x0 + 1, [U21](x0, x1) = x0 + x1, [isNatList](x0) = x0, [and](x0, x1) = x0 + x1 + 0, [isNatKind](x0) = 2x0 + 0, [isNat](x0) = x0 + 0, [isNatIListKind](x0) = 0, [a__isNatKind](x0) = 4x0 + 1, [a__isNatIListKind](x0) = 3x0 + 0, [length](x0) = x0 + 0, [a__and](x0, x1) = 1x0 + 4x1 + 0, [s](x0) = x0, [a__length](x0) = 0, [mark](x0) = x0 + 6, [a__U61](x0, x1) = x0 + 8x1 + 8, [a__U53](x0) = x0 + 0, [a__U52](x0, x1) = x0 + x1 + 8, [a__U51](x0, x1, x2) = x2 + 1, [a__U43](x0) = 1, [a__U22](x0) = x0 + 0, [a__isNat](x0) = 5, [a__U21](x0, x1) = x0 + x1 + 0, [a__isNatList](x0) = 3, [tt] = 0, [cons](x0, x1) = 1x0 + x1 + 1, [zeros] = 0, [0] = 3, [a__zeros] = 0 orientation: a__isNatKind#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNatKind#(length(V1)) = 0 >= 0 = a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__and#(tt(),X) = X + 0 >= X + 0 = mark#(X) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) mark#(isNatKind(X)) = 2X + 0 >= 0 = a__isNatKind#(X) mark#(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind#(X) mark#(and(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = a__and#(mark(X1),X2) mark#(and(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(length(X)) = X + 0 >= 0 = a__length#(mark(X)) a__U61#(tt(),L) = L + 0 >= 0 = a__length#(mark(L)) a__length#(cons(N,L)) = 0 >= 0 = a__isNatList#(L) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) = 0 >= 0 = a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) = 0 >= 0 = a__isNatList#(V2) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__U51#(tt(),V1,V2) = 0 >= 0 = a__isNat#(V1) a__isNat#(s(V1)) = 0 >= 0 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = 0 >= 0 = a__isNat#(V1) a__isNat#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__U61#(tt(),L) = L + 0 >= L + 0 = mark#(L) mark#(length(X)) = X + 0 >= X + 0 = mark#(X) mark#(U61(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = a__U61#(mark(X1),X2) mark#(U61(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(U53(X)) = X + 0 >= X + 0 = mark#(X) mark#(U52(X1,X2)) = X1 + X2 + 0 >= 0 = a__U52#(mark(X1),X2) mark#(U51(X1,X2,X3)) = X1 + X2 + X3 + 0 >= 0 = a__U51#(mark(X1),X2,X3) mark#(U51(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + 0 = mark#(X1) mark#(isNat(X)) = X + 0 >= 0 = a__isNat#(X) mark#(U22(X)) = 4X + 1 >= X + 0 = mark#(X) mark#(U21(X1,X2)) = X1 + X2 + 0 >= 0 = a__U21#(mark(X1),X2) mark#(U21(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(isNatList(X)) = X + 0 >= 0 = a__isNatList#(X) a__isNatKind(0()) = 7 >= 0 = tt() a__isNatKind(length(V1)) = 4V1 + 4 >= 3V1 + 0 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 4V1 + 1 >= 4V1 + 1 = a__isNatKind(V1) a__isNatKind(X) = 4X + 1 >= 2X + 0 = isNatKind(X) a__isNatIListKind(zeros()) = 3 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 4V1 + 3V2 + 4 >= 5V1 + 4 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(X) = 3X + 0 >= 0 = isNatIListKind(X) a__and(tt(),X) = 4X + 1 >= X + 6 = mark(X) a__and(X1,X2) = 1X1 + 4X2 + 0 >= X1 + X2 + 0 = and(X1,X2) mark(zeros()) = 6 >= 0 = a__zeros() mark(isNatList(X)) = X + 6 >= 3 = a__isNatList(X) mark(U21(X1,X2)) = X1 + X2 + 6 >= X1 + X2 + 6 = a__U21(mark(X1),X2) mark(U22(X)) = 4X + 6 >= X + 6 = a__U22(mark(X)) mark(isNat(X)) = X + 6 >= 5 = a__isNat(X) mark(U43(X)) = X + 6 >= 1 = a__U43(mark(X)) mark(U51(X1,X2,X3)) = X1 + X2 + X3 + 6 >= X3 + 1 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = X1 + X2 + 6 >= X1 + X2 + 8 = a__U52(mark(X1),X2) mark(U53(X)) = X + 6 >= X + 6 = a__U53(mark(X)) mark(U61(X1,X2)) = X1 + X2 + 6 >= X1 + 8X2 + 8 = a__U61(mark(X1),X2) mark(length(X)) = X + 6 >= 0 = a__length(mark(X)) mark(and(X1,X2)) = X1 + X2 + 6 >= 1X1 + 4X2 + 7 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 6 >= 3X + 0 = a__isNatIListKind(X) mark(isNatKind(X)) = 2X + 6 >= 4X + 1 = a__isNatKind(X) mark(cons(X1,X2)) = 1X1 + X2 + 6 >= 1X1 + X2 + 7 = cons(mark(X1),X2) mark(0()) = 6 >= 3 = 0() mark(tt()) = 6 >= 0 = tt() mark(s(X)) = X + 6 >= X + 6 = s(mark(X)) a__zeros() = 0 >= 4 = cons(0(),zeros()) a__zeros() = 0 >= 0 = zeros() a__isNatList(cons(V1,V2)) = 3 >= V2 + 1 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(X) = 3 >= X = isNatList(X) a__U51(tt(),V1,V2) = V2 + 1 >= V2 + 8 = a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) = X3 + 1 >= X1 + X2 + X3 + 0 = U51(X1,X2,X3) a__U52(tt(),V2) = V2 + 8 >= 3 = a__U53(a__isNatList(V2)) a__U52(X1,X2) = X1 + X2 + 8 >= X1 + X2 + 0 = U52(X1,X2) a__isNat(0()) = 5 >= 0 = tt() a__isNat(s(V1)) = 5 >= 4V1 + 1 = a__U21(a__isNatKind(V1),V1) a__isNat(X) = 5 >= X + 0 = isNat(X) a__U21(tt(),V1) = V1 + 0 >= 5 = a__U22(a__isNat(V1)) a__U21(X1,X2) = X1 + X2 + 0 >= X1 + X2 = U21(X1,X2) a__U22(X) = X + 0 >= 4X + 1 = U22(X) a__U53(X) = X + 0 >= X + 0 = U53(X) a__U43(X) = 1 >= X + 4 = U43(X) a__U61(tt(),L) = 8L + 8 >= 0 = s(a__length(mark(L))) a__U61(X1,X2) = X1 + 8X2 + 8 >= X1 + X2 = U61(X1,X2) a__length(cons(N,L)) = 0 >= 8L + 6N + 8 = a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__length(X) = 0 >= X + 0 = length(X) problem: DPs: a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__and#(tt(),X) -> mark#(X) mark#(s(X)) -> mark#(X) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) mark#(length(X)) -> a__length#(mark(X)) a__U61#(tt(),L) -> a__length#(mark(L)) a__length#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__U61#(tt(),L) -> mark#(L) mark#(length(X)) -> mark#(X) mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(U61(X1,X2)) -> mark#(X1) mark#(U53(X)) -> mark#(X) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(isNat(X)) -> a__isNat#(X) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U21(X1,X2)) -> mark#(X1) mark#(isNatList(X)) -> a__isNatList#(X) TRS: a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(X) -> isNatIListKind(X) a__and(tt(),X) -> mark(X) a__and(X1,X2) -> and(X1,X2) mark(zeros()) -> a__zeros() mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(X) -> isNatList(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__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__U53(X) -> U53(X) a__U43(X) -> U43(X) a__U61(tt(),L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__length(X) -> length(X) Restore Modifier: DPs: a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__and#(tt(),X) -> mark#(X) mark#(s(X)) -> mark#(X) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) mark#(length(X)) -> a__length#(mark(X)) a__U61#(tt(),L) -> a__length#(mark(L)) a__length#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__U61#(tt(),L) -> mark#(L) mark#(length(X)) -> mark#(X) mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(U61(X1,X2)) -> mark#(X1) mark#(U53(X)) -> mark#(X) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(isNat(X)) -> a__isNat#(X) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U21(X1,X2)) -> mark#(X1) mark#(isNatList(X)) -> a__isNatList#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U61(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() 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__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) mark(zeros()) -> a__zeros() mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U43(X) -> U43(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2) -> U61(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Usable Rule Processor: DPs: a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__and#(tt(),X) -> mark#(X) mark#(s(X)) -> mark#(X) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) mark#(length(X)) -> a__length#(mark(X)) a__U61#(tt(),L) -> a__length#(mark(L)) a__length#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__U61#(tt(),L) -> mark#(L) mark#(length(X)) -> mark#(X) mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(U61(X1,X2)) -> mark#(X1) mark#(U53(X)) -> mark#(X) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(isNat(X)) -> a__isNat#(X) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U21(X1,X2)) -> mark#(X1) mark#(isNatList(X)) -> a__isNatList#(X) TRS: a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(X) -> isNatIListKind(X) a__and(tt(),X) -> mark(X) a__and(X1,X2) -> and(X1,X2) mark(zeros()) -> a__zeros() mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(X) -> isNatList(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__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__U53(X) -> U53(X) a__U43(X) -> U43(X) a__U61(tt(),L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__length(X) -> length(X) Arctic Interpretation Processor: dimension: 1 usable rules: interpretation: [a__isNatIListKind#](x0) = 0, [a__isNatKind#](x0) = 0, [a__and#](x0, x1) = x1 + 0, [a__length#](x0) = 0, [mark#](x0) = x0, [a__U61#](x0, x1) = x1 + 0, [a__isNatList#](x0) = 0, [a__U52#](x0, x1) = 0, [a__U51#](x0, x1, x2) = 0, [a__isNat#](x0) = 0, [a__U21#](x0, x1) = 0, [U61](x0, x1) = x0 + 3x1 + 0, [U53](x0) = 6x0 + 0, [U52](x0, x1) = 0, [U51](x0, x1, x2) = x0 + x1 + 6x2 + 0, [U43](x0) = 2x0, [U22](x0) = 2x0 + 4, [U21](x0, x1) = x0 + 1x1 + 0, [isNatList](x0) = 1, [and](x0, x1) = x0 + 3x1 + 0, [isNatKind](x0) = 1x0 + 0, [isNat](x0) = 0, [isNatIListKind](x0) = 0, [a__isNatKind](x0) = x0, [a__isNatIListKind](x0) = 0, [length](x0) = x0 + 0, [a__and](x0, x1) = 2x0, [s](x0) = 4x0 + 1, [a__length](x0) = 1, [mark](x0) = 1x0 + 0, [a__U61](x0, x1) = 8x1, [a__U53](x0) = 4, [a__U52](x0, x1) = x1 + 0, [a__U51](x0, x1, x2) = 8x1 + 11x2 + 0, [a__U43](x0) = 4x0 + 0, [a__U22](x0) = x0 + 0, [a__isNat](x0) = 0, [a__U21](x0, x1) = 2x0 + 13x1 + 0, [a__isNatList](x0) = 0, [tt] = 0, [cons](x0, x1) = 4x0 + x1 + 0, [zeros] = 2, [0] = 0, [a__zeros] = 0 orientation: a__isNatKind#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNatKind#(length(V1)) = 0 >= 0 = a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__and#(tt(),X) = X + 0 >= X = mark#(X) mark#(s(X)) = 4X + 1 >= X = mark#(X) mark#(isNatKind(X)) = 1X + 0 >= 0 = a__isNatKind#(X) mark#(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind#(X) mark#(and(X1,X2)) = X1 + 3X2 + 0 >= X2 + 0 = a__and#(mark(X1),X2) mark#(and(X1,X2)) = X1 + 3X2 + 0 >= X1 = mark#(X1) mark#(length(X)) = X + 0 >= 0 = a__length#(mark(X)) a__U61#(tt(),L) = L + 0 >= 0 = a__length#(mark(L)) a__length#(cons(N,L)) = 0 >= 0 = a__isNatList#(L) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) = 0 >= 0 = a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) = 0 >= 0 = a__isNatList#(V2) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__U51#(tt(),V1,V2) = 0 >= 0 = a__isNat#(V1) a__isNat#(s(V1)) = 0 >= 0 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = 0 >= 0 = a__isNat#(V1) a__isNat#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__U61#(tt(),L) = L + 0 >= L = mark#(L) mark#(length(X)) = X + 0 >= X = mark#(X) mark#(U61(X1,X2)) = X1 + 3X2 + 0 >= X2 + 0 = a__U61#(mark(X1),X2) mark#(U61(X1,X2)) = X1 + 3X2 + 0 >= X1 = mark#(X1) mark#(U53(X)) = 6X + 0 >= X = mark#(X) mark#(U52(X1,X2)) = 0 >= 0 = a__U52#(mark(X1),X2) mark#(U51(X1,X2,X3)) = X1 + X2 + 6X3 + 0 >= 0 = a__U51#(mark(X1),X2,X3) mark#(U51(X1,X2,X3)) = X1 + X2 + 6X3 + 0 >= X1 = mark#(X1) mark#(isNat(X)) = 0 >= 0 = a__isNat#(X) mark#(U21(X1,X2)) = X1 + 1X2 + 0 >= 0 = a__U21#(mark(X1),X2) mark#(U21(X1,X2)) = X1 + 1X2 + 0 >= X1 = mark#(X1) mark#(isNatList(X)) = 1 >= 0 = a__isNatList#(X) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = V1 + 0 >= 0 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 4V1 + 1 >= V1 = a__isNatKind(V1) a__isNatKind(X) = X >= 1X + 0 = isNatKind(X) a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 0 >= 2V1 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(X) = 0 >= 0 = isNatIListKind(X) a__and(tt(),X) = 2 >= 1X + 0 = mark(X) a__and(X1,X2) = 2X1 >= X1 + 3X2 + 0 = and(X1,X2) mark(zeros()) = 3 >= 0 = a__zeros() mark(isNatList(X)) = 2 >= 0 = a__isNatList(X) mark(U21(X1,X2)) = 1X1 + 2X2 + 1 >= 3X1 + 13X2 + 2 = a__U21(mark(X1),X2) mark(U22(X)) = 3X + 5 >= 1X + 0 = a__U22(mark(X)) mark(isNat(X)) = 1 >= 0 = a__isNat(X) mark(U43(X)) = 3X + 0 >= 5X + 4 = a__U43(mark(X)) mark(U51(X1,X2,X3)) = 1X1 + 1X2 + 7X3 + 1 >= 8X2 + 11X3 + 0 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = 1 >= X2 + 0 = a__U52(mark(X1),X2) mark(U53(X)) = 7X + 1 >= 4 = a__U53(mark(X)) mark(U61(X1,X2)) = 1X1 + 4X2 + 1 >= 8X2 = a__U61(mark(X1),X2) mark(length(X)) = 1X + 1 >= 1 = a__length(mark(X)) mark(and(X1,X2)) = 1X1 + 4X2 + 1 >= 3X1 + 2 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 1 >= 0 = a__isNatIListKind(X) mark(isNatKind(X)) = 2X + 1 >= X = a__isNatKind(X) mark(cons(X1,X2)) = 5X1 + 1X2 + 1 >= 5X1 + X2 + 4 = cons(mark(X1),X2) mark(0()) = 1 >= 0 = 0() mark(tt()) = 1 >= 0 = tt() mark(s(X)) = 5X + 2 >= 5X + 4 = s(mark(X)) a__zeros() = 0 >= 4 = cons(0(),zeros()) a__zeros() = 0 >= 2 = zeros() a__isNatList(cons(V1,V2)) = 0 >= 8V1 + 11V2 + 0 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(X) = 0 >= 1 = isNatList(X) a__U51(tt(),V1,V2) = 8V1 + 11V2 + 0 >= V2 + 0 = a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) = 8X2 + 11X3 + 0 >= X1 + X2 + 6X3 + 0 = U51(X1,X2,X3) a__U52(tt(),V2) = V2 + 0 >= 4 = a__U53(a__isNatList(V2)) a__U52(X1,X2) = X2 + 0 >= 0 = U52(X1,X2) a__isNat(0()) = 0 >= 0 = tt() a__isNat(s(V1)) = 0 >= 13V1 + 0 = a__U21(a__isNatKind(V1),V1) a__isNat(X) = 0 >= 0 = isNat(X) a__U21(tt(),V1) = 13V1 + 2 >= 0 = a__U22(a__isNat(V1)) a__U21(X1,X2) = 2X1 + 13X2 + 0 >= X1 + 1X2 + 0 = U21(X1,X2) a__U22(X) = X + 0 >= 2X + 4 = U22(X) a__U53(X) = 4 >= 6X + 0 = U53(X) a__U43(X) = 4X + 0 >= 2X = U43(X) a__U61(tt(),L) = 8L >= 5 = s(a__length(mark(L))) a__U61(X1,X2) = 8X2 >= X1 + 3X2 + 0 = U61(X1,X2) a__length(cons(N,L)) = 1 >= 8L = a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__length(X) = 1 >= X + 0 = length(X) problem: DPs: a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__and#(tt(),X) -> mark#(X) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) mark#(length(X)) -> a__length#(mark(X)) a__U61#(tt(),L) -> a__length#(mark(L)) a__length#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)), V1,V2) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__U61#(tt(),L) -> mark#(L) mark#(length(X)) -> mark#(X) mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(U61(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(isNat(X)) -> a__isNat#(X) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U21(X1,X2)) -> mark#(X1) TRS: a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(X) -> isNatIListKind(X) a__and(tt(),X) -> mark(X) a__and(X1,X2) -> and(X1,X2) mark(zeros()) -> a__zeros() mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)), V1,V2) a__isNatList(X) -> isNatList(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__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__U53(X) -> U53(X) a__U43(X) -> U43(X) a__U61(tt(),L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__length(X) -> length(X) Restore Modifier: DPs: a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__and#(tt(),X) -> mark#(X) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) mark#(length(X)) -> a__length#(mark(X)) a__U61#(tt(),L) -> a__length#(mark(L)) a__length#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)), V1,V2) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__U61#(tt(),L) -> mark#(L) mark#(length(X)) -> mark#(X) mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(U61(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(isNat(X)) -> a__isNat#(X) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U21(X1,X2)) -> mark#(X1) TRS: a__zeros() -> cons(0(),zeros()) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U61(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)), V1,V2) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() 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__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) mark(zeros()) -> a__zeros() mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U43(X) -> U43(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2) -> U61(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Usable Rule Processor: DPs: a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__and#(tt(),X) -> mark#(X) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) mark#(length(X)) -> a__length#(mark(X)) a__U61#(tt(),L) -> a__length#(mark(L)) a__length#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)), V1,V2) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__U61#(tt(),L) -> mark#(L) mark#(length(X)) -> mark#(X) mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(U61(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(isNat(X)) -> a__isNat#(X) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U21(X1,X2)) -> mark#(X1) TRS: a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(X) -> isNatIListKind(X) a__and(tt(),X) -> mark(X) a__and(X1,X2) -> and(X1,X2) mark(zeros()) -> a__zeros() mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)), V1,V2) a__isNatList(X) -> isNatList(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__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__U53(X) -> U53(X) a__U43(X) -> U43(X) a__U61(tt(),L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__length(X) -> length(X) Arctic Interpretation Processor: dimension: 1 usable rules: interpretation: [a__isNatIListKind#](x0) = 0, [a__isNatKind#](x0) = 0, [a__and#](x0, x1) = x1 + 0, [a__length#](x0) = 0, [mark#](x0) = x0, [a__U61#](x0, x1) = x1 + 1, [a__isNatList#](x0) = 0, [a__U52#](x0, x1) = 0, [a__U51#](x0, x1, x2) = 0, [a__isNat#](x0) = 0, [a__U21#](x0, x1) = 0, [U61](x0, x1) = 4x0 + 2x1 + 1, [U53](x0) = x0 + 6, [U52](x0, x1) = x0 + x1 + 4, [U51](x0, x1, x2) = x0 + 2x1 + x2 + 0, [U43](x0) = x0 + 2, [U22](x0) = x0, [U21](x0, x1) = x0 + x1 + 0, [isNatList](x0) = 2x0 + 0, [and](x0, x1) = 4x0 + 1x1 + 3, [isNatKind](x0) = x0 + 3, [isNat](x0) = x0 + 0, [isNatIListKind](x0) = 0, [a__isNatKind](x0) = x0 + 0, [a__isNatIListKind](x0) = 3x0 + 0, [length](x0) = x0 + 2, [a__and](x0, x1) = x0 + 2x1 + 0, [s](x0) = x0 + 2, [a__length](x0) = 5, [mark](x0) = x0 + 0, [a__U61](x0, x1) = x0 + 0, [a__U53](x0) = 4x0 + 0, [a__U52](x0, x1) = 4x0 + x1 + 0, [a__U51](x0, x1, x2) = x0 + x1 + 8x2 + 0, [a__U43](x0) = 5x0 + 1, [a__U22](x0) = 3x0 + 1, [a__isNat](x0) = x0 + 1, [a__U21](x0, x1) = 7x0 + 12x1 + 8, [a__isNatList](x0) = 2, [tt] = 0, [cons](x0, x1) = x0 + x1 + 1, [zeros] = 4, [0] = 4, [a__zeros] = 0 orientation: a__isNatKind#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNatKind#(length(V1)) = 0 >= 0 = a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__and#(tt(),X) = X + 0 >= X = mark#(X) mark#(isNatKind(X)) = X + 3 >= 0 = a__isNatKind#(X) mark#(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind#(X) mark#(and(X1,X2)) = 4X1 + 1X2 + 3 >= X2 + 0 = a__and#(mark(X1),X2) mark#(and(X1,X2)) = 4X1 + 1X2 + 3 >= X1 = mark#(X1) mark#(length(X)) = X + 2 >= 0 = a__length#(mark(X)) a__U61#(tt(),L) = L + 1 >= 0 = a__length#(mark(L)) a__length#(cons(N,L)) = 0 >= 0 = a__isNatList#(L) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) = 0 >= 0 = a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) = 0 >= 0 = a__isNatList#(V2) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__U51#(tt(),V1,V2) = 0 >= 0 = a__isNat#(V1) a__isNat#(s(V1)) = 0 >= 0 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = 0 >= 0 = a__isNat#(V1) a__isNat#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__U61#(tt(),L) = L + 1 >= L = mark#(L) mark#(length(X)) = X + 2 >= X = mark#(X) mark#(U61(X1,X2)) = 4X1 + 2X2 + 1 >= X2 + 1 = a__U61#(mark(X1),X2) mark#(U61(X1,X2)) = 4X1 + 2X2 + 1 >= X1 = mark#(X1) mark#(U52(X1,X2)) = X1 + X2 + 4 >= 0 = a__U52#(mark(X1),X2) mark#(U51(X1,X2,X3)) = X1 + 2X2 + X3 + 0 >= 0 = a__U51#(mark(X1),X2,X3) mark#(U51(X1,X2,X3)) = X1 + 2X2 + X3 + 0 >= X1 = mark#(X1) mark#(isNat(X)) = X + 0 >= 0 = a__isNat#(X) mark#(U21(X1,X2)) = X1 + X2 + 0 >= 0 = a__U21#(mark(X1),X2) mark#(U21(X1,X2)) = X1 + X2 + 0 >= X1 = mark#(X1) a__isNatKind(0()) = 4 >= 0 = tt() a__isNatKind(length(V1)) = V1 + 2 >= 3V1 + 0 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = V1 + 2 >= V1 + 0 = a__isNatKind(V1) a__isNatKind(X) = X + 0 >= X + 3 = isNatKind(X) a__isNatIListKind(zeros()) = 7 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 3V1 + 3V2 + 4 >= V1 + 2 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(X) = 3X + 0 >= 0 = isNatIListKind(X) a__and(tt(),X) = 2X + 0 >= X + 0 = mark(X) a__and(X1,X2) = X1 + 2X2 + 0 >= 4X1 + 1X2 + 3 = and(X1,X2) mark(zeros()) = 4 >= 0 = a__zeros() mark(isNatList(X)) = 2X + 0 >= 2 = a__isNatList(X) mark(U21(X1,X2)) = X1 + X2 + 0 >= 7X1 + 12X2 + 8 = a__U21(mark(X1),X2) mark(U22(X)) = X + 0 >= 3X + 3 = a__U22(mark(X)) mark(isNat(X)) = X + 0 >= X + 1 = a__isNat(X) mark(U43(X)) = X + 2 >= 5X + 5 = a__U43(mark(X)) mark(U51(X1,X2,X3)) = X1 + 2X2 + X3 + 0 >= X1 + X2 + 8X3 + 0 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = X1 + X2 + 4 >= 4X1 + X2 + 4 = a__U52(mark(X1),X2) mark(U53(X)) = X + 6 >= 4X + 4 = a__U53(mark(X)) mark(U61(X1,X2)) = 4X1 + 2X2 + 1 >= X1 + 0 = a__U61(mark(X1),X2) mark(length(X)) = X + 2 >= 5 = a__length(mark(X)) mark(and(X1,X2)) = 4X1 + 1X2 + 3 >= X1 + 2X2 + 0 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 0 >= 3X + 0 = a__isNatIListKind(X) mark(isNatKind(X)) = X + 3 >= X + 0 = a__isNatKind(X) mark(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(mark(X1),X2) mark(0()) = 4 >= 4 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X + 2 >= X + 2 = s(mark(X)) a__zeros() = 0 >= 4 = cons(0(),zeros()) a__zeros() = 0 >= 4 = zeros() a__isNatList(cons(V1,V2)) = 2 >= V1 + 8V2 + 2 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(X) = 2 >= 2X + 0 = isNatList(X) a__U51(tt(),V1,V2) = V1 + 8V2 + 0 >= 4V1 + V2 + 5 = a__U52(a__isNat(V1),V2) a__U51(X1,X2,X3) = X1 + X2 + 8X3 + 0 >= X1 + 2X2 + X3 + 0 = U51(X1,X2,X3) a__U52(tt(),V2) = V2 + 4 >= 6 = a__U53(a__isNatList(V2)) a__U52(X1,X2) = 4X1 + X2 + 0 >= X1 + X2 + 4 = U52(X1,X2) a__isNat(0()) = 4 >= 0 = tt() a__isNat(s(V1)) = V1 + 2 >= 12V1 + 8 = a__U21(a__isNatKind(V1),V1) a__isNat(X) = X + 1 >= X + 0 = isNat(X) a__U21(tt(),V1) = 12V1 + 8 >= 3V1 + 4 = a__U22(a__isNat(V1)) a__U21(X1,X2) = 7X1 + 12X2 + 8 >= X1 + X2 + 0 = U21(X1,X2) a__U22(X) = 3X + 1 >= X = U22(X) a__U53(X) = 4X + 0 >= X + 6 = U53(X) a__U43(X) = 5X + 1 >= X + 2 = U43(X) a__U61(tt(),L) = 0 >= 5 = s(a__length(mark(L))) a__U61(X1,X2) = X1 + 0 >= 4X1 + 2X2 + 1 = U61(X1,X2) a__length(cons(N,L)) = 5 >= 6N + 6 = a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__length(X) = 5 >= X + 2 = length(X) problem: DPs: a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__and#(tt(),X) -> mark#(X) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) a__length#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)), V1,V2) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__U61#(tt(),L) -> mark#(L) mark#(length(X)) -> mark#(X) mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(isNat(X)) -> a__isNat#(X) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U21(X1,X2)) -> mark#(X1) TRS: a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(X) -> isNatIListKind(X) a__and(tt(),X) -> mark(X) a__and(X1,X2) -> and(X1,X2) mark(zeros()) -> a__zeros() mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)), V1,V2) a__isNatList(X) -> isNatList(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__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNat(X) -> isNat(X) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__U53(X) -> U53(X) a__U43(X) -> U43(X) a__U61(tt(),L) -> s(a__length(mark(L))) a__U61(X1,X2) -> U61(X1,X2) a__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L), isNatIListKind (L)), and(isNat(N),isNatKind(N))), L) a__length(X) -> length(X) Restore Modifier: DPs: a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__and#(tt(),X) -> mark#(X) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) a__length#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)), V1,V2) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__U61#(tt(),L) -> mark#(L) mark#(length(X)) -> mark#(X) mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(isNat(X)) -> a__isNat#(X) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U21(X1,X2)) -> mark#(X1) TRS: a__zeros() -> cons(0(),zeros()) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U61(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)), V1,V2) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() 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__length(cons(N,L)) -> a__U61(a__and(a__and(a__isNatList(L), isNatIListKind (L)), and(isNat(N),isNatKind(N))), L) mark(zeros()) -> a__zeros() mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U43(X) -> U43(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2) -> U61(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) SCC Processor: #sccs: 1 #rules: 21 #arcs: 324/484 DPs: a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__and#(tt(),X) -> mark#(X) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(length(X)) -> mark#(X) mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) a__U61#(tt(),L) -> mark#(L) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)), V1,V2) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(isNat(X)) -> a__isNat#(X) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U21(X1,X2)) -> mark#(X1) TRS: a__zeros() -> cons(0(),zeros()) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U61(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)), V1,V2) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() 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__length(cons(N,L)) -> a__U61(a__and(a__and( a__isNatList(L), isNatIListKind (L)), and(isNat(N),isNatKind(N))), L) mark(zeros()) -> a__zeros() mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U43(X) -> U43(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2) -> U61(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Subterm Criterion Processor: simple projection: pi(length) = 0 pi(isNatIListKind) = [0,0] pi(isNatKind) = [0,0] pi(and) = [0,0] pi(isNatList) = 0 pi(U22) = 0 pi(U51) = [0,0,0,1,2,2] pi(U52) = [1,1] pi(U53) = [0,0] pi(U61) = 1 pi(a__U21#) = [1,1,1] pi(a__isNat#) = 0 pi(a__U51#) = [1,1,2,2,2] pi(a__U52#) = 1 pi(a__isNatList#) = 0 pi(a__U61#) = [1,1] pi(mark#) = [0,0] pi(a__and#) = [1,1] pi(a__isNatKind#) = 0 pi(a__isNatIListKind#) = 0 problem: DPs: a__and#(tt(),X) -> mark#(X) mark#(length(X)) -> mark#(X) mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) a__U61#(tt(),L) -> mark#(L) a__U52#(tt(),V2) -> a__isNatList#(V2) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) TRS: a__zeros() -> cons(0(),zeros()) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U61(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)), V1,V2) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() 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__length(cons(N,L)) -> a__U61(a__and(a__and (a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) mark(zeros()) -> a__zeros() mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U43(X) -> U43(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2) -> U61(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) SCC Processor: #sccs: 1 #rules: 3 #arcs: 64/36 DPs: mark#(length(X)) -> mark#(X) mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) a__U61#(tt(),L) -> mark#(L) TRS: a__zeros() -> cons(0(),zeros()) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U61(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)), V1,V2) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() 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__length(cons(N,L)) -> a__U61(a__and(a__and (a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) mark(zeros()) -> a__zeros() mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U43(X) -> U43(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2) -> U61(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Size-Change Termination Processor: DPs: TRS: a__zeros() -> cons(0(),zeros()) a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U61(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1), isNatIListKind (V2)), V1,V2) a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() 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__length(cons(N,L)) -> a__U61(a__and(a__and (a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) mark(zeros()) -> a__zeros() mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) 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(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U43(X) -> U43(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2) -> U61(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) The DP: mark#(length(X)) -> mark#(X) has the edges: 0 > 0 The DP: mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) has the edges: 0 > 1 The DP: a__U61#(tt(),L) -> mark#(L) has the edges: 1 >= 0 Qed