Termination w.r.t. Q of the following Term Rewriting System could not be shown:
Q restricted rewrite system:
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:
U921(X1, X2, X3, mark(X4)) → U921(X1, X2, X3, X4)
ACTIVE(U62(tt)) → MARK(tt)
U411(X1, mark(X2)) → U411(X1, X2)
U911(X1, X2, mark(X3), X4) → U911(X1, X2, X3, X4)
CONS(X1, mark(X2)) → CONS(X1, X2)
U911(active(X1), X2, X3, X4) → U911(X1, X2, X3, X4)
TAKE(active(X1), X2) → TAKE(X1, X2)
MARK(U42(X)) → U421(mark(X))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(U41(X1, X2)) → MARK(X1)
U911(X1, X2, active(X3), X4) → U911(X1, X2, X3, X4)
U931(X1, X2, X3, mark(X4)) → U931(X1, X2, X3, X4)
ACTIVE(U72(tt, L)) → S(length(L))
U921(X1, active(X2), X3, X4) → U921(X1, X2, X3, X4)
MARK(U91(X1, X2, X3, X4)) → U911(mark(X1), X2, X3, X4)
MARK(take(X1, X2)) → MARK(X2)
CONS(mark(X1), X2) → CONS(X1, X2)
MARK(U62(X)) → ACTIVE(U62(mark(X)))
MARK(U93(X1, X2, X3, X4)) → U931(mark(X1), X2, X3, X4)
MARK(U91(X1, X2, X3, X4)) → MARK(X1)
MARK(U92(X1, X2, X3, X4)) → MARK(X1)
ACTIVE(U91(tt, IL, M, N)) → MARK(U92(isNat(M), IL, M, N))
ACTIVE(U11(tt)) → MARK(tt)
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(U21(tt)) → MARK(tt)
U611(X1, active(X2)) → U611(X1, X2)
LENGTH(active(X)) → LENGTH(X)
U511(mark(X1), X2) → U511(X1, X2)
S(mark(X)) → S(X)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(U91(tt, IL, M, N)) → U921(isNat(M), IL, M, N)
U211(active(X)) → U211(X)
MARK(U61(X1, X2)) → U611(mark(X1), X2)
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
ACTIVE(isNatList(nil)) → MARK(tt)
U711(mark(X1), X2, X3) → U711(X1, X2, X3)
U911(X1, X2, X3, active(X4)) → U911(X1, X2, X3, X4)
U921(X1, X2, X3, active(X4)) → U921(X1, X2, X3, X4)
U311(mark(X)) → U311(X)
MARK(U52(X)) → U521(mark(X))
ACTIVE(U72(tt, L)) → LENGTH(L)
MARK(U21(X)) → ACTIVE(U21(mark(X)))
U421(active(X)) → U421(X)
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(isNat(length(V1))) → U111(isNatList(V1))
MARK(U42(X)) → ACTIVE(U42(mark(X)))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
U411(mark(X1), X2) → U411(X1, X2)
ACTIVE(length(cons(N, L))) → U711(isNatList(L), L, N)
MARK(U62(X)) → U621(mark(X))
ACTIVE(U93(tt, IL, M, N)) → TAKE(M, IL)
U711(active(X1), X2, X3) → U711(X1, X2, X3)
ACTIVE(isNatList(take(V1, V2))) → U611(isNat(V1), V2)
U611(X1, mark(X2)) → U611(X1, X2)
ACTIVE(zeros) → CONS(0, zeros)
ACTIVE(isNatList(take(V1, V2))) → ISNAT(V1)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U93(X1, X2, X3, X4)) → ACTIVE(U93(mark(X1), X2, X3, X4))
MARK(take(X1, X2)) → MARK(X1)
ACTIVE(isNatIList(cons(V1, V2))) → ISNAT(V1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(U51(X1, X2)) → MARK(X1)
ACTIVE(take(0, IL)) → MARK(U81(isNatIList(IL)))
MARK(U31(X)) → U311(mark(X))
S(active(X)) → S(X)
MARK(cons(X1, X2)) → CONS(mark(X1), X2)
U311(active(X)) → U311(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U72(X1, X2)) → U721(mark(X1), X2)
ACTIVE(take(0, IL)) → ISNATILIST(IL)
U511(active(X1), X2) → U511(X1, X2)
ACTIVE(U71(tt, L, N)) → U721(isNat(N), L)
CONS(active(X1), X2) → CONS(X1, X2)
MARK(U81(X)) → U811(mark(X))
MARK(U62(X)) → MARK(X)
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
U921(X1, mark(X2), X3, X4) → U921(X1, X2, X3, X4)
ACTIVE(isNat(length(V1))) → ISNATLIST(V1)
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNatList(take(V1, V2))) → MARK(U61(isNat(V1), V2))
ACTIVE(U92(tt, IL, M, N)) → ISNAT(N)
MARK(U72(X1, X2)) → MARK(X1)
U921(X1, X2, active(X3), X4) → U921(X1, X2, X3, X4)
U511(X1, mark(X2)) → U511(X1, X2)
ACTIVE(take(s(M), cons(N, IL))) → ISNATILIST(IL)
U711(X1, active(X2), X3) → U711(X1, X2, X3)
ACTIVE(U91(tt, IL, M, N)) → ISNAT(M)
U111(active(X)) → U111(X)
U931(X1, X2, X3, active(X4)) → U931(X1, X2, X3, X4)
U811(mark(X)) → U811(X)
U611(mark(X1), X2) → U611(X1, X2)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
U521(active(X)) → U521(X)
ACTIVE(isNatIList(V)) → U311(isNatList(V))
ACTIVE(length(cons(N, L))) → ISNATLIST(L)
ISNAT(active(X)) → ISNAT(X)
MARK(U21(X)) → MARK(X)
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
ACTIVE(U41(tt, V2)) → ISNATILIST(V2)
U421(mark(X)) → U421(X)
ACTIVE(zeros) → MARK(cons(0, zeros))
U721(X1, active(X2)) → U721(X1, X2)
U931(mark(X1), X2, X3, X4) → U931(X1, X2, X3, X4)
ACTIVE(take(s(M), cons(N, IL))) → MARK(U91(isNatIList(IL), IL, M, N))
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIList(V2)))
ACTIVE(U61(tt, V2)) → U621(isNatIList(V2))
ACTIVE(U41(tt, V2)) → U421(isNatIList(V2))
ACTIVE(U51(tt, V2)) → U521(isNatList(V2))
U931(active(X1), X2, X3, X4) → U931(X1, X2, X3, X4)
ACTIVE(isNatIList(zeros)) → MARK(tt)
MARK(length(X)) → MARK(X)
MARK(U81(X)) → ACTIVE(U81(mark(X)))
MARK(U31(X)) → ACTIVE(U31(mark(X)))
ACTIVE(U61(tt, V2)) → ISNATILIST(V2)
MARK(take(X1, X2)) → TAKE(mark(X1), mark(X2))
U911(X1, mark(X2), X3, X4) → U911(X1, X2, X3, X4)
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
ACTIVE(isNatList(cons(V1, V2))) → U511(isNat(V1), V2)
U931(X1, mark(X2), X3, X4) → U931(X1, X2, X3, X4)
ACTIVE(U31(tt)) → MARK(tt)
ISNATILIST(mark(X)) → ISNATILIST(X)
U811(active(X)) → U811(X)
U911(X1, X2, X3, mark(X4)) → U911(X1, X2, X3, X4)
U611(active(X1), X2) → U611(X1, X2)
U711(X1, X2, active(X3)) → U711(X1, X2, X3)
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U41(X1, X2)) → U411(mark(X1), X2)
ACTIVE(isNatIList(cons(V1, V2))) → U411(isNat(V1), V2)
ACTIVE(isNat(0)) → MARK(tt)
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U71(X1, X2, X3)) → U711(mark(X1), X2, X3)
ACTIVE(isNat(s(V1))) → U211(isNat(V1))
MARK(U93(X1, X2, X3, X4)) → MARK(X1)
CONS(X1, active(X2)) → CONS(X1, X2)
ACTIVE(U92(tt, IL, M, N)) → U931(isNat(N), IL, M, N)
U921(X1, X2, mark(X3), X4) → U921(X1, X2, X3, X4)
U721(mark(X1), X2) → U721(X1, X2)
MARK(U52(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → ISNATLIST(V2)
U921(active(X1), X2, X3, X4) → U921(X1, X2, X3, X4)
LENGTH(mark(X)) → LENGTH(X)
MARK(U31(X)) → MARK(X)
ACTIVE(U93(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
ACTIVE(U52(tt)) → MARK(tt)
ISNAT(mark(X)) → ISNAT(X)
U111(mark(X)) → U111(X)
ACTIVE(length(nil)) → MARK(0)
MARK(U61(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(V1))) → ISNAT(V1)
U721(active(X1), X2) → U721(X1, X2)
MARK(U92(X1, X2, X3, X4)) → ACTIVE(U92(mark(X1), X2, X3, X4))
U931(X1, active(X2), X3, X4) → U931(X1, X2, X3, X4)
MARK(U11(X)) → MARK(X)
MARK(tt) → ACTIVE(tt)
TAKE(X1, active(X2)) → TAKE(X1, X2)
MARK(U91(X1, X2, X3, X4)) → ACTIVE(U91(mark(X1), X2, X3, X4))
MARK(cons(X1, X2)) → MARK(X1)
MARK(U92(X1, X2, X3, X4)) → U921(mark(X1), X2, X3, X4)
MARK(U21(X)) → U211(mark(X))
ACTIVE(U92(tt, IL, M, N)) → MARK(U93(isNat(N), IL, M, N))
U921(mark(X1), X2, X3, X4) → U921(X1, X2, X3, X4)
U411(active(X1), X2) → U411(X1, X2)
ACTIVE(isNatIList(V)) → ISNATLIST(V)
U911(mark(X1), X2, X3, X4) → U911(X1, X2, X3, X4)
MARK(U52(X)) → ACTIVE(U52(mark(X)))
U411(X1, active(X2)) → U411(X1, X2)
U931(X1, X2, active(X3), X4) → U931(X1, X2, X3, X4)
U711(X1, X2, mark(X3)) → U711(X1, X2, X3)
U211(mark(X)) → U211(X)
ACTIVE(U42(tt)) → MARK(tt)
ACTIVE(isNatList(cons(V1, V2))) → ISNAT(V1)
ACTIVE(U93(tt, IL, M, N)) → CONS(N, take(M, IL))
MARK(U11(X)) → ACTIVE(U11(mark(X)))
U911(X1, active(X2), X3, X4) → U911(X1, X2, X3, X4)
MARK(zeros) → ACTIVE(zeros)
MARK(length(X)) → LENGTH(mark(X))
TAKE(mark(X1), X2) → TAKE(X1, X2)
MARK(U42(X)) → MARK(X)
ISNATLIST(active(X)) → ISNATLIST(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
U521(mark(X)) → U521(X)
ACTIVE(U71(tt, L, N)) → ISNAT(N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U11(X)) → U111(mark(X))
ACTIVE(take(s(M), cons(N, IL))) → U911(isNatIList(IL), IL, M, N)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
ISNATLIST(mark(X)) → ISNATLIST(X)
MARK(U81(X)) → MARK(X)
U621(mark(X)) → U621(X)
U621(active(X)) → U621(X)
U711(X1, mark(X2), X3) → U711(X1, X2, X3)
ACTIVE(take(0, IL)) → U811(isNatIList(IL))
MARK(s(X)) → S(mark(X))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
U721(X1, mark(X2)) → U721(X1, X2)
MARK(U51(X1, X2)) → U511(mark(X1), X2)
ACTIVE(U81(tt)) → MARK(nil)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
TAKE(X1, mark(X2)) → TAKE(X1, X2)
MARK(0) → ACTIVE(0)
ISNATILIST(active(X)) → ISNATILIST(X)
U931(X1, X2, mark(X3), X4) → U931(X1, X2, X3, X4)
U511(X1, active(X2)) → U511(X1, X2)
MARK(nil) → ACTIVE(nil)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
U921(X1, X2, X3, mark(X4)) → U921(X1, X2, X3, X4)
ACTIVE(U62(tt)) → MARK(tt)
U411(X1, mark(X2)) → U411(X1, X2)
U911(X1, X2, mark(X3), X4) → U911(X1, X2, X3, X4)
CONS(X1, mark(X2)) → CONS(X1, X2)
U911(active(X1), X2, X3, X4) → U911(X1, X2, X3, X4)
TAKE(active(X1), X2) → TAKE(X1, X2)
MARK(U42(X)) → U421(mark(X))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(U41(X1, X2)) → MARK(X1)
U911(X1, X2, active(X3), X4) → U911(X1, X2, X3, X4)
U931(X1, X2, X3, mark(X4)) → U931(X1, X2, X3, X4)
ACTIVE(U72(tt, L)) → S(length(L))
U921(X1, active(X2), X3, X4) → U921(X1, X2, X3, X4)
MARK(U91(X1, X2, X3, X4)) → U911(mark(X1), X2, X3, X4)
MARK(take(X1, X2)) → MARK(X2)
CONS(mark(X1), X2) → CONS(X1, X2)
MARK(U62(X)) → ACTIVE(U62(mark(X)))
MARK(U93(X1, X2, X3, X4)) → U931(mark(X1), X2, X3, X4)
MARK(U91(X1, X2, X3, X4)) → MARK(X1)
MARK(U92(X1, X2, X3, X4)) → MARK(X1)
ACTIVE(U91(tt, IL, M, N)) → MARK(U92(isNat(M), IL, M, N))
ACTIVE(U11(tt)) → MARK(tt)
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(U21(tt)) → MARK(tt)
U611(X1, active(X2)) → U611(X1, X2)
LENGTH(active(X)) → LENGTH(X)
U511(mark(X1), X2) → U511(X1, X2)
S(mark(X)) → S(X)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(U91(tt, IL, M, N)) → U921(isNat(M), IL, M, N)
U211(active(X)) → U211(X)
MARK(U61(X1, X2)) → U611(mark(X1), X2)
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
ACTIVE(isNatList(nil)) → MARK(tt)
U711(mark(X1), X2, X3) → U711(X1, X2, X3)
U911(X1, X2, X3, active(X4)) → U911(X1, X2, X3, X4)
U921(X1, X2, X3, active(X4)) → U921(X1, X2, X3, X4)
U311(mark(X)) → U311(X)
MARK(U52(X)) → U521(mark(X))
ACTIVE(U72(tt, L)) → LENGTH(L)
MARK(U21(X)) → ACTIVE(U21(mark(X)))
U421(active(X)) → U421(X)
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(isNat(length(V1))) → U111(isNatList(V1))
MARK(U42(X)) → ACTIVE(U42(mark(X)))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
U411(mark(X1), X2) → U411(X1, X2)
ACTIVE(length(cons(N, L))) → U711(isNatList(L), L, N)
MARK(U62(X)) → U621(mark(X))
ACTIVE(U93(tt, IL, M, N)) → TAKE(M, IL)
U711(active(X1), X2, X3) → U711(X1, X2, X3)
ACTIVE(isNatList(take(V1, V2))) → U611(isNat(V1), V2)
U611(X1, mark(X2)) → U611(X1, X2)
ACTIVE(zeros) → CONS(0, zeros)
ACTIVE(isNatList(take(V1, V2))) → ISNAT(V1)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U93(X1, X2, X3, X4)) → ACTIVE(U93(mark(X1), X2, X3, X4))
MARK(take(X1, X2)) → MARK(X1)
ACTIVE(isNatIList(cons(V1, V2))) → ISNAT(V1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(U51(X1, X2)) → MARK(X1)
ACTIVE(take(0, IL)) → MARK(U81(isNatIList(IL)))
MARK(U31(X)) → U311(mark(X))
S(active(X)) → S(X)
MARK(cons(X1, X2)) → CONS(mark(X1), X2)
U311(active(X)) → U311(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U72(X1, X2)) → U721(mark(X1), X2)
ACTIVE(take(0, IL)) → ISNATILIST(IL)
U511(active(X1), X2) → U511(X1, X2)
ACTIVE(U71(tt, L, N)) → U721(isNat(N), L)
CONS(active(X1), X2) → CONS(X1, X2)
MARK(U81(X)) → U811(mark(X))
MARK(U62(X)) → MARK(X)
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
U921(X1, mark(X2), X3, X4) → U921(X1, X2, X3, X4)
ACTIVE(isNat(length(V1))) → ISNATLIST(V1)
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNatList(take(V1, V2))) → MARK(U61(isNat(V1), V2))
ACTIVE(U92(tt, IL, M, N)) → ISNAT(N)
MARK(U72(X1, X2)) → MARK(X1)
U921(X1, X2, active(X3), X4) → U921(X1, X2, X3, X4)
U511(X1, mark(X2)) → U511(X1, X2)
ACTIVE(take(s(M), cons(N, IL))) → ISNATILIST(IL)
U711(X1, active(X2), X3) → U711(X1, X2, X3)
ACTIVE(U91(tt, IL, M, N)) → ISNAT(M)
U111(active(X)) → U111(X)
U931(X1, X2, X3, active(X4)) → U931(X1, X2, X3, X4)
U811(mark(X)) → U811(X)
U611(mark(X1), X2) → U611(X1, X2)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
U521(active(X)) → U521(X)
ACTIVE(isNatIList(V)) → U311(isNatList(V))
ACTIVE(length(cons(N, L))) → ISNATLIST(L)
ISNAT(active(X)) → ISNAT(X)
MARK(U21(X)) → MARK(X)
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
ACTIVE(U41(tt, V2)) → ISNATILIST(V2)
U421(mark(X)) → U421(X)
ACTIVE(zeros) → MARK(cons(0, zeros))
U721(X1, active(X2)) → U721(X1, X2)
U931(mark(X1), X2, X3, X4) → U931(X1, X2, X3, X4)
ACTIVE(take(s(M), cons(N, IL))) → MARK(U91(isNatIList(IL), IL, M, N))
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIList(V2)))
ACTIVE(U61(tt, V2)) → U621(isNatIList(V2))
ACTIVE(U41(tt, V2)) → U421(isNatIList(V2))
ACTIVE(U51(tt, V2)) → U521(isNatList(V2))
U931(active(X1), X2, X3, X4) → U931(X1, X2, X3, X4)
ACTIVE(isNatIList(zeros)) → MARK(tt)
MARK(length(X)) → MARK(X)
MARK(U81(X)) → ACTIVE(U81(mark(X)))
MARK(U31(X)) → ACTIVE(U31(mark(X)))
ACTIVE(U61(tt, V2)) → ISNATILIST(V2)
MARK(take(X1, X2)) → TAKE(mark(X1), mark(X2))
U911(X1, mark(X2), X3, X4) → U911(X1, X2, X3, X4)
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
ACTIVE(isNatList(cons(V1, V2))) → U511(isNat(V1), V2)
U931(X1, mark(X2), X3, X4) → U931(X1, X2, X3, X4)
ACTIVE(U31(tt)) → MARK(tt)
ISNATILIST(mark(X)) → ISNATILIST(X)
U811(active(X)) → U811(X)
U911(X1, X2, X3, mark(X4)) → U911(X1, X2, X3, X4)
U611(active(X1), X2) → U611(X1, X2)
U711(X1, X2, active(X3)) → U711(X1, X2, X3)
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U41(X1, X2)) → U411(mark(X1), X2)
ACTIVE(isNatIList(cons(V1, V2))) → U411(isNat(V1), V2)
ACTIVE(isNat(0)) → MARK(tt)
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U71(X1, X2, X3)) → U711(mark(X1), X2, X3)
ACTIVE(isNat(s(V1))) → U211(isNat(V1))
MARK(U93(X1, X2, X3, X4)) → MARK(X1)
CONS(X1, active(X2)) → CONS(X1, X2)
ACTIVE(U92(tt, IL, M, N)) → U931(isNat(N), IL, M, N)
U921(X1, X2, mark(X3), X4) → U921(X1, X2, X3, X4)
U721(mark(X1), X2) → U721(X1, X2)
MARK(U52(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → ISNATLIST(V2)
U921(active(X1), X2, X3, X4) → U921(X1, X2, X3, X4)
LENGTH(mark(X)) → LENGTH(X)
MARK(U31(X)) → MARK(X)
ACTIVE(U93(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
ACTIVE(U52(tt)) → MARK(tt)
ISNAT(mark(X)) → ISNAT(X)
U111(mark(X)) → U111(X)
ACTIVE(length(nil)) → MARK(0)
MARK(U61(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(V1))) → ISNAT(V1)
U721(active(X1), X2) → U721(X1, X2)
MARK(U92(X1, X2, X3, X4)) → ACTIVE(U92(mark(X1), X2, X3, X4))
U931(X1, active(X2), X3, X4) → U931(X1, X2, X3, X4)
MARK(U11(X)) → MARK(X)
MARK(tt) → ACTIVE(tt)
TAKE(X1, active(X2)) → TAKE(X1, X2)
MARK(U91(X1, X2, X3, X4)) → ACTIVE(U91(mark(X1), X2, X3, X4))
MARK(cons(X1, X2)) → MARK(X1)
MARK(U92(X1, X2, X3, X4)) → U921(mark(X1), X2, X3, X4)
MARK(U21(X)) → U211(mark(X))
ACTIVE(U92(tt, IL, M, N)) → MARK(U93(isNat(N), IL, M, N))
U921(mark(X1), X2, X3, X4) → U921(X1, X2, X3, X4)
U411(active(X1), X2) → U411(X1, X2)
ACTIVE(isNatIList(V)) → ISNATLIST(V)
U911(mark(X1), X2, X3, X4) → U911(X1, X2, X3, X4)
MARK(U52(X)) → ACTIVE(U52(mark(X)))
U411(X1, active(X2)) → U411(X1, X2)
U931(X1, X2, active(X3), X4) → U931(X1, X2, X3, X4)
U711(X1, X2, mark(X3)) → U711(X1, X2, X3)
U211(mark(X)) → U211(X)
ACTIVE(U42(tt)) → MARK(tt)
ACTIVE(isNatList(cons(V1, V2))) → ISNAT(V1)
ACTIVE(U93(tt, IL, M, N)) → CONS(N, take(M, IL))
MARK(U11(X)) → ACTIVE(U11(mark(X)))
U911(X1, active(X2), X3, X4) → U911(X1, X2, X3, X4)
MARK(zeros) → ACTIVE(zeros)
MARK(length(X)) → LENGTH(mark(X))
TAKE(mark(X1), X2) → TAKE(X1, X2)
MARK(U42(X)) → MARK(X)
ISNATLIST(active(X)) → ISNATLIST(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
U521(mark(X)) → U521(X)
ACTIVE(U71(tt, L, N)) → ISNAT(N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U11(X)) → U111(mark(X))
ACTIVE(take(s(M), cons(N, IL))) → U911(isNatIList(IL), IL, M, N)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
ISNATLIST(mark(X)) → ISNATLIST(X)
MARK(U81(X)) → MARK(X)
U621(mark(X)) → U621(X)
U621(active(X)) → U621(X)
U711(X1, mark(X2), X3) → U711(X1, X2, X3)
ACTIVE(take(0, IL)) → U811(isNatIList(IL))
MARK(s(X)) → S(mark(X))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
U721(X1, mark(X2)) → U721(X1, X2)
MARK(U51(X1, X2)) → U511(mark(X1), X2)
ACTIVE(U81(tt)) → MARK(nil)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
TAKE(X1, mark(X2)) → TAKE(X1, X2)
MARK(0) → ACTIVE(0)
ISNATILIST(active(X)) → ISNATILIST(X)
U931(X1, X2, mark(X3), X4) → U931(X1, X2, X3, X4)
U511(X1, active(X2)) → U511(X1, X2)
MARK(nil) → ACTIVE(nil)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 23 SCCs with 68 less nodes.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
TAKE(X1, active(X2)) → TAKE(X1, X2)
TAKE(active(X1), X2) → TAKE(X1, X2)
TAKE(mark(X1), X2) → TAKE(X1, X2)
TAKE(X1, mark(X2)) → TAKE(X1, X2)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
TAKE(X1, active(X2)) → TAKE(X1, X2)
TAKE(mark(X1), X2) → TAKE(X1, X2)
TAKE(active(X1), X2) → TAKE(X1, X2)
TAKE(X1, mark(X2)) → TAKE(X1, X2)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- TAKE(X1, active(X2)) → TAKE(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
- TAKE(active(X1), X2) → TAKE(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
- TAKE(mark(X1), X2) → TAKE(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
- TAKE(X1, mark(X2)) → TAKE(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U931(X1, X2, active(X3), X4) → U931(X1, X2, X3, X4)
U931(X1, X2, X3, mark(X4)) → U931(X1, X2, X3, X4)
U931(X1, active(X2), X3, X4) → U931(X1, X2, X3, X4)
U931(X1, mark(X2), X3, X4) → U931(X1, X2, X3, X4)
U931(X1, X2, X3, active(X4)) → U931(X1, X2, X3, X4)
U931(X1, X2, mark(X3), X4) → U931(X1, X2, X3, X4)
U931(mark(X1), X2, X3, X4) → U931(X1, X2, X3, X4)
U931(active(X1), X2, X3, X4) → U931(X1, X2, X3, X4)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U931(X1, X2, active(X3), X4) → U931(X1, X2, X3, X4)
U931(X1, X2, X3, mark(X4)) → U931(X1, X2, X3, X4)
U931(X1, active(X2), X3, X4) → U931(X1, X2, X3, X4)
U931(X1, mark(X2), X3, X4) → U931(X1, X2, X3, X4)
U931(mark(X1), X2, X3, X4) → U931(X1, X2, X3, X4)
U931(X1, X2, mark(X3), X4) → U931(X1, X2, X3, X4)
U931(X1, X2, X3, active(X4)) → U931(X1, X2, X3, X4)
U931(active(X1), X2, X3, X4) → U931(X1, X2, X3, X4)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- U931(X1, X2, active(X3), X4) → U931(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4
- U931(X1, X2, X3, mark(X4)) → U931(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4
- U931(X1, active(X2), X3, X4) → U931(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4
- U931(X1, mark(X2), X3, X4) → U931(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4
- U931(X1, X2, X3, active(X4)) → U931(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4
- U931(X1, X2, mark(X3), X4) → U931(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4
- U931(mark(X1), X2, X3, X4) → U931(X1, X2, X3, X4)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4
- U931(active(X1), X2, X3, X4) → U931(X1, X2, X3, X4)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U921(X1, X2, active(X3), X4) → U921(X1, X2, X3, X4)
U921(X1, X2, X3, mark(X4)) → U921(X1, X2, X3, X4)
U921(X1, X2, mark(X3), X4) → U921(X1, X2, X3, X4)
U921(X1, X2, X3, active(X4)) → U921(X1, X2, X3, X4)
U921(active(X1), X2, X3, X4) → U921(X1, X2, X3, X4)
U921(X1, mark(X2), X3, X4) → U921(X1, X2, X3, X4)
U921(X1, active(X2), X3, X4) → U921(X1, X2, X3, X4)
U921(mark(X1), X2, X3, X4) → U921(X1, X2, X3, X4)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U921(X1, X2, X3, mark(X4)) → U921(X1, X2, X3, X4)
U921(X1, X2, active(X3), X4) → U921(X1, X2, X3, X4)
U921(X1, X2, mark(X3), X4) → U921(X1, X2, X3, X4)
U921(X1, X2, X3, active(X4)) → U921(X1, X2, X3, X4)
U921(active(X1), X2, X3, X4) → U921(X1, X2, X3, X4)
U921(X1, mark(X2), X3, X4) → U921(X1, X2, X3, X4)
U921(X1, active(X2), X3, X4) → U921(X1, X2, X3, X4)
U921(mark(X1), X2, X3, X4) → U921(X1, X2, X3, X4)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- U921(X1, X2, active(X3), X4) → U921(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4
- U921(X1, X2, X3, mark(X4)) → U921(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4
- U921(X1, X2, mark(X3), X4) → U921(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4
- U921(X1, X2, X3, active(X4)) → U921(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4
- U921(active(X1), X2, X3, X4) → U921(X1, X2, X3, X4)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4
- U921(X1, mark(X2), X3, X4) → U921(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4
- U921(X1, active(X2), X3, X4) → U921(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4
- U921(mark(X1), X2, X3, X4) → U921(X1, X2, X3, X4)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U911(X1, X2, X3, active(X4)) → U911(X1, X2, X3, X4)
U911(X1, active(X2), X3, X4) → U911(X1, X2, X3, X4)
U911(active(X1), X2, X3, X4) → U911(X1, X2, X3, X4)
U911(X1, X2, active(X3), X4) → U911(X1, X2, X3, X4)
U911(X1, X2, X3, mark(X4)) → U911(X1, X2, X3, X4)
U911(X1, X2, mark(X3), X4) → U911(X1, X2, X3, X4)
U911(X1, mark(X2), X3, X4) → U911(X1, X2, X3, X4)
U911(mark(X1), X2, X3, X4) → U911(X1, X2, X3, X4)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U911(X1, X2, X3, active(X4)) → U911(X1, X2, X3, X4)
U911(X1, active(X2), X3, X4) → U911(X1, X2, X3, X4)
U911(active(X1), X2, X3, X4) → U911(X1, X2, X3, X4)
U911(X1, X2, active(X3), X4) → U911(X1, X2, X3, X4)
U911(X1, X2, X3, mark(X4)) → U911(X1, X2, X3, X4)
U911(X1, X2, mark(X3), X4) → U911(X1, X2, X3, X4)
U911(X1, mark(X2), X3, X4) → U911(X1, X2, X3, X4)
U911(mark(X1), X2, X3, X4) → U911(X1, X2, X3, X4)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- U911(X1, X2, X3, active(X4)) → U911(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4
- U911(X1, active(X2), X3, X4) → U911(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4
- U911(active(X1), X2, X3, X4) → U911(X1, X2, X3, X4)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4
- U911(X1, X2, active(X3), X4) → U911(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4
- U911(X1, X2, X3, mark(X4)) → U911(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4
- U911(X1, X2, mark(X3), X4) → U911(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4
- U911(X1, mark(X2), X3, X4) → U911(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4
- U911(mark(X1), X2, X3, X4) → U911(X1, X2, X3, X4)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U811(active(X)) → U811(X)
U811(mark(X)) → U811(X)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U811(active(X)) → U811(X)
U811(mark(X)) → U811(X)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- U811(active(X)) → U811(X)
The graph contains the following edges 1 > 1
- U811(mark(X)) → U811(X)
The graph contains the following edges 1 > 1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
LENGTH(mark(X)) → LENGTH(X)
LENGTH(active(X)) → LENGTH(X)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
LENGTH(mark(X)) → LENGTH(X)
LENGTH(active(X)) → LENGTH(X)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- LENGTH(mark(X)) → LENGTH(X)
The graph contains the following edges 1 > 1
- LENGTH(active(X)) → LENGTH(X)
The graph contains the following edges 1 > 1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
S(mark(X)) → S(X)
S(active(X)) → S(X)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
S(active(X)) → S(X)
S(mark(X)) → S(X)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- S(mark(X)) → S(X)
The graph contains the following edges 1 > 1
- S(active(X)) → S(X)
The graph contains the following edges 1 > 1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNAT(active(X)) → ISNAT(X)
ISNAT(mark(X)) → ISNAT(X)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNAT(active(X)) → ISNAT(X)
ISNAT(mark(X)) → ISNAT(X)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- ISNAT(active(X)) → ISNAT(X)
The graph contains the following edges 1 > 1
- ISNAT(mark(X)) → ISNAT(X)
The graph contains the following edges 1 > 1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U721(active(X1), X2) → U721(X1, X2)
U721(mark(X1), X2) → U721(X1, X2)
U721(X1, mark(X2)) → U721(X1, X2)
U721(X1, active(X2)) → U721(X1, X2)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U721(active(X1), X2) → U721(X1, X2)
U721(mark(X1), X2) → U721(X1, X2)
U721(X1, mark(X2)) → U721(X1, X2)
U721(X1, active(X2)) → U721(X1, X2)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- U721(active(X1), X2) → U721(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
- U721(mark(X1), X2) → U721(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
- U721(X1, mark(X2)) → U721(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
- U721(X1, active(X2)) → U721(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U711(active(X1), X2, X3) → U711(X1, X2, X3)
U711(mark(X1), X2, X3) → U711(X1, X2, X3)
U711(X1, mark(X2), X3) → U711(X1, X2, X3)
U711(X1, active(X2), X3) → U711(X1, X2, X3)
U711(X1, X2, mark(X3)) → U711(X1, X2, X3)
U711(X1, X2, active(X3)) → U711(X1, X2, X3)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U711(active(X1), X2, X3) → U711(X1, X2, X3)
U711(X1, mark(X2), X3) → U711(X1, X2, X3)
U711(mark(X1), X2, X3) → U711(X1, X2, X3)
U711(X1, active(X2), X3) → U711(X1, X2, X3)
U711(X1, X2, mark(X3)) → U711(X1, X2, X3)
U711(X1, X2, active(X3)) → U711(X1, X2, X3)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- U711(active(X1), X2, X3) → U711(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
- U711(mark(X1), X2, X3) → U711(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
- U711(X1, mark(X2), X3) → U711(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
- U711(X1, active(X2), X3) → U711(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
- U711(X1, X2, mark(X3)) → U711(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
- U711(X1, X2, active(X3)) → U711(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U621(mark(X)) → U621(X)
U621(active(X)) → U621(X)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U621(mark(X)) → U621(X)
U621(active(X)) → U621(X)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- U621(mark(X)) → U621(X)
The graph contains the following edges 1 > 1
- U621(active(X)) → U621(X)
The graph contains the following edges 1 > 1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U611(X1, mark(X2)) → U611(X1, X2)
U611(X1, active(X2)) → U611(X1, X2)
U611(active(X1), X2) → U611(X1, X2)
U611(mark(X1), X2) → U611(X1, X2)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U611(X1, mark(X2)) → U611(X1, X2)
U611(active(X1), X2) → U611(X1, X2)
U611(X1, active(X2)) → U611(X1, X2)
U611(mark(X1), X2) → U611(X1, X2)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- U611(X1, mark(X2)) → U611(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
- U611(X1, active(X2)) → U611(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
- U611(active(X1), X2) → U611(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
- U611(mark(X1), X2) → U611(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(mark(X)) → ISNATLIST(X)
ISNATLIST(active(X)) → ISNATLIST(X)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(mark(X)) → ISNATLIST(X)
ISNATLIST(active(X)) → ISNATLIST(X)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- ISNATLIST(mark(X)) → ISNATLIST(X)
The graph contains the following edges 1 > 1
- ISNATLIST(active(X)) → ISNATLIST(X)
The graph contains the following edges 1 > 1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U521(mark(X)) → U521(X)
U521(active(X)) → U521(X)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U521(active(X)) → U521(X)
U521(mark(X)) → U521(X)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- U521(mark(X)) → U521(X)
The graph contains the following edges 1 > 1
- U521(active(X)) → U521(X)
The graph contains the following edges 1 > 1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U511(X1, mark(X2)) → U511(X1, X2)
U511(active(X1), X2) → U511(X1, X2)
U511(X1, active(X2)) → U511(X1, X2)
U511(mark(X1), X2) → U511(X1, X2)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U511(X1, mark(X2)) → U511(X1, X2)
U511(active(X1), X2) → U511(X1, X2)
U511(X1, active(X2)) → U511(X1, X2)
U511(mark(X1), X2) → U511(X1, X2)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- U511(X1, mark(X2)) → U511(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
- U511(active(X1), X2) → U511(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
- U511(X1, active(X2)) → U511(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
- U511(mark(X1), X2) → U511(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(mark(X)) → ISNATILIST(X)
ISNATILIST(active(X)) → ISNATILIST(X)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(mark(X)) → ISNATILIST(X)
ISNATILIST(active(X)) → ISNATILIST(X)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- ISNATILIST(mark(X)) → ISNATILIST(X)
The graph contains the following edges 1 > 1
- ISNATILIST(active(X)) → ISNATILIST(X)
The graph contains the following edges 1 > 1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U421(mark(X)) → U421(X)
U421(active(X)) → U421(X)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U421(active(X)) → U421(X)
U421(mark(X)) → U421(X)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- U421(mark(X)) → U421(X)
The graph contains the following edges 1 > 1
- U421(active(X)) → U421(X)
The graph contains the following edges 1 > 1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U411(X1, active(X2)) → U411(X1, X2)
U411(active(X1), X2) → U411(X1, X2)
U411(X1, mark(X2)) → U411(X1, X2)
U411(mark(X1), X2) → U411(X1, X2)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U411(X1, active(X2)) → U411(X1, X2)
U411(active(X1), X2) → U411(X1, X2)
U411(X1, mark(X2)) → U411(X1, X2)
U411(mark(X1), X2) → U411(X1, X2)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- U411(X1, active(X2)) → U411(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
- U411(active(X1), X2) → U411(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
- U411(X1, mark(X2)) → U411(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
- U411(mark(X1), X2) → U411(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U311(active(X)) → U311(X)
U311(mark(X)) → U311(X)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U311(active(X)) → U311(X)
U311(mark(X)) → U311(X)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- U311(active(X)) → U311(X)
The graph contains the following edges 1 > 1
- U311(mark(X)) → U311(X)
The graph contains the following edges 1 > 1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U211(mark(X)) → U211(X)
U211(active(X)) → U211(X)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U211(mark(X)) → U211(X)
U211(active(X)) → U211(X)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- U211(mark(X)) → U211(X)
The graph contains the following edges 1 > 1
- U211(active(X)) → U211(X)
The graph contains the following edges 1 > 1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U111(active(X)) → U111(X)
U111(mark(X)) → U111(X)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U111(active(X)) → U111(X)
U111(mark(X)) → U111(X)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- U111(active(X)) → U111(X)
The graph contains the following edges 1 > 1
- U111(mark(X)) → U111(X)
The graph contains the following edges 1 > 1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
CONS(X1, active(X2)) → CONS(X1, X2)
CONS(mark(X1), X2) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, mark(X2)) → CONS(X1, X2)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
CONS(mark(X1), X2) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)
CONS(X1, mark(X2)) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- CONS(X1, active(X2)) → CONS(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
- CONS(mark(X1), X2) → CONS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
- CONS(active(X1), X2) → CONS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
- CONS(X1, mark(X2)) → CONS(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
MARK(U21(X)) → ACTIVE(U21(mark(X)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U42(X)) → ACTIVE(U42(mark(X)))
MARK(U91(X1, X2, X3, X4)) → ACTIVE(U91(mark(X1), X2, X3, X4))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(U92(tt, IL, M, N)) → MARK(U93(isNat(N), IL, M, N))
MARK(length(X)) → MARK(X)
MARK(U81(X)) → ACTIVE(U81(mark(X)))
MARK(U31(X)) → ACTIVE(U31(mark(X)))
MARK(take(X1, X2)) → MARK(X1)
MARK(U93(X1, X2, X3, X4)) → ACTIVE(U93(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U51(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(take(0, IL)) → MARK(U81(isNatIList(IL)))
MARK(U52(X)) → ACTIVE(U52(mark(X)))
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U11(X)) → ACTIVE(U11(mark(X)))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(zeros) → ACTIVE(zeros)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
MARK(take(X1, X2)) → MARK(X2)
MARK(U62(X)) → ACTIVE(U62(mark(X)))
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U62(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(s(X)) → MARK(X)
MARK(U91(X1, X2, X3, X4)) → MARK(X1)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U92(X1, X2, X3, X4)) → MARK(X1)
MARK(U81(X)) → MARK(X)
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(U91(tt, IL, M, N)) → MARK(U92(isNat(M), IL, M, N))
ACTIVE(isNatList(take(V1, V2))) → MARK(U61(isNat(V1), V2))
MARK(U72(X1, X2)) → MARK(X1)
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U93(X1, X2, X3, X4)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
ACTIVE(U93(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
MARK(U31(X)) → MARK(X)
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(U21(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U61(X1, X2)) → MARK(X1)
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U92(X1, X2, X3, X4)) → ACTIVE(U92(mark(X1), X2, X3, X4))
ACTIVE(zeros) → MARK(cons(0, zeros))
ACTIVE(take(s(M), cons(N, IL))) → MARK(U91(isNatIList(IL), IL, M, N))
MARK(U11(X)) → MARK(X)
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIList(V2)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
MARK(U21(X)) → ACTIVE(U21(mark(X)))
MARK(U42(X)) → ACTIVE(U42(mark(X)))
MARK(U81(X)) → ACTIVE(U81(mark(X)))
MARK(U31(X)) → ACTIVE(U31(mark(X)))
MARK(U52(X)) → ACTIVE(U52(mark(X)))
MARK(U11(X)) → ACTIVE(U11(mark(X)))
MARK(U62(X)) → ACTIVE(U62(mark(X)))
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
The remaining pairs can at least be oriented weakly.
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U91(X1, X2, X3, X4)) → ACTIVE(U91(mark(X1), X2, X3, X4))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(U92(tt, IL, M, N)) → MARK(U93(isNat(N), IL, M, N))
MARK(length(X)) → MARK(X)
MARK(take(X1, X2)) → MARK(X1)
MARK(U93(X1, X2, X3, X4)) → ACTIVE(U93(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U51(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(take(0, IL)) → MARK(U81(isNatIList(IL)))
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(zeros) → ACTIVE(zeros)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
MARK(take(X1, X2)) → MARK(X2)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U62(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(s(X)) → MARK(X)
MARK(U91(X1, X2, X3, X4)) → MARK(X1)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U92(X1, X2, X3, X4)) → MARK(X1)
MARK(U81(X)) → MARK(X)
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(U91(tt, IL, M, N)) → MARK(U92(isNat(M), IL, M, N))
ACTIVE(isNatList(take(V1, V2))) → MARK(U61(isNat(V1), V2))
MARK(U72(X1, X2)) → MARK(X1)
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U93(X1, X2, X3, X4)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
ACTIVE(U93(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
MARK(U31(X)) → MARK(X)
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(U21(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U61(X1, X2)) → MARK(X1)
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U92(X1, X2, X3, X4)) → ACTIVE(U92(mark(X1), X2, X3, X4))
ACTIVE(zeros) → MARK(cons(0, zeros))
ACTIVE(take(s(M), cons(N, IL))) → MARK(U91(isNatIList(IL), IL, M, N))
MARK(U11(X)) → MARK(X)
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIList(V2)))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVE(x1)) = x1
POL(MARK(x1)) = 1
POL(U11(x1)) = 0
POL(U21(x1)) = 0
POL(U31(x1)) = 0
POL(U41(x1, x2)) = 1
POL(U42(x1)) = 0
POL(U51(x1, x2)) = 1
POL(U52(x1)) = 0
POL(U61(x1, x2)) = 1
POL(U62(x1)) = 0
POL(U71(x1, x2, x3)) = 1
POL(U72(x1, x2)) = 1
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 1
POL(U92(x1, x2, x3, x4)) = 1
POL(U93(x1, x2, x3, x4)) = 1
POL(active(x1)) = 0
POL(cons(x1, x2)) = 0
POL(isNat(x1)) = 1
POL(isNatIList(x1)) = 1
POL(isNatList(x1)) = 1
POL(length(x1)) = 1
POL(mark(x1)) = 0
POL(nil) = 0
POL(s(x1)) = 0
POL(take(x1, x2)) = 1
POL(tt) = 0
POL(zeros) = 1
The following usable rules [17] were oriented:
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
U11(active(X)) → U11(X)
U11(mark(X)) → U11(X)
U21(active(X)) → U21(X)
U21(mark(X)) → U21(X)
U31(active(X)) → U31(X)
U31(mark(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U42(active(X)) → U42(X)
U42(mark(X)) → U42(X)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
U52(active(X)) → U52(X)
U52(mark(X)) → U52(X)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
U81(active(X)) → U81(X)
U81(mark(X)) → U81(X)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
s(active(X)) → s(X)
s(mark(X)) → s(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U91(X1, X2, X3, X4)) → ACTIVE(U91(mark(X1), X2, X3, X4))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(U92(tt, IL, M, N)) → MARK(U93(isNat(N), IL, M, N))
MARK(length(X)) → MARK(X)
MARK(take(X1, X2)) → MARK(X1)
MARK(U93(X1, X2, X3, X4)) → ACTIVE(U93(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U51(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(take(0, IL)) → MARK(U81(isNatIList(IL)))
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(zeros) → ACTIVE(zeros)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
MARK(take(X1, X2)) → MARK(X2)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U62(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(U91(X1, X2, X3, X4)) → MARK(X1)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U92(X1, X2, X3, X4)) → MARK(X1)
MARK(U81(X)) → MARK(X)
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(U91(tt, IL, M, N)) → MARK(U92(isNat(M), IL, M, N))
ACTIVE(isNatList(take(V1, V2))) → MARK(U61(isNat(V1), V2))
MARK(U72(X1, X2)) → MARK(X1)
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U93(X1, X2, X3, X4)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
ACTIVE(U93(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
MARK(U31(X)) → MARK(X)
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(U21(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U61(X1, X2)) → MARK(X1)
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U92(X1, X2, X3, X4)) → ACTIVE(U92(mark(X1), X2, X3, X4))
ACTIVE(zeros) → MARK(cons(0, zeros))
ACTIVE(take(s(M), cons(N, IL))) → MARK(U91(isNatIList(IL), IL, M, N))
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIList(V2)))
MARK(U11(X)) → MARK(X)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
MARK(length(X)) → MARK(X)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U72(X1, X2)) → MARK(X1)
The remaining pairs can at least be oriented weakly.
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U91(X1, X2, X3, X4)) → ACTIVE(U91(mark(X1), X2, X3, X4))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(U92(tt, IL, M, N)) → MARK(U93(isNat(N), IL, M, N))
MARK(take(X1, X2)) → MARK(X1)
MARK(U93(X1, X2, X3, X4)) → ACTIVE(U93(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U51(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(take(0, IL)) → MARK(U81(isNatIList(IL)))
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(zeros) → ACTIVE(zeros)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
MARK(take(X1, X2)) → MARK(X2)
MARK(U62(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(U91(X1, X2, X3, X4)) → MARK(X1)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U92(X1, X2, X3, X4)) → MARK(X1)
MARK(U81(X)) → MARK(X)
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(U91(tt, IL, M, N)) → MARK(U92(isNat(M), IL, M, N))
ACTIVE(isNatList(take(V1, V2))) → MARK(U61(isNat(V1), V2))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U93(X1, X2, X3, X4)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
ACTIVE(U93(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
MARK(U31(X)) → MARK(X)
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(U21(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U61(X1, X2)) → MARK(X1)
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U92(X1, X2, X3, X4)) → ACTIVE(U92(mark(X1), X2, X3, X4))
ACTIVE(zeros) → MARK(cons(0, zeros))
ACTIVE(take(s(M), cons(N, IL))) → MARK(U91(isNatIList(IL), IL, M, N))
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIList(V2)))
MARK(U11(X)) → MARK(X)
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVE(x1)) = x1
POL(MARK(x1)) = x1
POL(U11(x1)) = x1
POL(U21(x1)) = x1
POL(U31(x1)) = x1
POL(U41(x1, x2)) = x1
POL(U42(x1)) = x1
POL(U51(x1, x2)) = x1
POL(U52(x1)) = x1
POL(U61(x1, x2)) = x1
POL(U62(x1)) = x1
POL(U71(x1, x2, x3)) = 1 + x1 + x2 + x3
POL(U72(x1, x2)) = 1 + x1 + x2
POL(U81(x1)) = x1
POL(U91(x1, x2, x3, x4)) = x1 + x2 + x3 + x4
POL(U92(x1, x2, x3, x4)) = x1 + x2 + x3 + x4
POL(U93(x1, x2, x3, x4)) = x1 + x2 + x3 + x4
POL(active(x1)) = x1
POL(cons(x1, x2)) = x1 + x2
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 1 + x1
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = x1 + x2
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
active(isNatList(nil)) → mark(tt)
active(isNatIList(zeros)) → mark(tt)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
active(length(nil)) → mark(0)
mark(0) → active(0)
mark(U42(X)) → active(U42(mark(X)))
mark(U62(X)) → active(U62(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(zeros) → active(zeros)
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(zeros) → mark(cons(0, zeros))
mark(U81(X)) → active(U81(mark(X)))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
mark(U31(X)) → active(U31(mark(X)))
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
mark(isNatList(X)) → active(isNatList(X))
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(isNat(X)) → active(isNat(X))
mark(U11(X)) → active(U11(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
mark(length(X)) → active(length(mark(X)))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(U21(X)) → active(U21(mark(X)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(U72(tt, L)) → mark(s(length(L)))
mark(s(X)) → active(s(mark(X)))
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
mark(tt) → active(tt)
U51(X1, mark(X2)) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
active(U52(tt)) → mark(tt)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
active(U42(tt)) → mark(tt)
U42(active(X)) → U42(X)
U42(mark(X)) → U42(X)
active(U62(tt)) → mark(tt)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
active(U81(tt)) → mark(nil)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
U52(active(X)) → U52(X)
U52(mark(X)) → U52(X)
active(isNat(0)) → mark(tt)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
U11(active(X)) → U11(X)
U11(mark(X)) → U11(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
U21(active(X)) → U21(X)
U21(mark(X)) → U21(X)
U31(active(X)) → U31(X)
U31(mark(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
active(U21(tt)) → mark(tt)
U81(active(X)) → U81(X)
U81(mark(X)) → U81(X)
active(U31(tt)) → mark(tt)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
active(U11(tt)) → mark(tt)
s(active(X)) → s(X)
s(mark(X)) → s(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
mark(nil) → active(nil)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U91(X1, X2, X3, X4)) → ACTIVE(U91(mark(X1), X2, X3, X4))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(U92(tt, IL, M, N)) → MARK(U93(isNat(N), IL, M, N))
MARK(take(X1, X2)) → MARK(X1)
MARK(U93(X1, X2, X3, X4)) → ACTIVE(U93(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U51(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(take(0, IL)) → MARK(U81(isNatIList(IL)))
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(zeros) → ACTIVE(zeros)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
MARK(take(X1, X2)) → MARK(X2)
MARK(U62(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U91(X1, X2, X3, X4)) → MARK(X1)
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U92(X1, X2, X3, X4)) → MARK(X1)
MARK(U81(X)) → MARK(X)
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(U91(tt, IL, M, N)) → MARK(U92(isNat(M), IL, M, N))
ACTIVE(isNatList(take(V1, V2))) → MARK(U61(isNat(V1), V2))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U93(X1, X2, X3, X4)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
ACTIVE(U93(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
MARK(U31(X)) → MARK(X)
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(U21(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U61(X1, X2)) → MARK(X1)
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U92(X1, X2, X3, X4)) → ACTIVE(U92(mark(X1), X2, X3, X4))
ACTIVE(zeros) → MARK(cons(0, zeros))
ACTIVE(take(s(M), cons(N, IL))) → MARK(U91(isNatIList(IL), IL, M, N))
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIList(V2)))
MARK(U11(X)) → MARK(X)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
ACTIVE(zeros) → MARK(cons(0, zeros))
The remaining pairs can at least be oriented weakly.
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U91(X1, X2, X3, X4)) → ACTIVE(U91(mark(X1), X2, X3, X4))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(U92(tt, IL, M, N)) → MARK(U93(isNat(N), IL, M, N))
MARK(take(X1, X2)) → MARK(X1)
MARK(U93(X1, X2, X3, X4)) → ACTIVE(U93(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U51(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(take(0, IL)) → MARK(U81(isNatIList(IL)))
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(zeros) → ACTIVE(zeros)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
MARK(take(X1, X2)) → MARK(X2)
MARK(U62(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U91(X1, X2, X3, X4)) → MARK(X1)
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U92(X1, X2, X3, X4)) → MARK(X1)
MARK(U81(X)) → MARK(X)
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(U91(tt, IL, M, N)) → MARK(U92(isNat(M), IL, M, N))
ACTIVE(isNatList(take(V1, V2))) → MARK(U61(isNat(V1), V2))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U93(X1, X2, X3, X4)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
ACTIVE(U93(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
MARK(U31(X)) → MARK(X)
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(U21(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U61(X1, X2)) → MARK(X1)
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U92(X1, X2, X3, X4)) → ACTIVE(U92(mark(X1), X2, X3, X4))
ACTIVE(take(s(M), cons(N, IL))) → MARK(U91(isNatIList(IL), IL, M, N))
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIList(V2)))
MARK(U11(X)) → MARK(X)
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVE(x1)) = x1
POL(MARK(x1)) = x1
POL(U11(x1)) = x1
POL(U21(x1)) = x1
POL(U31(x1)) = x1
POL(U41(x1, x2)) = x1
POL(U42(x1)) = x1
POL(U51(x1, x2)) = x1
POL(U52(x1)) = x1
POL(U61(x1, x2)) = x1
POL(U62(x1)) = x1
POL(U71(x1, x2, x3)) = 0
POL(U72(x1, x2)) = 0
POL(U81(x1)) = x1
POL(U91(x1, x2, x3, x4)) = x1 + x3 + x4
POL(U92(x1, x2, x3, x4)) = x1 + x3 + x4
POL(U93(x1, x2, x3, x4)) = x1 + x4
POL(active(x1)) = x1
POL(cons(x1, x2)) = x1
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 0
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = x1 + x2
POL(tt) = 0
POL(zeros) = 1
The following usable rules [17] were oriented:
active(isNatList(nil)) → mark(tt)
active(isNatIList(zeros)) → mark(tt)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
active(length(nil)) → mark(0)
mark(0) → active(0)
mark(U42(X)) → active(U42(mark(X)))
mark(U62(X)) → active(U62(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(zeros) → active(zeros)
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(zeros) → mark(cons(0, zeros))
mark(U81(X)) → active(U81(mark(X)))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
mark(U31(X)) → active(U31(mark(X)))
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
mark(isNatList(X)) → active(isNatList(X))
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(isNat(X)) → active(isNat(X))
mark(U11(X)) → active(U11(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
mark(length(X)) → active(length(mark(X)))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(U21(X)) → active(U21(mark(X)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(U72(tt, L)) → mark(s(length(L)))
mark(s(X)) → active(s(mark(X)))
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
mark(tt) → active(tt)
U51(X1, mark(X2)) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
active(U52(tt)) → mark(tt)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
active(U42(tt)) → mark(tt)
U42(active(X)) → U42(X)
U42(mark(X)) → U42(X)
active(U62(tt)) → mark(tt)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
active(U81(tt)) → mark(nil)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
U52(active(X)) → U52(X)
U52(mark(X)) → U52(X)
active(isNat(0)) → mark(tt)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
U11(active(X)) → U11(X)
U11(mark(X)) → U11(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
U21(active(X)) → U21(X)
U21(mark(X)) → U21(X)
U31(active(X)) → U31(X)
U31(mark(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
active(U21(tt)) → mark(tt)
U81(active(X)) → U81(X)
U81(mark(X)) → U81(X)
active(U31(tt)) → mark(tt)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
active(U11(tt)) → mark(tt)
s(active(X)) → s(X)
s(mark(X)) → s(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
mark(nil) → active(nil)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
MARK(take(X1, X2)) → MARK(X2)
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U62(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U91(X1, X2, X3, X4)) → MARK(X1)
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(U91(X1, X2, X3, X4)) → ACTIVE(U91(mark(X1), X2, X3, X4))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U92(X1, X2, X3, X4)) → MARK(X1)
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(U92(tt, IL, M, N)) → MARK(U93(isNat(N), IL, M, N))
MARK(U81(X)) → MARK(X)
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNatList(take(V1, V2))) → MARK(U61(isNat(V1), V2))
ACTIVE(U91(tt, IL, M, N)) → MARK(U92(isNat(M), IL, M, N))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U93(X1, X2, X3, X4)) → MARK(X1)
MARK(U52(X)) → MARK(X)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U93(X1, X2, X3, X4)) → ACTIVE(U93(mark(X1), X2, X3, X4))
MARK(take(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(U51(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
ACTIVE(take(0, IL)) → MARK(U81(isNatIList(IL)))
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
MARK(U31(X)) → MARK(X)
ACTIVE(U93(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
MARK(U61(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U92(X1, X2, X3, X4)) → ACTIVE(U92(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(zeros) → ACTIVE(zeros)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
ACTIVE(take(s(M), cons(N, IL))) → MARK(U91(isNatIList(IL), IL, M, N))
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
MARK(U11(X)) → MARK(X)
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIList(V2)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
MARK(take(X1, X2)) → MARK(X2)
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U62(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(U91(X1, X2, X3, X4)) → MARK(X1)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U91(X1, X2, X3, X4)) → ACTIVE(U91(mark(X1), X2, X3, X4))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(U92(X1, X2, X3, X4)) → MARK(X1)
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(U92(tt, IL, M, N)) → MARK(U93(isNat(N), IL, M, N))
MARK(U81(X)) → MARK(X)
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNatList(take(V1, V2))) → MARK(U61(isNat(V1), V2))
ACTIVE(U91(tt, IL, M, N)) → MARK(U92(isNat(M), IL, M, N))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U93(X1, X2, X3, X4)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(take(X1, X2)) → MARK(X1)
MARK(U93(X1, X2, X3, X4)) → ACTIVE(U93(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U51(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
ACTIVE(take(0, IL)) → MARK(U81(isNatIList(IL)))
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
MARK(U31(X)) → MARK(X)
ACTIVE(U93(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U61(X1, X2)) → MARK(X1)
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U92(X1, X2, X3, X4)) → ACTIVE(U92(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
ACTIVE(take(s(M), cons(N, IL))) → MARK(U91(isNatIList(IL), IL, M, N))
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIList(V2)))
MARK(U11(X)) → MARK(X)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
MARK(take(X1, X2)) → MARK(X2)
MARK(take(X1, X2)) → MARK(X1)
ACTIVE(take(0, IL)) → MARK(U81(isNatIList(IL)))
ACTIVE(take(s(M), cons(N, IL))) → MARK(U91(isNatIList(IL), IL, M, N))
The remaining pairs can at least be oriented weakly.
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U62(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(U91(X1, X2, X3, X4)) → MARK(X1)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U91(X1, X2, X3, X4)) → ACTIVE(U91(mark(X1), X2, X3, X4))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(U92(X1, X2, X3, X4)) → MARK(X1)
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(U92(tt, IL, M, N)) → MARK(U93(isNat(N), IL, M, N))
MARK(U81(X)) → MARK(X)
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNatList(take(V1, V2))) → MARK(U61(isNat(V1), V2))
ACTIVE(U91(tt, IL, M, N)) → MARK(U92(isNat(M), IL, M, N))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U93(X1, X2, X3, X4)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(U93(X1, X2, X3, X4)) → ACTIVE(U93(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U51(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
MARK(U31(X)) → MARK(X)
ACTIVE(U93(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U61(X1, X2)) → MARK(X1)
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U92(X1, X2, X3, X4)) → ACTIVE(U92(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIList(V2)))
MARK(U11(X)) → MARK(X)
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVE(x1)) = x1
POL(MARK(x1)) = x1
POL(U11(x1)) = x1
POL(U21(x1)) = x1
POL(U31(x1)) = x1
POL(U41(x1, x2)) = x1
POL(U42(x1)) = x1
POL(U51(x1, x2)) = x1
POL(U52(x1)) = x1
POL(U61(x1, x2)) = x1
POL(U62(x1)) = x1
POL(U71(x1, x2, x3)) = 0
POL(U72(x1, x2)) = 0
POL(U81(x1)) = x1
POL(U91(x1, x2, x3, x4)) = x1 + x3 + x4
POL(U92(x1, x2, x3, x4)) = x1 + x3 + x4
POL(U93(x1, x2, x3, x4)) = x1 + x4
POL(active(x1)) = x1
POL(cons(x1, x2)) = x1
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 0
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = 1 + x1 + x2
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
active(isNatList(nil)) → mark(tt)
active(isNatIList(zeros)) → mark(tt)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
active(length(nil)) → mark(0)
mark(0) → active(0)
mark(U42(X)) → active(U42(mark(X)))
mark(U62(X)) → active(U62(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(zeros) → active(zeros)
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(zeros) → mark(cons(0, zeros))
mark(U81(X)) → active(U81(mark(X)))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
mark(U31(X)) → active(U31(mark(X)))
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
mark(isNatList(X)) → active(isNatList(X))
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(isNat(X)) → active(isNat(X))
mark(U11(X)) → active(U11(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
mark(length(X)) → active(length(mark(X)))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(U21(X)) → active(U21(mark(X)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(U72(tt, L)) → mark(s(length(L)))
mark(s(X)) → active(s(mark(X)))
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
mark(tt) → active(tt)
U51(X1, mark(X2)) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
active(U52(tt)) → mark(tt)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
active(U42(tt)) → mark(tt)
U42(active(X)) → U42(X)
U42(mark(X)) → U42(X)
active(U62(tt)) → mark(tt)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
active(U81(tt)) → mark(nil)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
U52(active(X)) → U52(X)
U52(mark(X)) → U52(X)
active(isNat(0)) → mark(tt)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
U11(active(X)) → U11(X)
U11(mark(X)) → U11(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
U21(active(X)) → U21(X)
U21(mark(X)) → U21(X)
U31(active(X)) → U31(X)
U31(mark(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
active(U21(tt)) → mark(tt)
U81(active(X)) → U81(X)
U81(mark(X)) → U81(X)
active(U31(tt)) → mark(tt)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
active(U11(tt)) → mark(tt)
s(active(X)) → s(X)
s(mark(X)) → s(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
mark(nil) → active(nil)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
MARK(U62(X)) → MARK(X)
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U91(X1, X2, X3, X4)) → MARK(X1)
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U91(X1, X2, X3, X4)) → ACTIVE(U91(mark(X1), X2, X3, X4))
MARK(cons(X1, X2)) → MARK(X1)
MARK(U92(X1, X2, X3, X4)) → MARK(X1)
ACTIVE(U92(tt, IL, M, N)) → MARK(U93(isNat(N), IL, M, N))
MARK(U81(X)) → MARK(X)
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNatList(take(V1, V2))) → MARK(U61(isNat(V1), V2))
ACTIVE(U91(tt, IL, M, N)) → MARK(U92(isNat(M), IL, M, N))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U93(X1, X2, X3, X4)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(U93(X1, X2, X3, X4)) → ACTIVE(U93(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U51(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
ACTIVE(U93(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
MARK(U31(X)) → MARK(X)
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
MARK(U61(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U92(X1, X2, X3, X4)) → ACTIVE(U92(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
MARK(U11(X)) → MARK(X)
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIList(V2)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
MARK(U81(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.
MARK(U62(X)) → MARK(X)
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U91(X1, X2, X3, X4)) → MARK(X1)
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U91(X1, X2, X3, X4)) → ACTIVE(U91(mark(X1), X2, X3, X4))
MARK(cons(X1, X2)) → MARK(X1)
MARK(U92(X1, X2, X3, X4)) → MARK(X1)
ACTIVE(U92(tt, IL, M, N)) → MARK(U93(isNat(N), IL, M, N))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNatList(take(V1, V2))) → MARK(U61(isNat(V1), V2))
ACTIVE(U91(tt, IL, M, N)) → MARK(U92(isNat(M), IL, M, N))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U93(X1, X2, X3, X4)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(U93(X1, X2, X3, X4)) → ACTIVE(U93(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U51(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
ACTIVE(U93(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
MARK(U31(X)) → MARK(X)
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
MARK(U61(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U92(X1, X2, X3, X4)) → ACTIVE(U92(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
MARK(U11(X)) → MARK(X)
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIList(V2)))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVE(x1)) = x1
POL(MARK(x1)) = x1
POL(U11(x1)) = x1
POL(U21(x1)) = x1
POL(U31(x1)) = x1
POL(U41(x1, x2)) = x1
POL(U42(x1)) = x1
POL(U51(x1, x2)) = x1
POL(U52(x1)) = x1
POL(U61(x1, x2)) = x1
POL(U62(x1)) = x1
POL(U71(x1, x2, x3)) = 0
POL(U72(x1, x2)) = 0
POL(U81(x1)) = 1 + x1
POL(U91(x1, x2, x3, x4)) = x1 + x4
POL(U92(x1, x2, x3, x4)) = x1 + x4
POL(U93(x1, x2, x3, x4)) = x1 + x4
POL(active(x1)) = x1
POL(cons(x1, x2)) = x1
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 0
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = 1 + x2
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
active(isNatList(nil)) → mark(tt)
active(isNatIList(zeros)) → mark(tt)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
active(length(nil)) → mark(0)
mark(0) → active(0)
mark(U42(X)) → active(U42(mark(X)))
mark(U62(X)) → active(U62(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(zeros) → active(zeros)
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(zeros) → mark(cons(0, zeros))
mark(U81(X)) → active(U81(mark(X)))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
mark(U31(X)) → active(U31(mark(X)))
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
mark(isNatList(X)) → active(isNatList(X))
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(isNat(X)) → active(isNat(X))
mark(U11(X)) → active(U11(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
mark(length(X)) → active(length(mark(X)))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(U21(X)) → active(U21(mark(X)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(U72(tt, L)) → mark(s(length(L)))
mark(s(X)) → active(s(mark(X)))
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
mark(tt) → active(tt)
U51(X1, mark(X2)) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
active(U52(tt)) → mark(tt)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
active(U42(tt)) → mark(tt)
U42(active(X)) → U42(X)
U42(mark(X)) → U42(X)
active(U62(tt)) → mark(tt)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
active(U81(tt)) → mark(nil)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
U52(active(X)) → U52(X)
U52(mark(X)) → U52(X)
active(isNat(0)) → mark(tt)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
U11(active(X)) → U11(X)
U11(mark(X)) → U11(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
U21(active(X)) → U21(X)
U21(mark(X)) → U21(X)
U31(active(X)) → U31(X)
U31(mark(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U81(active(X)) → U81(X)
U81(mark(X)) → U81(X)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
active(U11(tt)) → mark(tt)
s(active(X)) → s(X)
s(mark(X)) → s(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
mark(nil) → active(nil)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U62(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(U91(X1, X2, X3, X4)) → MARK(X1)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U91(X1, X2, X3, X4)) → ACTIVE(U91(mark(X1), X2, X3, X4))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(U92(X1, X2, X3, X4)) → MARK(X1)
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(U92(tt, IL, M, N)) → MARK(U93(isNat(N), IL, M, N))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNatList(take(V1, V2))) → MARK(U61(isNat(V1), V2))
ACTIVE(U91(tt, IL, M, N)) → MARK(U92(isNat(M), IL, M, N))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U93(X1, X2, X3, X4)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(U93(X1, X2, X3, X4)) → ACTIVE(U93(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U51(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
MARK(U31(X)) → MARK(X)
ACTIVE(U93(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U61(X1, X2)) → MARK(X1)
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U92(X1, X2, X3, X4)) → ACTIVE(U92(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIList(V2)))
MARK(U11(X)) → MARK(X)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
MARK(U91(X1, X2, X3, X4)) → MARK(X1)
MARK(U92(X1, X2, X3, X4)) → MARK(X1)
MARK(cons(X1, X2)) → MARK(X1)
MARK(U93(X1, X2, X3, X4)) → MARK(X1)
The remaining pairs can at least be oriented weakly.
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U62(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U91(X1, X2, X3, X4)) → ACTIVE(U91(mark(X1), X2, X3, X4))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
ACTIVE(U92(tt, IL, M, N)) → MARK(U93(isNat(N), IL, M, N))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNatList(take(V1, V2))) → MARK(U61(isNat(V1), V2))
ACTIVE(U91(tt, IL, M, N)) → MARK(U92(isNat(M), IL, M, N))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U52(X)) → MARK(X)
MARK(U93(X1, X2, X3, X4)) → ACTIVE(U93(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U51(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
MARK(U31(X)) → MARK(X)
ACTIVE(U93(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U61(X1, X2)) → MARK(X1)
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U92(X1, X2, X3, X4)) → ACTIVE(U92(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIList(V2)))
MARK(U11(X)) → MARK(X)
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVE(x1)) = x1
POL(MARK(x1)) = x1
POL(U11(x1)) = x1
POL(U21(x1)) = x1
POL(U31(x1)) = x1
POL(U41(x1, x2)) = x1
POL(U42(x1)) = x1
POL(U51(x1, x2)) = x1
POL(U52(x1)) = x1
POL(U61(x1, x2)) = x1
POL(U62(x1)) = x1
POL(U71(x1, x2, x3)) = 0
POL(U72(x1, x2)) = 0
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 1 + x1 + x4
POL(U92(x1, x2, x3, x4)) = 1 + x1 + x4
POL(U93(x1, x2, x3, x4)) = 1 + x1 + x4
POL(active(x1)) = x1
POL(cons(x1, x2)) = 1 + x1
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 0
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = x2
POL(tt) = 0
POL(zeros) = 1
The following usable rules [17] were oriented:
active(isNatList(nil)) → mark(tt)
active(isNatIList(zeros)) → mark(tt)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
active(length(nil)) → mark(0)
mark(0) → active(0)
mark(U42(X)) → active(U42(mark(X)))
mark(U62(X)) → active(U62(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(zeros) → active(zeros)
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(zeros) → mark(cons(0, zeros))
mark(U81(X)) → active(U81(mark(X)))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
mark(U31(X)) → active(U31(mark(X)))
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
mark(isNatList(X)) → active(isNatList(X))
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(isNat(X)) → active(isNat(X))
mark(U11(X)) → active(U11(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
mark(length(X)) → active(length(mark(X)))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(U21(X)) → active(U21(mark(X)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(U72(tt, L)) → mark(s(length(L)))
mark(s(X)) → active(s(mark(X)))
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
mark(tt) → active(tt)
U51(X1, mark(X2)) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
active(U52(tt)) → mark(tt)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
active(U42(tt)) → mark(tt)
U42(active(X)) → U42(X)
U42(mark(X)) → U42(X)
active(U62(tt)) → mark(tt)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
active(U81(tt)) → mark(nil)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
U52(active(X)) → U52(X)
U52(mark(X)) → U52(X)
active(isNat(0)) → mark(tt)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
U11(active(X)) → U11(X)
U11(mark(X)) → U11(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
U21(active(X)) → U21(X)
U21(mark(X)) → U21(X)
U31(active(X)) → U31(X)
U31(mark(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U81(active(X)) → U81(X)
U81(mark(X)) → U81(X)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
active(U11(tt)) → mark(tt)
s(active(X)) → s(X)
s(mark(X)) → s(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
mark(nil) → active(nil)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
MARK(U62(X)) → MARK(X)
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U91(X1, X2, X3, X4)) → ACTIVE(U91(mark(X1), X2, X3, X4))
ACTIVE(U92(tt, IL, M, N)) → MARK(U93(isNat(N), IL, M, N))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(U91(tt, IL, M, N)) → MARK(U92(isNat(M), IL, M, N))
ACTIVE(isNatList(take(V1, V2))) → MARK(U61(isNat(V1), V2))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U52(X)) → MARK(X)
MARK(U93(X1, X2, X3, X4)) → ACTIVE(U93(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U51(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
MARK(U31(X)) → MARK(X)
ACTIVE(U93(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
MARK(U61(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U92(X1, X2, X3, X4)) → ACTIVE(U92(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
MARK(U11(X)) → MARK(X)
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIList(V2)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
The remaining pairs can at least be oriented weakly.
MARK(U62(X)) → MARK(X)
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U91(X1, X2, X3, X4)) → ACTIVE(U91(mark(X1), X2, X3, X4))
ACTIVE(U92(tt, IL, M, N)) → MARK(U93(isNat(N), IL, M, N))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(U91(tt, IL, M, N)) → MARK(U92(isNat(M), IL, M, N))
ACTIVE(isNatList(take(V1, V2))) → MARK(U61(isNat(V1), V2))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U52(X)) → MARK(X)
MARK(U93(X1, X2, X3, X4)) → ACTIVE(U93(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U51(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
MARK(U31(X)) → MARK(X)
ACTIVE(U93(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
MARK(U61(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U92(X1, X2, X3, X4)) → ACTIVE(U92(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
MARK(U11(X)) → MARK(X)
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIList(V2)))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVE(x1)) = x1
POL(MARK(x1)) = 1
POL(U11(x1)) = 0
POL(U21(x1)) = 0
POL(U31(x1)) = 0
POL(U41(x1, x2)) = 1
POL(U42(x1)) = 0
POL(U51(x1, x2)) = 1
POL(U52(x1)) = 0
POL(U61(x1, x2)) = 1
POL(U62(x1)) = 0
POL(U71(x1, x2, x3)) = 1
POL(U72(x1, x2)) = 1
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 1
POL(U92(x1, x2, x3, x4)) = 1
POL(U93(x1, x2, x3, x4)) = 1
POL(active(x1)) = 0
POL(cons(x1, x2)) = 0
POL(isNat(x1)) = 1
POL(isNatIList(x1)) = 1
POL(isNatList(x1)) = 1
POL(length(x1)) = 1
POL(mark(x1)) = 0
POL(nil) = 0
POL(s(x1)) = 0
POL(take(x1, x2)) = 0
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
length(active(X)) → length(X)
length(mark(X)) → length(X)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U62(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U91(X1, X2, X3, X4)) → ACTIVE(U91(mark(X1), X2, X3, X4))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
ACTIVE(U92(tt, IL, M, N)) → MARK(U93(isNat(N), IL, M, N))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNatList(take(V1, V2))) → MARK(U61(isNat(V1), V2))
ACTIVE(U91(tt, IL, M, N)) → MARK(U92(isNat(M), IL, M, N))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U52(X)) → MARK(X)
MARK(U93(X1, X2, X3, X4)) → ACTIVE(U93(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U51(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
MARK(U31(X)) → MARK(X)
ACTIVE(U93(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U61(X1, X2)) → MARK(X1)
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U92(X1, X2, X3, X4)) → ACTIVE(U92(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIList(V2)))
MARK(U11(X)) → MARK(X)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
MARK(U91(X1, X2, X3, X4)) → ACTIVE(U91(mark(X1), X2, X3, X4))
The remaining pairs can at least be oriented weakly.
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U62(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
ACTIVE(U92(tt, IL, M, N)) → MARK(U93(isNat(N), IL, M, N))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNatList(take(V1, V2))) → MARK(U61(isNat(V1), V2))
ACTIVE(U91(tt, IL, M, N)) → MARK(U92(isNat(M), IL, M, N))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U52(X)) → MARK(X)
MARK(U93(X1, X2, X3, X4)) → ACTIVE(U93(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U51(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
MARK(U31(X)) → MARK(X)
ACTIVE(U93(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U61(X1, X2)) → MARK(X1)
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U92(X1, X2, X3, X4)) → ACTIVE(U92(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIList(V2)))
MARK(U11(X)) → MARK(X)
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVE(x1)) = 0
POL(MARK(x1)) = x1
POL(U11(x1)) = x1
POL(U21(x1)) = x1
POL(U31(x1)) = x1
POL(U41(x1, x2)) = x1
POL(U42(x1)) = x1
POL(U51(x1, x2)) = x1
POL(U52(x1)) = x1
POL(U61(x1, x2)) = x1
POL(U62(x1)) = x1
POL(U71(x1, x2, x3)) = 0
POL(U72(x1, x2)) = 0
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 1 + x3 + x4
POL(U92(x1, x2, x3, x4)) = 0
POL(U93(x1, x2, x3, x4)) = 0
POL(active(x1)) = x1
POL(cons(x1, x2)) = 0
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 0
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = 0
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U42(active(X)) → U42(X)
U42(mark(X)) → U42(X)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
U52(active(X)) → U52(X)
U52(mark(X)) → U52(X)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
U11(active(X)) → U11(X)
U11(mark(X)) → U11(X)
U21(active(X)) → U21(X)
U21(mark(X)) → U21(X)
U31(active(X)) → U31(X)
U31(mark(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
MARK(U62(X)) → MARK(X)
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
ACTIVE(U92(tt, IL, M, N)) → MARK(U93(isNat(N), IL, M, N))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(U91(tt, IL, M, N)) → MARK(U92(isNat(M), IL, M, N))
ACTIVE(isNatList(take(V1, V2))) → MARK(U61(isNat(V1), V2))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U52(X)) → MARK(X)
MARK(U93(X1, X2, X3, X4)) → ACTIVE(U93(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U51(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
MARK(U31(X)) → MARK(X)
ACTIVE(U93(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
MARK(U61(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U92(X1, X2, X3, X4)) → ACTIVE(U92(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
MARK(U11(X)) → MARK(X)
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIList(V2)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
ACTIVE(U91(tt, IL, M, N)) → MARK(U92(isNat(M), IL, M, N))
The remaining pairs can at least be oriented weakly.
MARK(U62(X)) → MARK(X)
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
ACTIVE(U92(tt, IL, M, N)) → MARK(U93(isNat(N), IL, M, N))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNatList(take(V1, V2))) → MARK(U61(isNat(V1), V2))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U52(X)) → MARK(X)
MARK(U93(X1, X2, X3, X4)) → ACTIVE(U93(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U51(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
MARK(U31(X)) → MARK(X)
ACTIVE(U93(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
MARK(U61(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U92(X1, X2, X3, X4)) → ACTIVE(U92(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
MARK(U11(X)) → MARK(X)
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIList(V2)))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVE(x1)) = x1
POL(MARK(x1)) = 0
POL(U11(x1)) = 0
POL(U21(x1)) = 0
POL(U31(x1)) = 0
POL(U41(x1, x2)) = 0
POL(U42(x1)) = 0
POL(U51(x1, x2)) = 0
POL(U52(x1)) = 0
POL(U61(x1, x2)) = 0
POL(U62(x1)) = 0
POL(U71(x1, x2, x3)) = 0
POL(U72(x1, x2)) = 0
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 1 + x1 + x3 + x4
POL(U92(x1, x2, x3, x4)) = 0
POL(U93(x1, x2, x3, x4)) = 0
POL(active(x1)) = 0
POL(cons(x1, x2)) = 0
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 0
POL(mark(x1)) = 0
POL(nil) = 0
POL(s(x1)) = 0
POL(take(x1, x2)) = 0
POL(tt) = 1
POL(zeros) = 0
The following usable rules [17] were oriented:
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U62(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
ACTIVE(U92(tt, IL, M, N)) → MARK(U93(isNat(N), IL, M, N))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNatList(take(V1, V2))) → MARK(U61(isNat(V1), V2))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U52(X)) → MARK(X)
MARK(U93(X1, X2, X3, X4)) → ACTIVE(U93(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U51(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
MARK(U31(X)) → MARK(X)
ACTIVE(U93(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U61(X1, X2)) → MARK(X1)
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U92(X1, X2, X3, X4)) → ACTIVE(U92(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIList(V2)))
MARK(U11(X)) → MARK(X)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
MARK(U92(X1, X2, X3, X4)) → ACTIVE(U92(mark(X1), X2, X3, X4))
The remaining pairs can at least be oriented weakly.
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U62(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
ACTIVE(U92(tt, IL, M, N)) → MARK(U93(isNat(N), IL, M, N))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNatList(take(V1, V2))) → MARK(U61(isNat(V1), V2))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U52(X)) → MARK(X)
MARK(U93(X1, X2, X3, X4)) → ACTIVE(U93(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U51(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
MARK(U31(X)) → MARK(X)
ACTIVE(U93(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U61(X1, X2)) → MARK(X1)
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIList(V2)))
MARK(U11(X)) → MARK(X)
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVE(x1)) = 0
POL(MARK(x1)) = x1
POL(U11(x1)) = x1
POL(U21(x1)) = x1
POL(U31(x1)) = x1
POL(U41(x1, x2)) = x1
POL(U42(x1)) = x1
POL(U51(x1, x2)) = x1
POL(U52(x1)) = x1
POL(U61(x1, x2)) = x1
POL(U62(x1)) = x1
POL(U71(x1, x2, x3)) = 0
POL(U72(x1, x2)) = 0
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = x2 + x3
POL(U92(x1, x2, x3, x4)) = 1 + x2
POL(U93(x1, x2, x3, x4)) = 0
POL(active(x1)) = x1
POL(cons(x1, x2)) = 0
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 0
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = 0
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U42(active(X)) → U42(X)
U42(mark(X)) → U42(X)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
U52(active(X)) → U52(X)
U52(mark(X)) → U52(X)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
U11(active(X)) → U11(X)
U11(mark(X)) → U11(X)
U21(active(X)) → U21(X)
U21(mark(X)) → U21(X)
U31(active(X)) → U31(X)
U31(mark(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
MARK(U62(X)) → MARK(X)
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
ACTIVE(U92(tt, IL, M, N)) → MARK(U93(isNat(N), IL, M, N))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNatList(take(V1, V2))) → MARK(U61(isNat(V1), V2))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U52(X)) → MARK(X)
MARK(U93(X1, X2, X3, X4)) → ACTIVE(U93(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U51(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
MARK(U31(X)) → MARK(X)
ACTIVE(U93(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
MARK(U61(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
MARK(U11(X)) → MARK(X)
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIList(V2)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
ACTIVE(U92(tt, IL, M, N)) → MARK(U93(isNat(N), IL, M, N))
The remaining pairs can at least be oriented weakly.
MARK(U62(X)) → MARK(X)
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNatList(take(V1, V2))) → MARK(U61(isNat(V1), V2))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U52(X)) → MARK(X)
MARK(U93(X1, X2, X3, X4)) → ACTIVE(U93(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U51(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
MARK(U31(X)) → MARK(X)
ACTIVE(U93(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
MARK(U61(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
MARK(U11(X)) → MARK(X)
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIList(V2)))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVE(x1)) = x1
POL(MARK(x1)) = 0
POL(U11(x1)) = 0
POL(U21(x1)) = 0
POL(U31(x1)) = 0
POL(U41(x1, x2)) = 0
POL(U42(x1)) = 0
POL(U51(x1, x2)) = 0
POL(U52(x1)) = 0
POL(U61(x1, x2)) = 0
POL(U62(x1)) = 0
POL(U71(x1, x2, x3)) = 0
POL(U72(x1, x2)) = 0
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 0
POL(U92(x1, x2, x3, x4)) = 1 + x1 + x2 + x3
POL(U93(x1, x2, x3, x4)) = 0
POL(active(x1)) = 0
POL(cons(x1, x2)) = 0
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 0
POL(mark(x1)) = 0
POL(nil) = 0
POL(s(x1)) = 0
POL(take(x1, x2)) = 0
POL(tt) = 1
POL(zeros) = 0
The following usable rules [17] were oriented:
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U62(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNatList(take(V1, V2))) → MARK(U61(isNat(V1), V2))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U52(X)) → MARK(X)
MARK(U93(X1, X2, X3, X4)) → ACTIVE(U93(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U51(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
MARK(U31(X)) → MARK(X)
ACTIVE(U93(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U61(X1, X2)) → MARK(X1)
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIList(V2)))
MARK(U11(X)) → MARK(X)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
MARK(U93(X1, X2, X3, X4)) → ACTIVE(U93(mark(X1), X2, X3, X4))
The remaining pairs can at least be oriented weakly.
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U62(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNatList(take(V1, V2))) → MARK(U61(isNat(V1), V2))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U52(X)) → MARK(X)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U51(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
MARK(U31(X)) → MARK(X)
ACTIVE(U93(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U61(X1, X2)) → MARK(X1)
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIList(V2)))
MARK(U11(X)) → MARK(X)
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVE(x1)) = 0
POL(MARK(x1)) = x1
POL(U11(x1)) = x1
POL(U21(x1)) = x1
POL(U31(x1)) = x1
POL(U41(x1, x2)) = x1
POL(U42(x1)) = x1
POL(U51(x1, x2)) = x1
POL(U52(x1)) = x1
POL(U61(x1, x2)) = x1
POL(U62(x1)) = x1
POL(U71(x1, x2, x3)) = 0
POL(U72(x1, x2)) = 0
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 0
POL(U92(x1, x2, x3, x4)) = 0
POL(U93(x1, x2, x3, x4)) = 1
POL(active(x1)) = x1
POL(cons(x1, x2)) = 0
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 0
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = 0
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U42(active(X)) → U42(X)
U42(mark(X)) → U42(X)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
U52(active(X)) → U52(X)
U52(mark(X)) → U52(X)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
U11(active(X)) → U11(X)
U11(mark(X)) → U11(X)
U21(active(X)) → U21(X)
U21(mark(X)) → U21(X)
U31(active(X)) → U31(X)
U31(mark(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
MARK(U62(X)) → MARK(X)
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNatList(take(V1, V2))) → MARK(U61(isNat(V1), V2))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U52(X)) → MARK(X)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U51(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
MARK(U31(X)) → MARK(X)
ACTIVE(U93(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U61(X1, X2)) → MARK(X1)
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
MARK(U11(X)) → MARK(X)
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIList(V2)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
ACTIVE(U93(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
The remaining pairs can at least be oriented weakly.
MARK(U62(X)) → MARK(X)
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNatList(take(V1, V2))) → MARK(U61(isNat(V1), V2))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U52(X)) → MARK(X)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U51(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
MARK(U31(X)) → MARK(X)
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U61(X1, X2)) → MARK(X1)
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
MARK(U11(X)) → MARK(X)
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIList(V2)))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVE(x1)) = x1
POL(MARK(x1)) = 0
POL(U11(x1)) = 0
POL(U21(x1)) = 0
POL(U31(x1)) = 0
POL(U41(x1, x2)) = 0
POL(U42(x1)) = 0
POL(U51(x1, x2)) = 0
POL(U52(x1)) = 0
POL(U61(x1, x2)) = 0
POL(U62(x1)) = 0
POL(U71(x1, x2, x3)) = 0
POL(U72(x1, x2)) = 0
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 0
POL(U92(x1, x2, x3, x4)) = 0
POL(U93(x1, x2, x3, x4)) = 1 + x1 + x2 + x3 + x4
POL(active(x1)) = 0
POL(cons(x1, x2)) = 0
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 0
POL(mark(x1)) = 0
POL(nil) = 0
POL(s(x1)) = 0
POL(take(x1, x2)) = 0
POL(tt) = 1
POL(zeros) = 0
The following usable rules [17] were oriented:
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U62(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNatList(take(V1, V2))) → MARK(U61(isNat(V1), V2))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U52(X)) → MARK(X)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U51(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
MARK(U31(X)) → MARK(X)
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U61(X1, X2)) → MARK(X1)
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIList(V2)))
MARK(U11(X)) → MARK(X)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
ACTIVE(isNatList(take(V1, V2))) → MARK(U61(isNat(V1), V2))
MARK(U11(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U62(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U52(X)) → MARK(X)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U51(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
MARK(U31(X)) → MARK(X)
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U61(X1, X2)) → MARK(X1)
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIList(V2)))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVE(x1)) = x1
POL(MARK(x1)) = x1
POL(U11(x1)) = 1 + x1
POL(U21(x1)) = x1
POL(U31(x1)) = x1
POL(U41(x1, x2)) = x1 + x2
POL(U42(x1)) = x1
POL(U51(x1, x2)) = x1 + x2
POL(U52(x1)) = x1
POL(U61(x1, x2)) = x1 + x2
POL(U62(x1)) = x1
POL(U71(x1, x2, x3)) = 1 + x2 + x3
POL(U72(x1, x2)) = x1 + x2
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 1 + x2 + x3 + x4
POL(U92(x1, x2, x3, x4)) = 1 + x2 + x3 + x4
POL(U93(x1, x2, x3, x4)) = 1 + x2 + x3 + x4
POL(active(x1)) = x1
POL(cons(x1, x2)) = x1 + x2
POL(isNat(x1)) = 1 + x1
POL(isNatIList(x1)) = 1 + x1
POL(isNatList(x1)) = 1 + x1
POL(length(x1)) = 1 + x1
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = 1 + x1 + x2
POL(tt) = 1
POL(zeros) = 0
The following usable rules [17] were oriented:
active(isNatList(nil)) → mark(tt)
active(isNatIList(zeros)) → mark(tt)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
active(length(nil)) → mark(0)
mark(0) → active(0)
mark(U42(X)) → active(U42(mark(X)))
mark(U62(X)) → active(U62(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(zeros) → active(zeros)
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(zeros) → mark(cons(0, zeros))
mark(U81(X)) → active(U81(mark(X)))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
mark(U31(X)) → active(U31(mark(X)))
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
mark(isNatList(X)) → active(isNatList(X))
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(isNat(X)) → active(isNat(X))
mark(U11(X)) → active(U11(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
mark(length(X)) → active(length(mark(X)))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(U21(X)) → active(U21(mark(X)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(U72(tt, L)) → mark(s(length(L)))
mark(s(X)) → active(s(mark(X)))
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
mark(tt) → active(tt)
U51(X1, mark(X2)) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
active(U52(tt)) → mark(tt)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
active(U42(tt)) → mark(tt)
U42(active(X)) → U42(X)
U42(mark(X)) → U42(X)
active(U62(tt)) → mark(tt)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
active(U81(tt)) → mark(nil)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
U52(active(X)) → U52(X)
U52(mark(X)) → U52(X)
active(isNat(0)) → mark(tt)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
U11(active(X)) → U11(X)
U11(mark(X)) → U11(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
U21(active(X)) → U21(X)
U21(mark(X)) → U21(X)
U31(active(X)) → U31(X)
U31(mark(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U81(active(X)) → U81(X)
U81(mark(X)) → U81(X)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
active(U11(tt)) → mark(tt)
s(active(X)) → s(X)
s(mark(X)) → s(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
mark(nil) → active(nil)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
MARK(U62(X)) → MARK(X)
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U52(X)) → MARK(X)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U51(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
MARK(U31(X)) → MARK(X)
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U61(X1, X2)) → MARK(X1)
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIList(V2)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
MARK(U61(X1, X2)) → MARK(X1)
The remaining pairs can at least be oriented weakly.
MARK(U62(X)) → MARK(X)
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U52(X)) → MARK(X)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U51(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
MARK(U31(X)) → MARK(X)
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIList(V2)))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVE(x1)) = 0
POL(MARK(x1)) = x1
POL(U11(x1)) = 0
POL(U21(x1)) = x1
POL(U31(x1)) = x1
POL(U41(x1, x2)) = x1
POL(U42(x1)) = x1
POL(U51(x1, x2)) = x1
POL(U52(x1)) = x1
POL(U61(x1, x2)) = 1 + x1
POL(U62(x1)) = x1
POL(U71(x1, x2, x3)) = 0
POL(U72(x1, x2)) = 0
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 0
POL(U92(x1, x2, x3, x4)) = 0
POL(U93(x1, x2, x3, x4)) = 0
POL(active(x1)) = x1
POL(cons(x1, x2)) = 0
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 0
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = 0
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U42(active(X)) → U42(X)
U42(mark(X)) → U42(X)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
U52(active(X)) → U52(X)
U52(mark(X)) → U52(X)
U11(active(X)) → U11(X)
U11(mark(X)) → U11(X)
U21(active(X)) → U21(X)
U21(mark(X)) → U21(X)
U31(active(X)) → U31(X)
U31(mark(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U62(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U52(X)) → MARK(X)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(U51(X1, X2)) → MARK(X1)
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
MARK(U31(X)) → MARK(X)
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIList(V2)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIList(V2)))
The remaining pairs can at least be oriented weakly.
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U62(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U52(X)) → MARK(X)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(U51(X1, X2)) → MARK(X1)
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
MARK(U31(X)) → MARK(X)
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVE(x1)) = x1
POL(MARK(x1)) = 1
POL(U11(x1)) = 0
POL(U21(x1)) = 0
POL(U31(x1)) = 0
POL(U41(x1, x2)) = 1
POL(U42(x1)) = 0
POL(U51(x1, x2)) = 1
POL(U52(x1)) = 0
POL(U61(x1, x2)) = 1 + x1
POL(U62(x1)) = 0
POL(U71(x1, x2, x3)) = 1
POL(U72(x1, x2)) = 1
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 0
POL(U92(x1, x2, x3, x4)) = 0
POL(U93(x1, x2, x3, x4)) = 0
POL(active(x1)) = 1
POL(cons(x1, x2)) = 0
POL(isNat(x1)) = 1
POL(isNatIList(x1)) = 1
POL(isNatList(x1)) = 1
POL(length(x1)) = 1
POL(mark(x1)) = 1
POL(nil) = 0
POL(s(x1)) = 0
POL(take(x1, x2)) = 0
POL(tt) = 1
POL(zeros) = 0
The following usable rules [17] were oriented:
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
MARK(U62(X)) → MARK(X)
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U52(X)) → MARK(X)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U51(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
MARK(U31(X)) → MARK(X)
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
MARK(U62(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U52(X)) → MARK(X)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U51(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
MARK(U31(X)) → MARK(X)
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVE(x1)) = 0
POL(MARK(x1)) = x1
POL(U11(x1)) = 0
POL(U21(x1)) = x1
POL(U31(x1)) = x1
POL(U41(x1, x2)) = x1
POL(U42(x1)) = x1
POL(U51(x1, x2)) = x1
POL(U52(x1)) = x1
POL(U61(x1, x2)) = 0
POL(U62(x1)) = 1 + x1
POL(U71(x1, x2, x3)) = 0
POL(U72(x1, x2)) = 0
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 0
POL(U92(x1, x2, x3, x4)) = 0
POL(U93(x1, x2, x3, x4)) = 0
POL(active(x1)) = x1
POL(cons(x1, x2)) = 0
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 0
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = 0
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U42(active(X)) → U42(X)
U42(mark(X)) → U42(X)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
U52(active(X)) → U52(X)
U52(mark(X)) → U52(X)
U11(active(X)) → U11(X)
U11(mark(X)) → U11(X)
U21(active(X)) → U21(X)
U21(mark(X)) → U21(X)
U31(active(X)) → U31(X)
U31(mark(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U52(X)) → MARK(X)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(U51(X1, X2)) → MARK(X1)
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
MARK(U31(X)) → MARK(X)
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(U41(X1, X2)) → MARK(X1)
The remaining pairs can at least be oriented weakly.
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U52(X)) → MARK(X)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(U51(X1, X2)) → MARK(X1)
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
MARK(U31(X)) → MARK(X)
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVE(x1)) = x1
POL(MARK(x1)) = x1
POL(U11(x1)) = x1
POL(U21(x1)) = x1
POL(U31(x1)) = x1
POL(U41(x1, x2)) = 1 + x1
POL(U42(x1)) = x1
POL(U51(x1, x2)) = x1
POL(U52(x1)) = x1
POL(U61(x1, x2)) = 0
POL(U62(x1)) = 0
POL(U71(x1, x2, x3)) = 0
POL(U72(x1, x2)) = 0
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = x3
POL(U92(x1, x2, x3, x4)) = x3
POL(U93(x1, x2, x3, x4)) = 0
POL(active(x1)) = x1
POL(cons(x1, x2)) = 0
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 1
POL(isNatList(x1)) = 0
POL(length(x1)) = 0
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = x1
POL(tt) = 0
POL(zeros) = 1
The following usable rules [17] were oriented:
active(isNatList(nil)) → mark(tt)
active(isNatIList(zeros)) → mark(tt)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
active(length(nil)) → mark(0)
mark(0) → active(0)
mark(U42(X)) → active(U42(mark(X)))
mark(U62(X)) → active(U62(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(zeros) → active(zeros)
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(zeros) → mark(cons(0, zeros))
mark(U81(X)) → active(U81(mark(X)))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
mark(U31(X)) → active(U31(mark(X)))
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
mark(isNatList(X)) → active(isNatList(X))
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(isNat(X)) → active(isNat(X))
mark(U11(X)) → active(U11(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
mark(length(X)) → active(length(mark(X)))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(U21(X)) → active(U21(mark(X)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(U72(tt, L)) → mark(s(length(L)))
mark(s(X)) → active(s(mark(X)))
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
mark(tt) → active(tt)
U51(X1, mark(X2)) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
active(U52(tt)) → mark(tt)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
active(U42(tt)) → mark(tt)
U42(active(X)) → U42(X)
U42(mark(X)) → U42(X)
active(U62(tt)) → mark(tt)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
active(U81(tt)) → mark(nil)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
U52(active(X)) → U52(X)
U52(mark(X)) → U52(X)
active(isNat(0)) → mark(tt)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
U11(active(X)) → U11(X)
U11(mark(X)) → U11(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
U21(active(X)) → U21(X)
U21(mark(X)) → U21(X)
U31(active(X)) → U31(X)
U31(mark(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U81(active(X)) → U81(X)
U81(mark(X)) → U81(X)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
active(U11(tt)) → mark(tt)
s(active(X)) → s(X)
s(mark(X)) → s(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
mark(nil) → active(nil)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
MARK(U31(X)) → MARK(X)
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U52(X)) → MARK(X)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(U51(X1, X2)) → MARK(X1)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
MARK(U31(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U52(X)) → MARK(X)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(U51(X1, X2)) → MARK(X1)
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVE(x1)) = 0
POL(MARK(x1)) = x1
POL(U11(x1)) = 0
POL(U21(x1)) = x1
POL(U31(x1)) = 1 + x1
POL(U41(x1, x2)) = 0
POL(U42(x1)) = x1
POL(U51(x1, x2)) = x1
POL(U52(x1)) = x1
POL(U61(x1, x2)) = 0
POL(U62(x1)) = 0
POL(U71(x1, x2, x3)) = 0
POL(U72(x1, x2)) = 0
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 0
POL(U92(x1, x2, x3, x4)) = 0
POL(U93(x1, x2, x3, x4)) = 0
POL(active(x1)) = x1
POL(cons(x1, x2)) = 0
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 0
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = 0
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U42(active(X)) → U42(X)
U42(mark(X)) → U42(X)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
U52(active(X)) → U52(X)
U52(mark(X)) → U52(X)
U11(active(X)) → U11(X)
U11(mark(X)) → U11(X)
U21(active(X)) → U21(X)
U21(mark(X)) → U21(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U52(X)) → MARK(X)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U42(X)) → MARK(X)
MARK(U51(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
The remaining pairs can at least be oriented weakly.
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U52(X)) → MARK(X)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U42(X)) → MARK(X)
MARK(U51(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVE(x1)) = x1
POL(MARK(x1)) = x1
POL(U11(x1)) = 0
POL(U21(x1)) = x1
POL(U31(x1)) = 0
POL(U41(x1, x2)) = 0
POL(U42(x1)) = x1
POL(U51(x1, x2)) = x1 + x2
POL(U52(x1)) = x1
POL(U61(x1, x2)) = 0
POL(U62(x1)) = 0
POL(U71(x1, x2, x3)) = 1
POL(U72(x1, x2)) = 1
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = x1 + x2 + x4
POL(U92(x1, x2, x3, x4)) = x2 + x4
POL(U93(x1, x2, x3, x4)) = x2 + x4
POL(active(x1)) = x1
POL(cons(x1, x2)) = x1 + x2
POL(isNat(x1)) = x1
POL(isNatIList(x1)) = 0
POL(isNatList(x1)) = x1
POL(length(x1)) = 1
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = x2
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
active(isNatList(nil)) → mark(tt)
active(isNatIList(zeros)) → mark(tt)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
active(length(nil)) → mark(0)
mark(0) → active(0)
mark(U42(X)) → active(U42(mark(X)))
mark(U62(X)) → active(U62(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(zeros) → active(zeros)
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(zeros) → mark(cons(0, zeros))
mark(U81(X)) → active(U81(mark(X)))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
mark(U31(X)) → active(U31(mark(X)))
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
mark(isNatList(X)) → active(isNatList(X))
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(isNat(X)) → active(isNat(X))
mark(U11(X)) → active(U11(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
mark(length(X)) → active(length(mark(X)))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(U21(X)) → active(U21(mark(X)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(U72(tt, L)) → mark(s(length(L)))
mark(s(X)) → active(s(mark(X)))
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
mark(tt) → active(tt)
U51(X1, mark(X2)) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
active(U52(tt)) → mark(tt)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
active(U42(tt)) → mark(tt)
U42(active(X)) → U42(X)
U42(mark(X)) → U42(X)
active(U62(tt)) → mark(tt)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
active(U81(tt)) → mark(nil)
U52(active(X)) → U52(X)
U52(mark(X)) → U52(X)
active(isNat(0)) → mark(tt)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
U11(active(X)) → U11(X)
U11(mark(X)) → U11(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
U21(active(X)) → U21(X)
U21(mark(X)) → U21(X)
U31(active(X)) → U31(X)
U31(mark(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U81(active(X)) → U81(X)
U81(mark(X)) → U81(X)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
active(U11(tt)) → mark(tt)
s(active(X)) → s(X)
s(mark(X)) → s(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
mark(nil) → active(nil)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U52(X)) → MARK(X)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(U51(X1, X2)) → MARK(X1)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
MARK(U51(X1, X2)) → MARK(X1)
The remaining pairs can at least be oriented weakly.
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U52(X)) → MARK(X)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVE(x1)) = x1
POL(MARK(x1)) = x1
POL(U11(x1)) = 0
POL(U21(x1)) = x1
POL(U31(x1)) = 0
POL(U41(x1, x2)) = 0
POL(U42(x1)) = x1
POL(U51(x1, x2)) = 1 + x1
POL(U52(x1)) = x1
POL(U61(x1, x2)) = 0
POL(U62(x1)) = 0
POL(U71(x1, x2, x3)) = 0
POL(U72(x1, x2)) = 0
POL(U81(x1)) = 1
POL(U91(x1, x2, x3, x4)) = 1
POL(U92(x1, x2, x3, x4)) = 1
POL(U93(x1, x2, x3, x4)) = 1
POL(active(x1)) = x1
POL(cons(x1, x2)) = 0
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatList(x1)) = 1
POL(length(x1)) = 0
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = 1 + x2
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
active(isNatList(nil)) → mark(tt)
active(isNatIList(zeros)) → mark(tt)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
active(length(nil)) → mark(0)
mark(0) → active(0)
mark(U42(X)) → active(U42(mark(X)))
mark(U62(X)) → active(U62(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(zeros) → active(zeros)
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(zeros) → mark(cons(0, zeros))
mark(U81(X)) → active(U81(mark(X)))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
mark(U31(X)) → active(U31(mark(X)))
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
mark(isNatList(X)) → active(isNatList(X))
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(isNat(X)) → active(isNat(X))
mark(U11(X)) → active(U11(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
mark(length(X)) → active(length(mark(X)))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(U21(X)) → active(U21(mark(X)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(U72(tt, L)) → mark(s(length(L)))
mark(s(X)) → active(s(mark(X)))
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
mark(tt) → active(tt)
U51(X1, mark(X2)) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
active(U52(tt)) → mark(tt)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
active(U42(tt)) → mark(tt)
U42(active(X)) → U42(X)
U42(mark(X)) → U42(X)
active(U62(tt)) → mark(tt)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
active(U81(tt)) → mark(nil)
U52(active(X)) → U52(X)
U52(mark(X)) → U52(X)
active(isNat(0)) → mark(tt)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
U11(active(X)) → U11(X)
U11(mark(X)) → U11(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
U21(active(X)) → U21(X)
U21(mark(X)) → U21(X)
U31(active(X)) → U31(X)
U31(mark(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U81(active(X)) → U81(X)
U81(mark(X)) → U81(X)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
active(U11(tt)) → mark(tt)
s(active(X)) → s(X)
s(mark(X)) → s(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
mark(nil) → active(nil)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U52(X)) → MARK(X)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U42(X)) → MARK(X)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
The remaining pairs can at least be oriented weakly.
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U52(X)) → MARK(X)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U42(X)) → MARK(X)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVE(x1)) = x1
POL(MARK(x1)) = x1
POL(U11(x1)) = 0
POL(U21(x1)) = x1
POL(U31(x1)) = 0
POL(U41(x1, x2)) = 0
POL(U42(x1)) = x1
POL(U51(x1, x2)) = x2
POL(U52(x1)) = x1
POL(U61(x1, x2)) = 0
POL(U62(x1)) = 0
POL(U71(x1, x2, x3)) = 0
POL(U72(x1, x2)) = 0
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 0
POL(U92(x1, x2, x3, x4)) = 0
POL(U93(x1, x2, x3, x4)) = 0
POL(active(x1)) = x1
POL(cons(x1, x2)) = 1 + x1 + x2
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatList(x1)) = x1
POL(length(x1)) = 0
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = x1
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U42(active(X)) → U42(X)
U42(mark(X)) → U42(X)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
U52(active(X)) → U52(X)
U52(mark(X)) → U52(X)
U21(active(X)) → U21(X)
U21(mark(X)) → U21(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U52(X)) → MARK(X)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
MARK(isNatList(X)) → ACTIVE(isNatList(X))
The remaining pairs can at least be oriented weakly.
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U52(X)) → MARK(X)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVE(x1)) = x1
POL(MARK(x1)) = 1
POL(U11(x1)) = 0
POL(U21(x1)) = 0
POL(U31(x1)) = 0
POL(U41(x1, x2)) = 1
POL(U42(x1)) = 0
POL(U51(x1, x2)) = 1
POL(U52(x1)) = 0
POL(U61(x1, x2)) = 0
POL(U62(x1)) = 0
POL(U71(x1, x2, x3)) = 1
POL(U72(x1, x2)) = 1
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 0
POL(U92(x1, x2, x3, x4)) = 0
POL(U93(x1, x2, x3, x4)) = 0
POL(active(x1)) = 0
POL(cons(x1, x2)) = 0
POL(isNat(x1)) = 1
POL(isNatIList(x1)) = 1
POL(isNatList(x1)) = 0
POL(length(x1)) = 1
POL(mark(x1)) = 0
POL(nil) = 0
POL(s(x1)) = 0
POL(take(x1, x2)) = 0
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U52(X)) → MARK(X)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
The remaining pairs can at least be oriented weakly.
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U52(X)) → MARK(X)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVE(x1)) = 0
POL(MARK(x1)) = x1
POL(U11(x1)) = 0
POL(U21(x1)) = x1
POL(U31(x1)) = 0
POL(U41(x1, x2)) = 0
POL(U42(x1)) = x1
POL(U51(x1, x2)) = 1
POL(U52(x1)) = x1
POL(U61(x1, x2)) = 0
POL(U62(x1)) = 0
POL(U71(x1, x2, x3)) = 0
POL(U72(x1, x2)) = 0
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 0
POL(U92(x1, x2, x3, x4)) = 0
POL(U93(x1, x2, x3, x4)) = 0
POL(active(x1)) = x1
POL(cons(x1, x2)) = 0
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 0
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = 0
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U42(active(X)) → U42(X)
U42(mark(X)) → U42(X)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
U52(active(X)) → U52(X)
U52(mark(X)) → U52(X)
U21(active(X)) → U21(X)
U21(mark(X)) → U21(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U52(X)) → MARK(X)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U42(X)) → MARK(X)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
The remaining pairs can at least be oriented weakly.
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U52(X)) → MARK(X)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U42(X)) → MARK(X)
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVE(x1)) = x1
POL(MARK(x1)) = 0
POL(U11(x1)) = 0
POL(U21(x1)) = 0
POL(U31(x1)) = 0
POL(U41(x1, x2)) = 0
POL(U42(x1)) = 0
POL(U51(x1, x2)) = x1 + x2
POL(U52(x1)) = 0
POL(U61(x1, x2)) = 0
POL(U62(x1)) = 0
POL(U71(x1, x2, x3)) = 0
POL(U72(x1, x2)) = 0
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 0
POL(U92(x1, x2, x3, x4)) = 0
POL(U93(x1, x2, x3, x4)) = 0
POL(active(x1)) = 0
POL(cons(x1, x2)) = 0
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 0
POL(mark(x1)) = 0
POL(nil) = 0
POL(s(x1)) = 0
POL(take(x1, x2)) = 0
POL(tt) = 1
POL(zeros) = 0
The following usable rules [17] were oriented:
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U52(X)) → MARK(X)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U42(X)) → MARK(X)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
MARK(U52(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U42(X)) → MARK(X)
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVE(x1)) = 0
POL(MARK(x1)) = x1
POL(U11(x1)) = 0
POL(U21(x1)) = x1
POL(U31(x1)) = 0
POL(U41(x1, x2)) = 0
POL(U42(x1)) = x1
POL(U51(x1, x2)) = 0
POL(U52(x1)) = 1 + x1
POL(U61(x1, x2)) = 0
POL(U62(x1)) = 0
POL(U71(x1, x2, x3)) = 0
POL(U72(x1, x2)) = 0
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 0
POL(U92(x1, x2, x3, x4)) = 0
POL(U93(x1, x2, x3, x4)) = 0
POL(active(x1)) = x1
POL(cons(x1, x2)) = 0
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 0
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = 0
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U42(active(X)) → U42(X)
U42(mark(X)) → U42(X)
U21(active(X)) → U21(X)
U21(mark(X)) → U21(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U42(X)) → MARK(X)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
The remaining pairs can at least be oriented weakly.
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U42(X)) → MARK(X)
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVE(x1)) = x1
POL(MARK(x1)) = x1
POL(U11(x1)) = 0
POL(U21(x1)) = x1
POL(U31(x1)) = 0
POL(U41(x1, x2)) = 1 + x2
POL(U42(x1)) = x1
POL(U51(x1, x2)) = 0
POL(U52(x1)) = 0
POL(U61(x1, x2)) = 0
POL(U62(x1)) = 0
POL(U71(x1, x2, x3)) = 0
POL(U72(x1, x2)) = 0
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 0
POL(U92(x1, x2, x3, x4)) = 0
POL(U93(x1, x2, x3, x4)) = 0
POL(active(x1)) = x1
POL(cons(x1, x2)) = 1 + x2
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 1 + x1
POL(isNatList(x1)) = 0
POL(length(x1)) = 0
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = 0
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U42(active(X)) → U42(X)
U42(mark(X)) → U42(X)
U21(active(X)) → U21(X)
U21(mark(X)) → U21(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U42(X)) → MARK(X)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
The remaining pairs can at least be oriented weakly.
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U42(X)) → MARK(X)
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVE(x1)) = x1
POL(MARK(x1)) = 1
POL(U11(x1)) = 0
POL(U21(x1)) = 0
POL(U31(x1)) = 0
POL(U41(x1, x2)) = 1
POL(U42(x1)) = 0
POL(U51(x1, x2)) = 0
POL(U52(x1)) = 0
POL(U61(x1, x2)) = 0
POL(U62(x1)) = 0
POL(U71(x1, x2, x3)) = 1
POL(U72(x1, x2)) = 1
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 0
POL(U92(x1, x2, x3, x4)) = 0
POL(U93(x1, x2, x3, x4)) = 0
POL(active(x1)) = 0
POL(cons(x1, x2)) = 0
POL(isNat(x1)) = 1
POL(isNatIList(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 1
POL(mark(x1)) = 0
POL(nil) = 0
POL(s(x1)) = 0
POL(take(x1, x2)) = 0
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U42(X)) → MARK(X)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
The remaining pairs can at least be oriented weakly.
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U42(X)) → MARK(X)
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVE(x1)) = 0
POL(MARK(x1)) = x1
POL(U11(x1)) = 0
POL(U21(x1)) = x1
POL(U31(x1)) = 0
POL(U41(x1, x2)) = 1
POL(U42(x1)) = x1
POL(U51(x1, x2)) = 0
POL(U52(x1)) = 0
POL(U61(x1, x2)) = 0
POL(U62(x1)) = 0
POL(U71(x1, x2, x3)) = 0
POL(U72(x1, x2)) = 0
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 0
POL(U92(x1, x2, x3, x4)) = 0
POL(U93(x1, x2, x3, x4)) = 0
POL(active(x1)) = x1
POL(cons(x1, x2)) = 0
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 0
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = 0
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U42(active(X)) → U42(X)
U42(mark(X)) → U42(X)
U21(active(X)) → U21(X)
U21(mark(X)) → U21(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(s(X)) → MARK(X)
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(U42(X)) → MARK(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U21(X)) → MARK(X)
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
The remaining pairs can at least be oriented weakly.
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(s(X)) → MARK(X)
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(U42(X)) → MARK(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U21(X)) → MARK(X)
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVE(x1)) = x1
POL(MARK(x1)) = 0
POL(U11(x1)) = 0
POL(U21(x1)) = 0
POL(U31(x1)) = 0
POL(U41(x1, x2)) = x1 + x2
POL(U42(x1)) = 0
POL(U51(x1, x2)) = 0
POL(U52(x1)) = 0
POL(U61(x1, x2)) = 0
POL(U62(x1)) = 0
POL(U71(x1, x2, x3)) = 0
POL(U72(x1, x2)) = 0
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 0
POL(U92(x1, x2, x3, x4)) = 0
POL(U93(x1, x2, x3, x4)) = 0
POL(active(x1)) = 0
POL(cons(x1, x2)) = 0
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 0
POL(mark(x1)) = 0
POL(nil) = 0
POL(s(x1)) = 0
POL(take(x1, x2)) = 0
POL(tt) = 1
POL(zeros) = 0
The following usable rules [17] were oriented:
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U42(X)) → MARK(X)
MARK(U21(X)) → MARK(X)
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
MARK(U42(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVE(x1)) = 0
POL(MARK(x1)) = x1
POL(U11(x1)) = 0
POL(U21(x1)) = x1
POL(U31(x1)) = 0
POL(U41(x1, x2)) = 0
POL(U42(x1)) = 1 + x1
POL(U51(x1, x2)) = 0
POL(U52(x1)) = 0
POL(U61(x1, x2)) = 0
POL(U62(x1)) = 0
POL(U71(x1, x2, x3)) = 0
POL(U72(x1, x2)) = 0
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 0
POL(U92(x1, x2, x3, x4)) = 0
POL(U93(x1, x2, x3, x4)) = 0
POL(active(x1)) = x1
POL(cons(x1, x2)) = 0
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatList(x1)) = x1
POL(length(x1)) = 0
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = 0
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U21(active(X)) → U21(X)
U21(mark(X)) → U21(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(s(X)) → MARK(X)
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U21(X)) → MARK(X)
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
The remaining pairs can at least be oriented weakly.
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(s(X)) → MARK(X)
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U21(X)) → MARK(X)
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
| M( U92(x1, ..., x4) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
| M( take(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( U51(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( isNatList(x1) ) = | | + | | · | x1 |
| M( isNatIList(x1) ) = | | + | | · | x1 |
| M( U91(x1, ..., x4) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
| M( U93(x1, ..., x4) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
| M( U72(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( cons(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( U61(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( U41(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( U71(x1, ..., x3) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
Tuple symbols:
Matrix type:
We used a basic matrix type which is not further parametrizeable.
As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U21(active(X)) → U21(X)
U21(mark(X)) → U21(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
MARK(isNat(X)) → ACTIVE(isNat(X))
The remaining pairs can at least be oriented weakly.
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U21(X)) → MARK(X)
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVE(x1)) = x1
POL(MARK(x1)) = 1
POL(U11(x1)) = 0
POL(U21(x1)) = 0
POL(U31(x1)) = 0
POL(U41(x1, x2)) = 0
POL(U42(x1)) = 0
POL(U51(x1, x2)) = 0
POL(U52(x1)) = 0
POL(U61(x1, x2)) = 0
POL(U62(x1)) = 0
POL(U71(x1, x2, x3)) = 1
POL(U72(x1, x2)) = 1
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 0
POL(U92(x1, x2, x3, x4)) = 0
POL(U93(x1, x2, x3, x4)) = 0
POL(active(x1)) = 0
POL(cons(x1, x2)) = 0
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 1
POL(mark(x1)) = 0
POL(nil) = 0
POL(s(x1)) = 0
POL(take(x1, x2)) = 0
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(s(X)) → MARK(X)
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U21(X)) → MARK(X)
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
MARK(U21(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(s(X)) → MARK(X)
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVE(x1)) = 0
POL(MARK(x1)) = x1
POL(U11(x1)) = 0
POL(U21(x1)) = 1 + x1
POL(U31(x1)) = 0
POL(U41(x1, x2)) = 0
POL(U42(x1)) = 0
POL(U51(x1, x2)) = 0
POL(U52(x1)) = 0
POL(U61(x1, x2)) = 0
POL(U62(x1)) = 0
POL(U71(x1, x2, x3)) = 0
POL(U72(x1, x2)) = 0
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 0
POL(U92(x1, x2, x3, x4)) = 0
POL(U93(x1, x2, x3, x4)) = 0
POL(active(x1)) = x1
POL(cons(x1, x2)) = 0
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 0
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = 0
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(s(X)) → MARK(X)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3)) at position [0] we obtained the following new rules:
MARK(U71(isNat(x0), y1, y2)) → ACTIVE(U71(active(isNat(x0)), y1, y2))
MARK(U71(y0, mark(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(s(x0), y1, y2)) → ACTIVE(U71(active(s(mark(x0))), y1, y2))
MARK(U71(length(x0), y1, y2)) → ACTIVE(U71(active(length(mark(x0))), y1, y2))
MARK(U71(U41(x0, x1), y1, y2)) → ACTIVE(U71(active(U41(mark(x0), x1)), y1, y2))
MARK(U71(U61(x0, x1), y1, y2)) → ACTIVE(U71(active(U61(mark(x0), x1)), y1, y2))
MARK(U71(U62(x0), y1, y2)) → ACTIVE(U71(active(U62(mark(x0))), y1, y2))
MARK(U71(x0, x1, x2)) → ACTIVE(U71(x0, x1, x2))
MARK(U71(y0, x1, mark(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(zeros, y1, y2)) → ACTIVE(U71(active(zeros), y1, y2))
MARK(U71(U11(x0), y1, y2)) → ACTIVE(U71(active(U11(mark(x0))), y1, y2))
MARK(U71(y0, active(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(U21(x0), y1, y2)) → ACTIVE(U71(active(U21(mark(x0))), y1, y2))
MARK(U71(take(x0, x1), y1, y2)) → ACTIVE(U71(active(take(mark(x0), mark(x1))), y1, y2))
MARK(U71(y0, x1, active(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(U72(x0, x1), y1, y2)) → ACTIVE(U71(active(U72(mark(x0), x1)), y1, y2))
MARK(U71(cons(x0, x1), y1, y2)) → ACTIVE(U71(active(cons(mark(x0), x1)), y1, y2))
MARK(U71(U31(x0), y1, y2)) → ACTIVE(U71(active(U31(mark(x0))), y1, y2))
MARK(U71(U91(x0, x1, x2, x3), y1, y2)) → ACTIVE(U71(active(U91(mark(x0), x1, x2, x3)), y1, y2))
MARK(U71(U93(x0, x1, x2, x3), y1, y2)) → ACTIVE(U71(active(U93(mark(x0), x1, x2, x3)), y1, y2))
MARK(U71(U81(x0), y1, y2)) → ACTIVE(U71(active(U81(mark(x0))), y1, y2))
MARK(U71(isNatIList(x0), y1, y2)) → ACTIVE(U71(active(isNatIList(x0)), y1, y2))
MARK(U71(U42(x0), y1, y2)) → ACTIVE(U71(active(U42(mark(x0))), y1, y2))
MARK(U71(U52(x0), y1, y2)) → ACTIVE(U71(active(U52(mark(x0))), y1, y2))
MARK(U71(nil, y1, y2)) → ACTIVE(U71(active(nil), y1, y2))
MARK(U71(tt, y1, y2)) → ACTIVE(U71(active(tt), y1, y2))
MARK(U71(isNatList(x0), y1, y2)) → ACTIVE(U71(active(isNatList(x0)), y1, y2))
MARK(U71(0, y1, y2)) → ACTIVE(U71(active(0), y1, y2))
MARK(U71(U51(x0, x1), y1, y2)) → ACTIVE(U71(active(U51(mark(x0), x1)), y1, y2))
MARK(U71(U71(x0, x1, x2), y1, y2)) → ACTIVE(U71(active(U71(mark(x0), x1, x2)), y1, y2))
MARK(U71(U92(x0, x1, x2, x3), y1, y2)) → ACTIVE(U71(active(U92(mark(x0), x1, x2, x3)), y1, y2))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
MARK(U71(isNat(x0), y1, y2)) → ACTIVE(U71(active(isNat(x0)), y1, y2))
MARK(U71(s(x0), y1, y2)) → ACTIVE(U71(active(s(mark(x0))), y1, y2))
MARK(U71(y0, mark(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(s(X)) → MARK(X)
MARK(U71(length(x0), y1, y2)) → ACTIVE(U71(active(length(mark(x0))), y1, y2))
MARK(U71(U41(x0, x1), y1, y2)) → ACTIVE(U71(active(U41(mark(x0), x1)), y1, y2))
MARK(U71(U61(x0, x1), y1, y2)) → ACTIVE(U71(active(U61(mark(x0), x1)), y1, y2))
MARK(U71(U62(x0), y1, y2)) → ACTIVE(U71(active(U62(mark(x0))), y1, y2))
MARK(U71(x0, x1, x2)) → ACTIVE(U71(x0, x1, x2))
MARK(U71(y0, x1, mark(x2))) → ACTIVE(U71(mark(y0), x1, x2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U71(zeros, y1, y2)) → ACTIVE(U71(active(zeros), y1, y2))
MARK(U71(U11(x0), y1, y2)) → ACTIVE(U71(active(U11(mark(x0))), y1, y2))
MARK(U71(y0, active(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(U21(x0), y1, y2)) → ACTIVE(U71(active(U21(mark(x0))), y1, y2))
MARK(U71(take(x0, x1), y1, y2)) → ACTIVE(U71(active(take(mark(x0), mark(x1))), y1, y2))
MARK(U71(y0, x1, active(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(U72(x0, x1), y1, y2)) → ACTIVE(U71(active(U72(mark(x0), x1)), y1, y2))
MARK(U71(cons(x0, x1), y1, y2)) → ACTIVE(U71(active(cons(mark(x0), x1)), y1, y2))
MARK(U71(U31(x0), y1, y2)) → ACTIVE(U71(active(U31(mark(x0))), y1, y2))
MARK(U71(U91(x0, x1, x2, x3), y1, y2)) → ACTIVE(U71(active(U91(mark(x0), x1, x2, x3)), y1, y2))
MARK(U71(isNatIList(x0), y1, y2)) → ACTIVE(U71(active(isNatIList(x0)), y1, y2))
MARK(U71(U81(x0), y1, y2)) → ACTIVE(U71(active(U81(mark(x0))), y1, y2))
MARK(U71(U93(x0, x1, x2, x3), y1, y2)) → ACTIVE(U71(active(U93(mark(x0), x1, x2, x3)), y1, y2))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(U71(U42(x0), y1, y2)) → ACTIVE(U71(active(U42(mark(x0))), y1, y2))
MARK(U71(U52(x0), y1, y2)) → ACTIVE(U71(active(U52(mark(x0))), y1, y2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U71(nil, y1, y2)) → ACTIVE(U71(active(nil), y1, y2))
MARK(U71(tt, y1, y2)) → ACTIVE(U71(active(tt), y1, y2))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U71(isNatList(x0), y1, y2)) → ACTIVE(U71(active(isNatList(x0)), y1, y2))
MARK(U71(U51(x0, x1), y1, y2)) → ACTIVE(U71(active(U51(mark(x0), x1)), y1, y2))
MARK(U71(0, y1, y2)) → ACTIVE(U71(active(0), y1, y2))
MARK(U71(U71(x0, x1, x2), y1, y2)) → ACTIVE(U71(active(U71(mark(x0), x1, x2)), y1, y2))
MARK(U71(U92(x0, x1, x2, x3), y1, y2)) → ACTIVE(U71(active(U92(mark(x0), x1, x2, x3)), y1, y2))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2)) at position [0] we obtained the following new rules:
MARK(U72(U31(x0), y1)) → ACTIVE(U72(active(U31(mark(x0))), y1))
MARK(U72(nil, y1)) → ACTIVE(U72(active(nil), y1))
MARK(U72(tt, y1)) → ACTIVE(U72(active(tt), y1))
MARK(U72(U62(x0), y1)) → ACTIVE(U72(active(U62(mark(x0))), y1))
MARK(U72(0, y1)) → ACTIVE(U72(active(0), y1))
MARK(U72(length(x0), y1)) → ACTIVE(U72(active(length(mark(x0))), y1))
MARK(U72(U91(x0, x1, x2, x3), y1)) → ACTIVE(U72(active(U91(mark(x0), x1, x2, x3)), y1))
MARK(U72(U51(x0, x1), y1)) → ACTIVE(U72(active(U51(mark(x0), x1)), y1))
MARK(U72(U93(x0, x1, x2, x3), y1)) → ACTIVE(U72(active(U93(mark(x0), x1, x2, x3)), y1))
MARK(U72(x0, x1)) → ACTIVE(U72(x0, x1))
MARK(U72(U71(x0, x1, x2), y1)) → ACTIVE(U72(active(U71(mark(x0), x1, x2)), y1))
MARK(U72(cons(x0, x1), y1)) → ACTIVE(U72(active(cons(mark(x0), x1)), y1))
MARK(U72(U41(x0, x1), y1)) → ACTIVE(U72(active(U41(mark(x0), x1)), y1))
MARK(U72(U92(x0, x1, x2, x3), y1)) → ACTIVE(U72(active(U92(mark(x0), x1, x2, x3)), y1))
MARK(U72(U52(x0), y1)) → ACTIVE(U72(active(U52(mark(x0))), y1))
MARK(U72(U42(x0), y1)) → ACTIVE(U72(active(U42(mark(x0))), y1))
MARK(U72(U11(x0), y1)) → ACTIVE(U72(active(U11(mark(x0))), y1))
MARK(U72(U81(x0), y1)) → ACTIVE(U72(active(U81(mark(x0))), y1))
MARK(U72(take(x0, x1), y1)) → ACTIVE(U72(active(take(mark(x0), mark(x1))), y1))
MARK(U72(isNatList(x0), y1)) → ACTIVE(U72(active(isNatList(x0)), y1))
MARK(U72(isNat(x0), y1)) → ACTIVE(U72(active(isNat(x0)), y1))
MARK(U72(s(x0), y1)) → ACTIVE(U72(active(s(mark(x0))), y1))
MARK(U72(zeros, y1)) → ACTIVE(U72(active(zeros), y1))
MARK(U72(U61(x0, x1), y1)) → ACTIVE(U72(active(U61(mark(x0), x1)), y1))
MARK(U72(y0, active(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U72(isNatIList(x0), y1)) → ACTIVE(U72(active(isNatIList(x0)), y1))
MARK(U72(y0, mark(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U72(U21(x0), y1)) → ACTIVE(U72(active(U21(mark(x0))), y1))
MARK(U72(U72(x0, x1), y1)) → ACTIVE(U72(active(U72(mark(x0), x1)), y1))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
MARK(U71(isNat(x0), y1, y2)) → ACTIVE(U71(active(isNat(x0)), y1, y2))
MARK(U72(U31(x0), y1)) → ACTIVE(U72(active(U31(mark(x0))), y1))
MARK(U72(nil, y1)) → ACTIVE(U72(active(nil), y1))
MARK(U71(y0, mark(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(length(x0), y1, y2)) → ACTIVE(U71(active(length(mark(x0))), y1, y2))
MARK(U71(U41(x0, x1), y1, y2)) → ACTIVE(U71(active(U41(mark(x0), x1)), y1, y2))
MARK(U72(U62(x0), y1)) → ACTIVE(U72(active(U62(mark(x0))), y1))
MARK(U72(0, y1)) → ACTIVE(U72(active(0), y1))
MARK(U72(length(x0), y1)) → ACTIVE(U72(active(length(mark(x0))), y1))
MARK(U71(y0, x1, mark(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U72(U51(x0, x1), y1)) → ACTIVE(U72(active(U51(mark(x0), x1)), y1))
MARK(U71(U21(x0), y1, y2)) → ACTIVE(U71(active(U21(mark(x0))), y1, y2))
MARK(U71(take(x0, x1), y1, y2)) → ACTIVE(U71(active(take(mark(x0), mark(x1))), y1, y2))
MARK(U72(U71(x0, x1, x2), y1)) → ACTIVE(U72(active(U71(mark(x0), x1, x2)), y1))
MARK(U71(y0, x1, active(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(U72(x0, x1), y1, y2)) → ACTIVE(U71(active(U72(mark(x0), x1)), y1, y2))
MARK(U72(cons(x0, x1), y1)) → ACTIVE(U72(active(cons(mark(x0), x1)), y1))
MARK(U71(cons(x0, x1), y1, y2)) → ACTIVE(U71(active(cons(mark(x0), x1)), y1, y2))
MARK(U72(U41(x0, x1), y1)) → ACTIVE(U72(active(U41(mark(x0), x1)), y1))
MARK(U72(U92(x0, x1, x2, x3), y1)) → ACTIVE(U72(active(U92(mark(x0), x1, x2, x3)), y1))
MARK(U72(U52(x0), y1)) → ACTIVE(U72(active(U52(mark(x0))), y1))
MARK(U71(U81(x0), y1, y2)) → ACTIVE(U71(active(U81(mark(x0))), y1, y2))
MARK(U72(U42(x0), y1)) → ACTIVE(U72(active(U42(mark(x0))), y1))
MARK(U72(U81(x0), y1)) → ACTIVE(U72(active(U81(mark(x0))), y1))
MARK(U72(take(x0, x1), y1)) → ACTIVE(U72(active(take(mark(x0), mark(x1))), y1))
MARK(U71(U42(x0), y1, y2)) → ACTIVE(U71(active(U42(mark(x0))), y1, y2))
MARK(U71(U52(x0), y1, y2)) → ACTIVE(U71(active(U52(mark(x0))), y1, y2))
MARK(U72(s(x0), y1)) → ACTIVE(U72(active(s(mark(x0))), y1))
MARK(U72(isNatList(x0), y1)) → ACTIVE(U72(active(isNatList(x0)), y1))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U71(tt, y1, y2)) → ACTIVE(U71(active(tt), y1, y2))
MARK(U72(U61(x0, x1), y1)) → ACTIVE(U72(active(U61(mark(x0), x1)), y1))
MARK(U71(U51(x0, x1), y1, y2)) → ACTIVE(U71(active(U51(mark(x0), x1)), y1, y2))
MARK(U71(0, y1, y2)) → ACTIVE(U71(active(0), y1, y2))
MARK(U72(y0, mark(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U71(U92(x0, x1, x2, x3), y1, y2)) → ACTIVE(U71(active(U92(mark(x0), x1, x2, x3)), y1, y2))
MARK(U72(U72(x0, x1), y1)) → ACTIVE(U72(active(U72(mark(x0), x1)), y1))
MARK(U71(s(x0), y1, y2)) → ACTIVE(U71(active(s(mark(x0))), y1, y2))
MARK(s(X)) → MARK(X)
MARK(U72(tt, y1)) → ACTIVE(U72(active(tt), y1))
MARK(U71(U61(x0, x1), y1, y2)) → ACTIVE(U71(active(U61(mark(x0), x1)), y1, y2))
MARK(U71(U62(x0), y1, y2)) → ACTIVE(U71(active(U62(mark(x0))), y1, y2))
MARK(U71(x0, x1, x2)) → ACTIVE(U71(x0, x1, x2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U71(zeros, y1, y2)) → ACTIVE(U71(active(zeros), y1, y2))
MARK(U71(U11(x0), y1, y2)) → ACTIVE(U71(active(U11(mark(x0))), y1, y2))
MARK(U71(y0, active(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U72(U91(x0, x1, x2, x3), y1)) → ACTIVE(U72(active(U91(mark(x0), x1, x2, x3)), y1))
MARK(U72(U93(x0, x1, x2, x3), y1)) → ACTIVE(U72(active(U93(mark(x0), x1, x2, x3)), y1))
MARK(U72(x0, x1)) → ACTIVE(U72(x0, x1))
MARK(U71(U31(x0), y1, y2)) → ACTIVE(U71(active(U31(mark(x0))), y1, y2))
MARK(U71(U91(x0, x1, x2, x3), y1, y2)) → ACTIVE(U71(active(U91(mark(x0), x1, x2, x3)), y1, y2))
MARK(U71(isNatIList(x0), y1, y2)) → ACTIVE(U71(active(isNatIList(x0)), y1, y2))
MARK(U71(U93(x0, x1, x2, x3), y1, y2)) → ACTIVE(U71(active(U93(mark(x0), x1, x2, x3)), y1, y2))
MARK(U72(U11(x0), y1)) → ACTIVE(U72(active(U11(mark(x0))), y1))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(U72(isNat(x0), y1)) → ACTIVE(U72(active(isNat(x0)), y1))
MARK(U71(nil, y1, y2)) → ACTIVE(U71(active(nil), y1, y2))
MARK(U72(zeros, y1)) → ACTIVE(U72(active(zeros), y1))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U71(isNatList(x0), y1, y2)) → ACTIVE(U71(active(isNatList(x0)), y1, y2))
MARK(U72(y0, active(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U72(isNatIList(x0), y1)) → ACTIVE(U72(active(isNatIList(x0)), y1))
MARK(U71(U71(x0, x1, x2), y1, y2)) → ACTIVE(U71(active(U71(mark(x0), x1, x2)), y1, y2))
MARK(U72(U21(x0), y1)) → ACTIVE(U72(active(U21(mark(x0))), y1))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(length(X)) → ACTIVE(length(mark(X))) at position [0] we obtained the following new rules:
MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(length(tt)) → ACTIVE(length(active(tt)))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(length(U62(x0))) → ACTIVE(length(active(U62(mark(x0)))))
MARK(length(U91(x0, x1, x2, x3))) → ACTIVE(length(active(U91(mark(x0), x1, x2, x3))))
MARK(length(U81(x0))) → ACTIVE(length(active(U81(mark(x0)))))
MARK(length(U42(x0))) → ACTIVE(length(active(U42(mark(x0)))))
MARK(length(U41(x0, x1))) → ACTIVE(length(active(U41(mark(x0), x1))))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(length(U52(x0))) → ACTIVE(length(active(U52(mark(x0)))))
MARK(length(nil)) → ACTIVE(length(active(nil)))
MARK(length(U61(x0, x1))) → ACTIVE(length(active(U61(mark(x0), x1))))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(length(U92(x0, x1, x2, x3))) → ACTIVE(length(active(U92(mark(x0), x1, x2, x3))))
MARK(length(U93(x0, x1, x2, x3))) → ACTIVE(length(active(U93(mark(x0), x1, x2, x3))))
MARK(length(U71(x0, x1, x2))) → ACTIVE(length(active(U71(mark(x0), x1, x2))))
MARK(length(U31(x0))) → ACTIVE(length(active(U31(mark(x0)))))
MARK(length(U51(x0, x1))) → ACTIVE(length(active(U51(mark(x0), x1))))
MARK(length(0)) → ACTIVE(length(active(0)))
MARK(length(U11(x0))) → ACTIVE(length(active(U11(mark(x0)))))
MARK(length(U21(x0))) → ACTIVE(length(active(U21(mark(x0)))))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(U72(x0, x1))) → ACTIVE(length(active(U72(mark(x0), x1))))
MARK(length(x0)) → ACTIVE(length(x0))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(length(tt)) → ACTIVE(length(active(tt)))
MARK(U72(nil, y1)) → ACTIVE(U72(active(nil), y1))
MARK(U72(U31(x0), y1)) → ACTIVE(U72(active(U31(mark(x0))), y1))
MARK(U71(isNat(x0), y1, y2)) → ACTIVE(U71(active(isNat(x0)), y1, y2))
MARK(length(U62(x0))) → ACTIVE(length(active(U62(mark(x0)))))
MARK(U71(y0, mark(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(length(x0), y1, y2)) → ACTIVE(U71(active(length(mark(x0))), y1, y2))
MARK(U72(U62(x0), y1)) → ACTIVE(U72(active(U62(mark(x0))), y1))
MARK(U71(U41(x0, x1), y1, y2)) → ACTIVE(U71(active(U41(mark(x0), x1)), y1, y2))
MARK(U72(length(x0), y1)) → ACTIVE(U72(active(length(mark(x0))), y1))
MARK(U72(0, y1)) → ACTIVE(U72(active(0), y1))
MARK(U71(y0, x1, mark(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U72(U51(x0, x1), y1)) → ACTIVE(U72(active(U51(mark(x0), x1)), y1))
MARK(U71(U21(x0), y1, y2)) → ACTIVE(U71(active(U21(mark(x0))), y1, y2))
MARK(U72(U71(x0, x1, x2), y1)) → ACTIVE(U72(active(U71(mark(x0), x1, x2)), y1))
MARK(U71(take(x0, x1), y1, y2)) → ACTIVE(U71(active(take(mark(x0), mark(x1))), y1, y2))
MARK(U71(y0, x1, active(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(length(U41(x0, x1))) → ACTIVE(length(active(U41(mark(x0), x1))))
MARK(U72(cons(x0, x1), y1)) → ACTIVE(U72(active(cons(mark(x0), x1)), y1))
MARK(U71(U72(x0, x1), y1, y2)) → ACTIVE(U71(active(U72(mark(x0), x1)), y1, y2))
MARK(length(U52(x0))) → ACTIVE(length(active(U52(mark(x0)))))
MARK(U71(cons(x0, x1), y1, y2)) → ACTIVE(U71(active(cons(mark(x0), x1)), y1, y2))
MARK(length(nil)) → ACTIVE(length(active(nil)))
MARK(U72(U41(x0, x1), y1)) → ACTIVE(U72(active(U41(mark(x0), x1)), y1))
MARK(length(U61(x0, x1))) → ACTIVE(length(active(U61(mark(x0), x1))))
MARK(U72(U92(x0, x1, x2, x3), y1)) → ACTIVE(U72(active(U92(mark(x0), x1, x2, x3)), y1))
MARK(length(U93(x0, x1, x2, x3))) → ACTIVE(length(active(U93(mark(x0), x1, x2, x3))))
MARK(U72(U52(x0), y1)) → ACTIVE(U72(active(U52(mark(x0))), y1))
MARK(U71(U81(x0), y1, y2)) → ACTIVE(U71(active(U81(mark(x0))), y1, y2))
MARK(length(U31(x0))) → ACTIVE(length(active(U31(mark(x0)))))
MARK(U72(U42(x0), y1)) → ACTIVE(U72(active(U42(mark(x0))), y1))
MARK(length(U51(x0, x1))) → ACTIVE(length(active(U51(mark(x0), x1))))
MARK(U72(U81(x0), y1)) → ACTIVE(U72(active(U81(mark(x0))), y1))
MARK(U71(U42(x0), y1, y2)) → ACTIVE(U71(active(U42(mark(x0))), y1, y2))
MARK(U72(take(x0, x1), y1)) → ACTIVE(U72(active(take(mark(x0), mark(x1))), y1))
MARK(length(U11(x0))) → ACTIVE(length(active(U11(mark(x0)))))
MARK(length(0)) → ACTIVE(length(active(0)))
MARK(U71(U52(x0), y1, y2)) → ACTIVE(U71(active(U52(mark(x0))), y1, y2))
MARK(length(U21(x0))) → ACTIVE(length(active(U21(mark(x0)))))
MARK(U72(isNatList(x0), y1)) → ACTIVE(U72(active(isNatList(x0)), y1))
MARK(U72(s(x0), y1)) → ACTIVE(U72(active(s(mark(x0))), y1))
MARK(U71(tt, y1, y2)) → ACTIVE(U71(active(tt), y1, y2))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(U72(U61(x0, x1), y1)) → ACTIVE(U72(active(U61(mark(x0), x1)), y1))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(U71(0, y1, y2)) → ACTIVE(U71(active(0), y1, y2))
MARK(U71(U51(x0, x1), y1, y2)) → ACTIVE(U71(active(U51(mark(x0), x1)), y1, y2))
MARK(U72(y0, mark(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U71(U92(x0, x1, x2, x3), y1, y2)) → ACTIVE(U71(active(U92(mark(x0), x1, x2, x3)), y1, y2))
MARK(U72(U72(x0, x1), y1)) → ACTIVE(U72(active(U72(mark(x0), x1)), y1))
MARK(length(U72(x0, x1))) → ACTIVE(length(active(U72(mark(x0), x1))))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(U72(tt, y1)) → ACTIVE(U72(active(tt), y1))
MARK(s(X)) → MARK(X)
MARK(U71(s(x0), y1, y2)) → ACTIVE(U71(active(s(mark(x0))), y1, y2))
MARK(length(U91(x0, x1, x2, x3))) → ACTIVE(length(active(U91(mark(x0), x1, x2, x3))))
MARK(length(U81(x0))) → ACTIVE(length(active(U81(mark(x0)))))
MARK(U71(U62(x0), y1, y2)) → ACTIVE(U71(active(U62(mark(x0))), y1, y2))
MARK(U71(U61(x0, x1), y1, y2)) → ACTIVE(U71(active(U61(mark(x0), x1)), y1, y2))
MARK(U71(x0, x1, x2)) → ACTIVE(U71(x0, x1, x2))
MARK(U71(zeros, y1, y2)) → ACTIVE(U71(active(zeros), y1, y2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U71(y0, active(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(U11(x0), y1, y2)) → ACTIVE(U71(active(U11(mark(x0))), y1, y2))
MARK(length(U42(x0))) → ACTIVE(length(active(U42(mark(x0)))))
MARK(U72(U91(x0, x1, x2, x3), y1)) → ACTIVE(U72(active(U91(mark(x0), x1, x2, x3)), y1))
MARK(U72(U93(x0, x1, x2, x3), y1)) → ACTIVE(U72(active(U93(mark(x0), x1, x2, x3)), y1))
MARK(U72(x0, x1)) → ACTIVE(U72(x0, x1))
MARK(U71(U31(x0), y1, y2)) → ACTIVE(U71(active(U31(mark(x0))), y1, y2))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(length(U92(x0, x1, x2, x3))) → ACTIVE(length(active(U92(mark(x0), x1, x2, x3))))
MARK(U71(U91(x0, x1, x2, x3), y1, y2)) → ACTIVE(U71(active(U91(mark(x0), x1, x2, x3)), y1, y2))
MARK(length(U71(x0, x1, x2))) → ACTIVE(length(active(U71(mark(x0), x1, x2))))
MARK(U71(U93(x0, x1, x2, x3), y1, y2)) → ACTIVE(U71(active(U93(mark(x0), x1, x2, x3)), y1, y2))
MARK(U71(isNatIList(x0), y1, y2)) → ACTIVE(U71(active(isNatIList(x0)), y1, y2))
MARK(U72(U11(x0), y1)) → ACTIVE(U72(active(U11(mark(x0))), y1))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(U72(isNat(x0), y1)) → ACTIVE(U72(active(isNat(x0)), y1))
MARK(U71(nil, y1, y2)) → ACTIVE(U71(active(nil), y1, y2))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U72(zeros, y1)) → ACTIVE(U72(active(zeros), y1))
MARK(U71(isNatList(x0), y1, y2)) → ACTIVE(U71(active(isNatList(x0)), y1, y2))
MARK(U72(y0, active(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U72(isNatIList(x0), y1)) → ACTIVE(U72(active(isNatIList(x0)), y1))
MARK(U71(U71(x0, x1, x2), y1, y2)) → ACTIVE(U71(active(U71(mark(x0), x1, x2)), y1, y2))
MARK(U72(U21(x0), y1)) → ACTIVE(U72(active(U21(mark(x0))), y1))
MARK(length(x0)) → ACTIVE(length(x0))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
MARK(U72(U51(x0, x1), y1)) → ACTIVE(U72(active(U51(mark(x0), x1)), y1))
MARK(U72(cons(x0, x1), y1)) → ACTIVE(U72(active(cons(mark(x0), x1)), y1))
MARK(U72(U41(x0, x1), y1)) → ACTIVE(U72(active(U41(mark(x0), x1)), y1))
MARK(U72(U92(x0, x1, x2, x3), y1)) → ACTIVE(U72(active(U92(mark(x0), x1, x2, x3)), y1))
MARK(U72(U42(x0), y1)) → ACTIVE(U72(active(U42(mark(x0))), y1))
MARK(U72(U61(x0, x1), y1)) → ACTIVE(U72(active(U61(mark(x0), x1)), y1))
MARK(U72(tt, y1)) → ACTIVE(U72(active(tt), y1))
MARK(U72(U93(x0, x1, x2, x3), y1)) → ACTIVE(U72(active(U93(mark(x0), x1, x2, x3)), y1))
MARK(U72(isNatIList(x0), y1)) → ACTIVE(U72(active(isNatIList(x0)), y1))
The remaining pairs can at least be oriented weakly.
MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(length(tt)) → ACTIVE(length(active(tt)))
MARK(U72(nil, y1)) → ACTIVE(U72(active(nil), y1))
MARK(U72(U31(x0), y1)) → ACTIVE(U72(active(U31(mark(x0))), y1))
MARK(U71(isNat(x0), y1, y2)) → ACTIVE(U71(active(isNat(x0)), y1, y2))
MARK(length(U62(x0))) → ACTIVE(length(active(U62(mark(x0)))))
MARK(U71(y0, mark(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(length(x0), y1, y2)) → ACTIVE(U71(active(length(mark(x0))), y1, y2))
MARK(U72(U62(x0), y1)) → ACTIVE(U72(active(U62(mark(x0))), y1))
MARK(U71(U41(x0, x1), y1, y2)) → ACTIVE(U71(active(U41(mark(x0), x1)), y1, y2))
MARK(U72(length(x0), y1)) → ACTIVE(U72(active(length(mark(x0))), y1))
MARK(U72(0, y1)) → ACTIVE(U72(active(0), y1))
MARK(U71(y0, x1, mark(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(U21(x0), y1, y2)) → ACTIVE(U71(active(U21(mark(x0))), y1, y2))
MARK(U72(U71(x0, x1, x2), y1)) → ACTIVE(U72(active(U71(mark(x0), x1, x2)), y1))
MARK(U71(take(x0, x1), y1, y2)) → ACTIVE(U71(active(take(mark(x0), mark(x1))), y1, y2))
MARK(U71(y0, x1, active(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(length(U41(x0, x1))) → ACTIVE(length(active(U41(mark(x0), x1))))
MARK(U71(U72(x0, x1), y1, y2)) → ACTIVE(U71(active(U72(mark(x0), x1)), y1, y2))
MARK(length(U52(x0))) → ACTIVE(length(active(U52(mark(x0)))))
MARK(U71(cons(x0, x1), y1, y2)) → ACTIVE(U71(active(cons(mark(x0), x1)), y1, y2))
MARK(length(nil)) → ACTIVE(length(active(nil)))
MARK(length(U61(x0, x1))) → ACTIVE(length(active(U61(mark(x0), x1))))
MARK(length(U93(x0, x1, x2, x3))) → ACTIVE(length(active(U93(mark(x0), x1, x2, x3))))
MARK(U72(U52(x0), y1)) → ACTIVE(U72(active(U52(mark(x0))), y1))
MARK(U71(U81(x0), y1, y2)) → ACTIVE(U71(active(U81(mark(x0))), y1, y2))
MARK(length(U31(x0))) → ACTIVE(length(active(U31(mark(x0)))))
MARK(length(U51(x0, x1))) → ACTIVE(length(active(U51(mark(x0), x1))))
MARK(U72(U81(x0), y1)) → ACTIVE(U72(active(U81(mark(x0))), y1))
MARK(U71(U42(x0), y1, y2)) → ACTIVE(U71(active(U42(mark(x0))), y1, y2))
MARK(U72(take(x0, x1), y1)) → ACTIVE(U72(active(take(mark(x0), mark(x1))), y1))
MARK(length(U11(x0))) → ACTIVE(length(active(U11(mark(x0)))))
MARK(length(0)) → ACTIVE(length(active(0)))
MARK(U71(U52(x0), y1, y2)) → ACTIVE(U71(active(U52(mark(x0))), y1, y2))
MARK(length(U21(x0))) → ACTIVE(length(active(U21(mark(x0)))))
MARK(U72(isNatList(x0), y1)) → ACTIVE(U72(active(isNatList(x0)), y1))
MARK(U72(s(x0), y1)) → ACTIVE(U72(active(s(mark(x0))), y1))
MARK(U71(tt, y1, y2)) → ACTIVE(U71(active(tt), y1, y2))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(U71(0, y1, y2)) → ACTIVE(U71(active(0), y1, y2))
MARK(U71(U51(x0, x1), y1, y2)) → ACTIVE(U71(active(U51(mark(x0), x1)), y1, y2))
MARK(U72(y0, mark(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U71(U92(x0, x1, x2, x3), y1, y2)) → ACTIVE(U71(active(U92(mark(x0), x1, x2, x3)), y1, y2))
MARK(U72(U72(x0, x1), y1)) → ACTIVE(U72(active(U72(mark(x0), x1)), y1))
MARK(length(U72(x0, x1))) → ACTIVE(length(active(U72(mark(x0), x1))))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
MARK(U71(s(x0), y1, y2)) → ACTIVE(U71(active(s(mark(x0))), y1, y2))
MARK(length(U91(x0, x1, x2, x3))) → ACTIVE(length(active(U91(mark(x0), x1, x2, x3))))
MARK(length(U81(x0))) → ACTIVE(length(active(U81(mark(x0)))))
MARK(U71(U62(x0), y1, y2)) → ACTIVE(U71(active(U62(mark(x0))), y1, y2))
MARK(U71(U61(x0, x1), y1, y2)) → ACTIVE(U71(active(U61(mark(x0), x1)), y1, y2))
MARK(U71(x0, x1, x2)) → ACTIVE(U71(x0, x1, x2))
MARK(U71(zeros, y1, y2)) → ACTIVE(U71(active(zeros), y1, y2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U71(y0, active(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(U11(x0), y1, y2)) → ACTIVE(U71(active(U11(mark(x0))), y1, y2))
MARK(length(U42(x0))) → ACTIVE(length(active(U42(mark(x0)))))
MARK(U72(U91(x0, x1, x2, x3), y1)) → ACTIVE(U72(active(U91(mark(x0), x1, x2, x3)), y1))
MARK(U72(x0, x1)) → ACTIVE(U72(x0, x1))
MARK(U71(U31(x0), y1, y2)) → ACTIVE(U71(active(U31(mark(x0))), y1, y2))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(length(U92(x0, x1, x2, x3))) → ACTIVE(length(active(U92(mark(x0), x1, x2, x3))))
MARK(U71(U91(x0, x1, x2, x3), y1, y2)) → ACTIVE(U71(active(U91(mark(x0), x1, x2, x3)), y1, y2))
MARK(length(U71(x0, x1, x2))) → ACTIVE(length(active(U71(mark(x0), x1, x2))))
MARK(U71(U93(x0, x1, x2, x3), y1, y2)) → ACTIVE(U71(active(U93(mark(x0), x1, x2, x3)), y1, y2))
MARK(U71(isNatIList(x0), y1, y2)) → ACTIVE(U71(active(isNatIList(x0)), y1, y2))
MARK(U72(U11(x0), y1)) → ACTIVE(U72(active(U11(mark(x0))), y1))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(U72(isNat(x0), y1)) → ACTIVE(U72(active(isNat(x0)), y1))
MARK(U71(nil, y1, y2)) → ACTIVE(U71(active(nil), y1, y2))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U72(zeros, y1)) → ACTIVE(U72(active(zeros), y1))
MARK(U71(isNatList(x0), y1, y2)) → ACTIVE(U71(active(isNatList(x0)), y1, y2))
MARK(U72(y0, active(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U71(U71(x0, x1, x2), y1, y2)) → ACTIVE(U71(active(U71(mark(x0), x1, x2)), y1, y2))
MARK(U72(U21(x0), y1)) → ACTIVE(U72(active(U21(mark(x0))), y1))
MARK(length(x0)) → ACTIVE(length(x0))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVE(x1)) = 0
POL(MARK(x1)) = x1
POL(U11(x1)) = 0
POL(U21(x1)) = 0
POL(U31(x1)) = x1
POL(U41(x1, x2)) = 1 + x1 + x2
POL(U42(x1)) = 1
POL(U51(x1, x2)) = 1 + x1 + x2
POL(U52(x1)) = 0
POL(U61(x1, x2)) = 1 + x1
POL(U62(x1)) = 0
POL(U71(x1, x2, x3)) = 0
POL(U72(x1, x2)) = x1
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = x2
POL(U92(x1, x2, x3, x4)) = 1 + x2
POL(U93(x1, x2, x3, x4)) = 1
POL(active(x1)) = x1
POL(cons(x1, x2)) = 1
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 1 + x1
POL(isNatList(x1)) = 0
POL(length(x1)) = 0
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = 0
POL(tt) = 1
POL(zeros) = 0
The following usable rules [17] were oriented:
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(length(tt)) → ACTIVE(length(active(tt)))
MARK(U72(nil, y1)) → ACTIVE(U72(active(nil), y1))
MARK(U72(U31(x0), y1)) → ACTIVE(U72(active(U31(mark(x0))), y1))
MARK(U71(isNat(x0), y1, y2)) → ACTIVE(U71(active(isNat(x0)), y1, y2))
MARK(length(U62(x0))) → ACTIVE(length(active(U62(mark(x0)))))
MARK(U71(y0, mark(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(length(x0), y1, y2)) → ACTIVE(U71(active(length(mark(x0))), y1, y2))
MARK(U72(U62(x0), y1)) → ACTIVE(U72(active(U62(mark(x0))), y1))
MARK(U71(U41(x0, x1), y1, y2)) → ACTIVE(U71(active(U41(mark(x0), x1)), y1, y2))
MARK(U72(length(x0), y1)) → ACTIVE(U72(active(length(mark(x0))), y1))
MARK(U72(0, y1)) → ACTIVE(U72(active(0), y1))
MARK(U71(y0, x1, mark(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(U21(x0), y1, y2)) → ACTIVE(U71(active(U21(mark(x0))), y1, y2))
MARK(U71(take(x0, x1), y1, y2)) → ACTIVE(U71(active(take(mark(x0), mark(x1))), y1, y2))
MARK(U72(U71(x0, x1, x2), y1)) → ACTIVE(U72(active(U71(mark(x0), x1, x2)), y1))
MARK(U71(y0, x1, active(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(U72(x0, x1), y1, y2)) → ACTIVE(U71(active(U72(mark(x0), x1)), y1, y2))
MARK(length(U41(x0, x1))) → ACTIVE(length(active(U41(mark(x0), x1))))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(U71(cons(x0, x1), y1, y2)) → ACTIVE(U71(active(cons(mark(x0), x1)), y1, y2))
MARK(length(U52(x0))) → ACTIVE(length(active(U52(mark(x0)))))
MARK(length(nil)) → ACTIVE(length(active(nil)))
MARK(length(U61(x0, x1))) → ACTIVE(length(active(U61(mark(x0), x1))))
MARK(length(U93(x0, x1, x2, x3))) → ACTIVE(length(active(U93(mark(x0), x1, x2, x3))))
MARK(U72(U52(x0), y1)) → ACTIVE(U72(active(U52(mark(x0))), y1))
MARK(U71(U81(x0), y1, y2)) → ACTIVE(U71(active(U81(mark(x0))), y1, y2))
MARK(length(U31(x0))) → ACTIVE(length(active(U31(mark(x0)))))
MARK(length(U51(x0, x1))) → ACTIVE(length(active(U51(mark(x0), x1))))
MARK(U72(U81(x0), y1)) → ACTIVE(U72(active(U81(mark(x0))), y1))
MARK(U71(U42(x0), y1, y2)) → ACTIVE(U71(active(U42(mark(x0))), y1, y2))
MARK(U72(take(x0, x1), y1)) → ACTIVE(U72(active(take(mark(x0), mark(x1))), y1))
MARK(length(U11(x0))) → ACTIVE(length(active(U11(mark(x0)))))
MARK(length(0)) → ACTIVE(length(active(0)))
MARK(U71(U52(x0), y1, y2)) → ACTIVE(U71(active(U52(mark(x0))), y1, y2))
MARK(length(U21(x0))) → ACTIVE(length(active(U21(mark(x0)))))
MARK(U72(isNatList(x0), y1)) → ACTIVE(U72(active(isNatList(x0)), y1))
MARK(U72(s(x0), y1)) → ACTIVE(U72(active(s(mark(x0))), y1))
MARK(U71(tt, y1, y2)) → ACTIVE(U71(active(tt), y1, y2))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(U71(0, y1, y2)) → ACTIVE(U71(active(0), y1, y2))
MARK(U71(U51(x0, x1), y1, y2)) → ACTIVE(U71(active(U51(mark(x0), x1)), y1, y2))
MARK(U72(y0, mark(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U71(U92(x0, x1, x2, x3), y1, y2)) → ACTIVE(U71(active(U92(mark(x0), x1, x2, x3)), y1, y2))
MARK(U72(U72(x0, x1), y1)) → ACTIVE(U72(active(U72(mark(x0), x1)), y1))
MARK(length(U72(x0, x1))) → ACTIVE(length(active(U72(mark(x0), x1))))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(U71(s(x0), y1, y2)) → ACTIVE(U71(active(s(mark(x0))), y1, y2))
MARK(s(X)) → MARK(X)
MARK(length(U91(x0, x1, x2, x3))) → ACTIVE(length(active(U91(mark(x0), x1, x2, x3))))
MARK(length(U81(x0))) → ACTIVE(length(active(U81(mark(x0)))))
MARK(U71(U61(x0, x1), y1, y2)) → ACTIVE(U71(active(U61(mark(x0), x1)), y1, y2))
MARK(U71(U62(x0), y1, y2)) → ACTIVE(U71(active(U62(mark(x0))), y1, y2))
MARK(U71(x0, x1, x2)) → ACTIVE(U71(x0, x1, x2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U71(zeros, y1, y2)) → ACTIVE(U71(active(zeros), y1, y2))
MARK(U71(U11(x0), y1, y2)) → ACTIVE(U71(active(U11(mark(x0))), y1, y2))
MARK(U71(y0, active(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U72(U91(x0, x1, x2, x3), y1)) → ACTIVE(U72(active(U91(mark(x0), x1, x2, x3)), y1))
MARK(length(U42(x0))) → ACTIVE(length(active(U42(mark(x0)))))
MARK(U72(x0, x1)) → ACTIVE(U72(x0, x1))
MARK(U71(U31(x0), y1, y2)) → ACTIVE(U71(active(U31(mark(x0))), y1, y2))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(length(U92(x0, x1, x2, x3))) → ACTIVE(length(active(U92(mark(x0), x1, x2, x3))))
MARK(U71(U91(x0, x1, x2, x3), y1, y2)) → ACTIVE(U71(active(U91(mark(x0), x1, x2, x3)), y1, y2))
MARK(length(U71(x0, x1, x2))) → ACTIVE(length(active(U71(mark(x0), x1, x2))))
MARK(U71(isNatIList(x0), y1, y2)) → ACTIVE(U71(active(isNatIList(x0)), y1, y2))
MARK(U71(U93(x0, x1, x2, x3), y1, y2)) → ACTIVE(U71(active(U93(mark(x0), x1, x2, x3)), y1, y2))
MARK(U72(U11(x0), y1)) → ACTIVE(U72(active(U11(mark(x0))), y1))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(U72(isNat(x0), y1)) → ACTIVE(U72(active(isNat(x0)), y1))
MARK(U71(nil, y1, y2)) → ACTIVE(U71(active(nil), y1, y2))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
MARK(U72(zeros, y1)) → ACTIVE(U72(active(zeros), y1))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U71(isNatList(x0), y1, y2)) → ACTIVE(U71(active(isNatList(x0)), y1, y2))
MARK(U72(y0, active(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U71(U71(x0, x1, x2), y1, y2)) → ACTIVE(U71(active(U71(mark(x0), x1, x2)), y1, y2))
MARK(U72(U21(x0), y1)) → ACTIVE(U72(active(U21(mark(x0))), y1))
MARK(length(x0)) → ACTIVE(length(x0))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(length(tt)) → ACTIVE(length(active(tt))) at position [0] we obtained the following new rules:
MARK(length(tt)) → ACTIVE(length(tt))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(U71(isNat(x0), y1, y2)) → ACTIVE(U71(active(isNat(x0)), y1, y2))
MARK(U72(U31(x0), y1)) → ACTIVE(U72(active(U31(mark(x0))), y1))
MARK(U72(nil, y1)) → ACTIVE(U72(active(nil), y1))
MARK(length(U62(x0))) → ACTIVE(length(active(U62(mark(x0)))))
MARK(U71(y0, mark(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(length(x0), y1, y2)) → ACTIVE(U71(active(length(mark(x0))), y1, y2))
MARK(U71(U41(x0, x1), y1, y2)) → ACTIVE(U71(active(U41(mark(x0), x1)), y1, y2))
MARK(U72(U62(x0), y1)) → ACTIVE(U72(active(U62(mark(x0))), y1))
MARK(U72(0, y1)) → ACTIVE(U72(active(0), y1))
MARK(U72(length(x0), y1)) → ACTIVE(U72(active(length(mark(x0))), y1))
MARK(U71(y0, x1, mark(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(U21(x0), y1, y2)) → ACTIVE(U71(active(U21(mark(x0))), y1, y2))
MARK(U72(U71(x0, x1, x2), y1)) → ACTIVE(U72(active(U71(mark(x0), x1, x2)), y1))
MARK(U71(take(x0, x1), y1, y2)) → ACTIVE(U71(active(take(mark(x0), mark(x1))), y1, y2))
MARK(U71(y0, x1, active(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(length(U41(x0, x1))) → ACTIVE(length(active(U41(mark(x0), x1))))
MARK(U71(U72(x0, x1), y1, y2)) → ACTIVE(U71(active(U72(mark(x0), x1)), y1, y2))
MARK(length(U52(x0))) → ACTIVE(length(active(U52(mark(x0)))))
MARK(U71(cons(x0, x1), y1, y2)) → ACTIVE(U71(active(cons(mark(x0), x1)), y1, y2))
MARK(length(nil)) → ACTIVE(length(active(nil)))
MARK(length(U61(x0, x1))) → ACTIVE(length(active(U61(mark(x0), x1))))
MARK(U72(U52(x0), y1)) → ACTIVE(U72(active(U52(mark(x0))), y1))
MARK(length(U93(x0, x1, x2, x3))) → ACTIVE(length(active(U93(mark(x0), x1, x2, x3))))
MARK(U71(U81(x0), y1, y2)) → ACTIVE(U71(active(U81(mark(x0))), y1, y2))
MARK(length(U31(x0))) → ACTIVE(length(active(U31(mark(x0)))))
MARK(U72(U81(x0), y1)) → ACTIVE(U72(active(U81(mark(x0))), y1))
MARK(length(U51(x0, x1))) → ACTIVE(length(active(U51(mark(x0), x1))))
MARK(U72(take(x0, x1), y1)) → ACTIVE(U72(active(take(mark(x0), mark(x1))), y1))
MARK(U71(U42(x0), y1, y2)) → ACTIVE(U71(active(U42(mark(x0))), y1, y2))
MARK(length(0)) → ACTIVE(length(active(0)))
MARK(length(U11(x0))) → ACTIVE(length(active(U11(mark(x0)))))
MARK(length(tt)) → ACTIVE(length(tt))
MARK(U71(U52(x0), y1, y2)) → ACTIVE(U71(active(U52(mark(x0))), y1, y2))
MARK(length(U21(x0))) → ACTIVE(length(active(U21(mark(x0)))))
MARK(U72(s(x0), y1)) → ACTIVE(U72(active(s(mark(x0))), y1))
MARK(U72(isNatList(x0), y1)) → ACTIVE(U72(active(isNatList(x0)), y1))
MARK(U71(tt, y1, y2)) → ACTIVE(U71(active(tt), y1, y2))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(U71(U51(x0, x1), y1, y2)) → ACTIVE(U71(active(U51(mark(x0), x1)), y1, y2))
MARK(U71(0, y1, y2)) → ACTIVE(U71(active(0), y1, y2))
MARK(U72(y0, mark(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U71(U92(x0, x1, x2, x3), y1, y2)) → ACTIVE(U71(active(U92(mark(x0), x1, x2, x3)), y1, y2))
MARK(U72(U72(x0, x1), y1)) → ACTIVE(U72(active(U72(mark(x0), x1)), y1))
MARK(length(U72(x0, x1))) → ACTIVE(length(active(U72(mark(x0), x1))))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
MARK(U71(s(x0), y1, y2)) → ACTIVE(U71(active(s(mark(x0))), y1, y2))
MARK(length(U91(x0, x1, x2, x3))) → ACTIVE(length(active(U91(mark(x0), x1, x2, x3))))
MARK(length(U81(x0))) → ACTIVE(length(active(U81(mark(x0)))))
MARK(U71(U62(x0), y1, y2)) → ACTIVE(U71(active(U62(mark(x0))), y1, y2))
MARK(U71(U61(x0, x1), y1, y2)) → ACTIVE(U71(active(U61(mark(x0), x1)), y1, y2))
MARK(U71(x0, x1, x2)) → ACTIVE(U71(x0, x1, x2))
MARK(U71(zeros, y1, y2)) → ACTIVE(U71(active(zeros), y1, y2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U71(y0, active(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(U11(x0), y1, y2)) → ACTIVE(U71(active(U11(mark(x0))), y1, y2))
MARK(length(U42(x0))) → ACTIVE(length(active(U42(mark(x0)))))
MARK(U72(U91(x0, x1, x2, x3), y1)) → ACTIVE(U72(active(U91(mark(x0), x1, x2, x3)), y1))
MARK(U72(x0, x1)) → ACTIVE(U72(x0, x1))
MARK(U71(U31(x0), y1, y2)) → ACTIVE(U71(active(U31(mark(x0))), y1, y2))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(length(U92(x0, x1, x2, x3))) → ACTIVE(length(active(U92(mark(x0), x1, x2, x3))))
MARK(U71(U91(x0, x1, x2, x3), y1, y2)) → ACTIVE(U71(active(U91(mark(x0), x1, x2, x3)), y1, y2))
MARK(length(U71(x0, x1, x2))) → ACTIVE(length(active(U71(mark(x0), x1, x2))))
MARK(U71(U93(x0, x1, x2, x3), y1, y2)) → ACTIVE(U71(active(U93(mark(x0), x1, x2, x3)), y1, y2))
MARK(U71(isNatIList(x0), y1, y2)) → ACTIVE(U71(active(isNatIList(x0)), y1, y2))
MARK(U72(U11(x0), y1)) → ACTIVE(U72(active(U11(mark(x0))), y1))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(U72(isNat(x0), y1)) → ACTIVE(U72(active(isNat(x0)), y1))
MARK(U71(nil, y1, y2)) → ACTIVE(U71(active(nil), y1, y2))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U72(zeros, y1)) → ACTIVE(U72(active(zeros), y1))
MARK(U71(isNatList(x0), y1, y2)) → ACTIVE(U71(active(isNatList(x0)), y1, y2))
MARK(U72(y0, active(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U71(U71(x0, x1, x2), y1, y2)) → ACTIVE(U71(active(U71(mark(x0), x1, x2)), y1, y2))
MARK(U72(U21(x0), y1)) → ACTIVE(U72(active(U21(mark(x0))), y1))
MARK(length(x0)) → ACTIVE(length(x0))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(U71(isNat(x0), y1, y2)) → ACTIVE(U71(active(isNat(x0)), y1, y2))
MARK(U72(U31(x0), y1)) → ACTIVE(U72(active(U31(mark(x0))), y1))
MARK(U72(nil, y1)) → ACTIVE(U72(active(nil), y1))
MARK(length(U62(x0))) → ACTIVE(length(active(U62(mark(x0)))))
MARK(U71(y0, mark(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(length(x0), y1, y2)) → ACTIVE(U71(active(length(mark(x0))), y1, y2))
MARK(U71(U41(x0, x1), y1, y2)) → ACTIVE(U71(active(U41(mark(x0), x1)), y1, y2))
MARK(U72(U62(x0), y1)) → ACTIVE(U72(active(U62(mark(x0))), y1))
MARK(U72(0, y1)) → ACTIVE(U72(active(0), y1))
MARK(U72(length(x0), y1)) → ACTIVE(U72(active(length(mark(x0))), y1))
MARK(U71(y0, x1, mark(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(U21(x0), y1, y2)) → ACTIVE(U71(active(U21(mark(x0))), y1, y2))
MARK(U71(take(x0, x1), y1, y2)) → ACTIVE(U71(active(take(mark(x0), mark(x1))), y1, y2))
MARK(U72(U71(x0, x1, x2), y1)) → ACTIVE(U72(active(U71(mark(x0), x1, x2)), y1))
MARK(U71(y0, x1, active(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(length(U41(x0, x1))) → ACTIVE(length(active(U41(mark(x0), x1))))
MARK(U71(U72(x0, x1), y1, y2)) → ACTIVE(U71(active(U72(mark(x0), x1)), y1, y2))
MARK(length(U52(x0))) → ACTIVE(length(active(U52(mark(x0)))))
MARK(U71(cons(x0, x1), y1, y2)) → ACTIVE(U71(active(cons(mark(x0), x1)), y1, y2))
MARK(length(nil)) → ACTIVE(length(active(nil)))
MARK(length(U61(x0, x1))) → ACTIVE(length(active(U61(mark(x0), x1))))
MARK(length(U93(x0, x1, x2, x3))) → ACTIVE(length(active(U93(mark(x0), x1, x2, x3))))
MARK(U72(U52(x0), y1)) → ACTIVE(U72(active(U52(mark(x0))), y1))
MARK(U71(U81(x0), y1, y2)) → ACTIVE(U71(active(U81(mark(x0))), y1, y2))
MARK(length(U31(x0))) → ACTIVE(length(active(U31(mark(x0)))))
MARK(length(U51(x0, x1))) → ACTIVE(length(active(U51(mark(x0), x1))))
MARK(U72(U81(x0), y1)) → ACTIVE(U72(active(U81(mark(x0))), y1))
MARK(U71(U42(x0), y1, y2)) → ACTIVE(U71(active(U42(mark(x0))), y1, y2))
MARK(U72(take(x0, x1), y1)) → ACTIVE(U72(active(take(mark(x0), mark(x1))), y1))
MARK(length(U11(x0))) → ACTIVE(length(active(U11(mark(x0)))))
MARK(length(0)) → ACTIVE(length(active(0)))
MARK(U71(U52(x0), y1, y2)) → ACTIVE(U71(active(U52(mark(x0))), y1, y2))
MARK(length(U21(x0))) → ACTIVE(length(active(U21(mark(x0)))))
MARK(U72(s(x0), y1)) → ACTIVE(U72(active(s(mark(x0))), y1))
MARK(U72(isNatList(x0), y1)) → ACTIVE(U72(active(isNatList(x0)), y1))
MARK(U71(tt, y1, y2)) → ACTIVE(U71(active(tt), y1, y2))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(U71(0, y1, y2)) → ACTIVE(U71(active(0), y1, y2))
MARK(U71(U51(x0, x1), y1, y2)) → ACTIVE(U71(active(U51(mark(x0), x1)), y1, y2))
MARK(U72(y0, mark(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U71(U92(x0, x1, x2, x3), y1, y2)) → ACTIVE(U71(active(U92(mark(x0), x1, x2, x3)), y1, y2))
MARK(U72(U72(x0, x1), y1)) → ACTIVE(U72(active(U72(mark(x0), x1)), y1))
MARK(length(U72(x0, x1))) → ACTIVE(length(active(U72(mark(x0), x1))))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
MARK(U71(s(x0), y1, y2)) → ACTIVE(U71(active(s(mark(x0))), y1, y2))
MARK(length(U91(x0, x1, x2, x3))) → ACTIVE(length(active(U91(mark(x0), x1, x2, x3))))
MARK(length(U81(x0))) → ACTIVE(length(active(U81(mark(x0)))))
MARK(U71(U61(x0, x1), y1, y2)) → ACTIVE(U71(active(U61(mark(x0), x1)), y1, y2))
MARK(U71(U62(x0), y1, y2)) → ACTIVE(U71(active(U62(mark(x0))), y1, y2))
MARK(U71(x0, x1, x2)) → ACTIVE(U71(x0, x1, x2))
MARK(U71(zeros, y1, y2)) → ACTIVE(U71(active(zeros), y1, y2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U71(y0, active(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(U11(x0), y1, y2)) → ACTIVE(U71(active(U11(mark(x0))), y1, y2))
MARK(length(U42(x0))) → ACTIVE(length(active(U42(mark(x0)))))
MARK(U72(U91(x0, x1, x2, x3), y1)) → ACTIVE(U72(active(U91(mark(x0), x1, x2, x3)), y1))
MARK(U72(x0, x1)) → ACTIVE(U72(x0, x1))
MARK(U71(U31(x0), y1, y2)) → ACTIVE(U71(active(U31(mark(x0))), y1, y2))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(length(U92(x0, x1, x2, x3))) → ACTIVE(length(active(U92(mark(x0), x1, x2, x3))))
MARK(U71(U91(x0, x1, x2, x3), y1, y2)) → ACTIVE(U71(active(U91(mark(x0), x1, x2, x3)), y1, y2))
MARK(length(U71(x0, x1, x2))) → ACTIVE(length(active(U71(mark(x0), x1, x2))))
MARK(U71(U93(x0, x1, x2, x3), y1, y2)) → ACTIVE(U71(active(U93(mark(x0), x1, x2, x3)), y1, y2))
MARK(U71(isNatIList(x0), y1, y2)) → ACTIVE(U71(active(isNatIList(x0)), y1, y2))
MARK(U72(U11(x0), y1)) → ACTIVE(U72(active(U11(mark(x0))), y1))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(U72(isNat(x0), y1)) → ACTIVE(U72(active(isNat(x0)), y1))
MARK(U71(nil, y1, y2)) → ACTIVE(U71(active(nil), y1, y2))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
MARK(U72(zeros, y1)) → ACTIVE(U72(active(zeros), y1))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U71(isNatList(x0), y1, y2)) → ACTIVE(U71(active(isNatList(x0)), y1, y2))
MARK(U72(y0, active(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U71(U71(x0, x1, x2), y1, y2)) → ACTIVE(U71(active(U71(mark(x0), x1, x2)), y1, y2))
MARK(U72(U21(x0), y1)) → ACTIVE(U72(active(U21(mark(x0))), y1))
MARK(length(x0)) → ACTIVE(length(x0))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(length(nil)) → ACTIVE(length(active(nil))) at position [0] we obtained the following new rules:
MARK(length(nil)) → ACTIVE(length(nil))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(U72(nil, y1)) → ACTIVE(U72(active(nil), y1))
MARK(U72(U31(x0), y1)) → ACTIVE(U72(active(U31(mark(x0))), y1))
MARK(U71(isNat(x0), y1, y2)) → ACTIVE(U71(active(isNat(x0)), y1, y2))
MARK(length(U62(x0))) → ACTIVE(length(active(U62(mark(x0)))))
MARK(U71(y0, mark(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(length(x0), y1, y2)) → ACTIVE(U71(active(length(mark(x0))), y1, y2))
MARK(U72(U62(x0), y1)) → ACTIVE(U72(active(U62(mark(x0))), y1))
MARK(U71(U41(x0, x1), y1, y2)) → ACTIVE(U71(active(U41(mark(x0), x1)), y1, y2))
MARK(U72(length(x0), y1)) → ACTIVE(U72(active(length(mark(x0))), y1))
MARK(U72(0, y1)) → ACTIVE(U72(active(0), y1))
MARK(U71(y0, x1, mark(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(U21(x0), y1, y2)) → ACTIVE(U71(active(U21(mark(x0))), y1, y2))
MARK(U72(U71(x0, x1, x2), y1)) → ACTIVE(U72(active(U71(mark(x0), x1, x2)), y1))
MARK(U71(take(x0, x1), y1, y2)) → ACTIVE(U71(active(take(mark(x0), mark(x1))), y1, y2))
MARK(length(nil)) → ACTIVE(length(nil))
MARK(U71(y0, x1, active(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(U72(x0, x1), y1, y2)) → ACTIVE(U71(active(U72(mark(x0), x1)), y1, y2))
MARK(length(U41(x0, x1))) → ACTIVE(length(active(U41(mark(x0), x1))))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(U71(cons(x0, x1), y1, y2)) → ACTIVE(U71(active(cons(mark(x0), x1)), y1, y2))
MARK(length(U52(x0))) → ACTIVE(length(active(U52(mark(x0)))))
MARK(length(U61(x0, x1))) → ACTIVE(length(active(U61(mark(x0), x1))))
MARK(U72(U52(x0), y1)) → ACTIVE(U72(active(U52(mark(x0))), y1))
MARK(length(U93(x0, x1, x2, x3))) → ACTIVE(length(active(U93(mark(x0), x1, x2, x3))))
MARK(U71(U81(x0), y1, y2)) → ACTIVE(U71(active(U81(mark(x0))), y1, y2))
MARK(length(U31(x0))) → ACTIVE(length(active(U31(mark(x0)))))
MARK(U72(U81(x0), y1)) → ACTIVE(U72(active(U81(mark(x0))), y1))
MARK(length(U51(x0, x1))) → ACTIVE(length(active(U51(mark(x0), x1))))
MARK(U72(take(x0, x1), y1)) → ACTIVE(U72(active(take(mark(x0), mark(x1))), y1))
MARK(U71(U42(x0), y1, y2)) → ACTIVE(U71(active(U42(mark(x0))), y1, y2))
MARK(length(0)) → ACTIVE(length(active(0)))
MARK(length(U11(x0))) → ACTIVE(length(active(U11(mark(x0)))))
MARK(U71(U52(x0), y1, y2)) → ACTIVE(U71(active(U52(mark(x0))), y1, y2))
MARK(length(U21(x0))) → ACTIVE(length(active(U21(mark(x0)))))
MARK(U72(isNatList(x0), y1)) → ACTIVE(U72(active(isNatList(x0)), y1))
MARK(U72(s(x0), y1)) → ACTIVE(U72(active(s(mark(x0))), y1))
MARK(U71(tt, y1, y2)) → ACTIVE(U71(active(tt), y1, y2))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(U71(U51(x0, x1), y1, y2)) → ACTIVE(U71(active(U51(mark(x0), x1)), y1, y2))
MARK(U71(0, y1, y2)) → ACTIVE(U71(active(0), y1, y2))
MARK(U72(y0, mark(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U71(U92(x0, x1, x2, x3), y1, y2)) → ACTIVE(U71(active(U92(mark(x0), x1, x2, x3)), y1, y2))
MARK(U72(U72(x0, x1), y1)) → ACTIVE(U72(active(U72(mark(x0), x1)), y1))
MARK(length(U72(x0, x1))) → ACTIVE(length(active(U72(mark(x0), x1))))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(U71(s(x0), y1, y2)) → ACTIVE(U71(active(s(mark(x0))), y1, y2))
MARK(s(X)) → MARK(X)
MARK(length(U91(x0, x1, x2, x3))) → ACTIVE(length(active(U91(mark(x0), x1, x2, x3))))
MARK(length(U81(x0))) → ACTIVE(length(active(U81(mark(x0)))))
MARK(U71(U62(x0), y1, y2)) → ACTIVE(U71(active(U62(mark(x0))), y1, y2))
MARK(U71(U61(x0, x1), y1, y2)) → ACTIVE(U71(active(U61(mark(x0), x1)), y1, y2))
MARK(U71(x0, x1, x2)) → ACTIVE(U71(x0, x1, x2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U71(zeros, y1, y2)) → ACTIVE(U71(active(zeros), y1, y2))
MARK(U71(U11(x0), y1, y2)) → ACTIVE(U71(active(U11(mark(x0))), y1, y2))
MARK(U71(y0, active(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U72(U91(x0, x1, x2, x3), y1)) → ACTIVE(U72(active(U91(mark(x0), x1, x2, x3)), y1))
MARK(length(U42(x0))) → ACTIVE(length(active(U42(mark(x0)))))
MARK(U72(x0, x1)) → ACTIVE(U72(x0, x1))
MARK(U71(U31(x0), y1, y2)) → ACTIVE(U71(active(U31(mark(x0))), y1, y2))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(length(U92(x0, x1, x2, x3))) → ACTIVE(length(active(U92(mark(x0), x1, x2, x3))))
MARK(U71(U91(x0, x1, x2, x3), y1, y2)) → ACTIVE(U71(active(U91(mark(x0), x1, x2, x3)), y1, y2))
MARK(length(U71(x0, x1, x2))) → ACTIVE(length(active(U71(mark(x0), x1, x2))))
MARK(U71(isNatIList(x0), y1, y2)) → ACTIVE(U71(active(isNatIList(x0)), y1, y2))
MARK(U71(U93(x0, x1, x2, x3), y1, y2)) → ACTIVE(U71(active(U93(mark(x0), x1, x2, x3)), y1, y2))
MARK(U72(U11(x0), y1)) → ACTIVE(U72(active(U11(mark(x0))), y1))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(U72(isNat(x0), y1)) → ACTIVE(U72(active(isNat(x0)), y1))
MARK(U71(nil, y1, y2)) → ACTIVE(U71(active(nil), y1, y2))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U72(zeros, y1)) → ACTIVE(U72(active(zeros), y1))
MARK(U71(isNatList(x0), y1, y2)) → ACTIVE(U71(active(isNatList(x0)), y1, y2))
MARK(U72(y0, active(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U71(U71(x0, x1, x2), y1, y2)) → ACTIVE(U71(active(U71(mark(x0), x1, x2)), y1, y2))
MARK(U72(U21(x0), y1)) → ACTIVE(U72(active(U21(mark(x0))), y1))
MARK(length(x0)) → ACTIVE(length(x0))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(U71(isNat(x0), y1, y2)) → ACTIVE(U71(active(isNat(x0)), y1, y2))
MARK(U72(U31(x0), y1)) → ACTIVE(U72(active(U31(mark(x0))), y1))
MARK(U72(nil, y1)) → ACTIVE(U72(active(nil), y1))
MARK(length(U62(x0))) → ACTIVE(length(active(U62(mark(x0)))))
MARK(U71(y0, mark(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(length(x0), y1, y2)) → ACTIVE(U71(active(length(mark(x0))), y1, y2))
MARK(U71(U41(x0, x1), y1, y2)) → ACTIVE(U71(active(U41(mark(x0), x1)), y1, y2))
MARK(U72(U62(x0), y1)) → ACTIVE(U72(active(U62(mark(x0))), y1))
MARK(U72(0, y1)) → ACTIVE(U72(active(0), y1))
MARK(U72(length(x0), y1)) → ACTIVE(U72(active(length(mark(x0))), y1))
MARK(U71(y0, x1, mark(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(U21(x0), y1, y2)) → ACTIVE(U71(active(U21(mark(x0))), y1, y2))
MARK(U71(take(x0, x1), y1, y2)) → ACTIVE(U71(active(take(mark(x0), mark(x1))), y1, y2))
MARK(U72(U71(x0, x1, x2), y1)) → ACTIVE(U72(active(U71(mark(x0), x1, x2)), y1))
MARK(U71(y0, x1, active(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(length(U41(x0, x1))) → ACTIVE(length(active(U41(mark(x0), x1))))
MARK(U71(U72(x0, x1), y1, y2)) → ACTIVE(U71(active(U72(mark(x0), x1)), y1, y2))
MARK(length(U52(x0))) → ACTIVE(length(active(U52(mark(x0)))))
MARK(U71(cons(x0, x1), y1, y2)) → ACTIVE(U71(active(cons(mark(x0), x1)), y1, y2))
MARK(length(U61(x0, x1))) → ACTIVE(length(active(U61(mark(x0), x1))))
MARK(length(U93(x0, x1, x2, x3))) → ACTIVE(length(active(U93(mark(x0), x1, x2, x3))))
MARK(U72(U52(x0), y1)) → ACTIVE(U72(active(U52(mark(x0))), y1))
MARK(U71(U81(x0), y1, y2)) → ACTIVE(U71(active(U81(mark(x0))), y1, y2))
MARK(length(U31(x0))) → ACTIVE(length(active(U31(mark(x0)))))
MARK(length(U51(x0, x1))) → ACTIVE(length(active(U51(mark(x0), x1))))
MARK(U72(U81(x0), y1)) → ACTIVE(U72(active(U81(mark(x0))), y1))
MARK(U71(U42(x0), y1, y2)) → ACTIVE(U71(active(U42(mark(x0))), y1, y2))
MARK(U72(take(x0, x1), y1)) → ACTIVE(U72(active(take(mark(x0), mark(x1))), y1))
MARK(length(U11(x0))) → ACTIVE(length(active(U11(mark(x0)))))
MARK(length(0)) → ACTIVE(length(active(0)))
MARK(U71(U52(x0), y1, y2)) → ACTIVE(U71(active(U52(mark(x0))), y1, y2))
MARK(length(U21(x0))) → ACTIVE(length(active(U21(mark(x0)))))
MARK(U72(s(x0), y1)) → ACTIVE(U72(active(s(mark(x0))), y1))
MARK(U72(isNatList(x0), y1)) → ACTIVE(U72(active(isNatList(x0)), y1))
MARK(U71(tt, y1, y2)) → ACTIVE(U71(active(tt), y1, y2))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(U71(0, y1, y2)) → ACTIVE(U71(active(0), y1, y2))
MARK(U71(U51(x0, x1), y1, y2)) → ACTIVE(U71(active(U51(mark(x0), x1)), y1, y2))
MARK(U72(y0, mark(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U71(U92(x0, x1, x2, x3), y1, y2)) → ACTIVE(U71(active(U92(mark(x0), x1, x2, x3)), y1, y2))
MARK(U72(U72(x0, x1), y1)) → ACTIVE(U72(active(U72(mark(x0), x1)), y1))
MARK(length(U72(x0, x1))) → ACTIVE(length(active(U72(mark(x0), x1))))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
MARK(U71(s(x0), y1, y2)) → ACTIVE(U71(active(s(mark(x0))), y1, y2))
MARK(length(U91(x0, x1, x2, x3))) → ACTIVE(length(active(U91(mark(x0), x1, x2, x3))))
MARK(length(U81(x0))) → ACTIVE(length(active(U81(mark(x0)))))
MARK(U71(U61(x0, x1), y1, y2)) → ACTIVE(U71(active(U61(mark(x0), x1)), y1, y2))
MARK(U71(U62(x0), y1, y2)) → ACTIVE(U71(active(U62(mark(x0))), y1, y2))
MARK(U71(x0, x1, x2)) → ACTIVE(U71(x0, x1, x2))
MARK(U71(zeros, y1, y2)) → ACTIVE(U71(active(zeros), y1, y2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U71(y0, active(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(U11(x0), y1, y2)) → ACTIVE(U71(active(U11(mark(x0))), y1, y2))
MARK(length(U42(x0))) → ACTIVE(length(active(U42(mark(x0)))))
MARK(U72(U91(x0, x1, x2, x3), y1)) → ACTIVE(U72(active(U91(mark(x0), x1, x2, x3)), y1))
MARK(U72(x0, x1)) → ACTIVE(U72(x0, x1))
MARK(U71(U31(x0), y1, y2)) → ACTIVE(U71(active(U31(mark(x0))), y1, y2))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(length(U92(x0, x1, x2, x3))) → ACTIVE(length(active(U92(mark(x0), x1, x2, x3))))
MARK(U71(U91(x0, x1, x2, x3), y1, y2)) → ACTIVE(U71(active(U91(mark(x0), x1, x2, x3)), y1, y2))
MARK(length(U71(x0, x1, x2))) → ACTIVE(length(active(U71(mark(x0), x1, x2))))
MARK(U71(U93(x0, x1, x2, x3), y1, y2)) → ACTIVE(U71(active(U93(mark(x0), x1, x2, x3)), y1, y2))
MARK(U71(isNatIList(x0), y1, y2)) → ACTIVE(U71(active(isNatIList(x0)), y1, y2))
MARK(U72(U11(x0), y1)) → ACTIVE(U72(active(U11(mark(x0))), y1))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(U72(isNat(x0), y1)) → ACTIVE(U72(active(isNat(x0)), y1))
MARK(U71(nil, y1, y2)) → ACTIVE(U71(active(nil), y1, y2))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
MARK(U72(zeros, y1)) → ACTIVE(U72(active(zeros), y1))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U71(isNatList(x0), y1, y2)) → ACTIVE(U71(active(isNatList(x0)), y1, y2))
MARK(U72(y0, active(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U71(U71(x0, x1, x2), y1, y2)) → ACTIVE(U71(active(U71(mark(x0), x1, x2)), y1, y2))
MARK(U72(U21(x0), y1)) → ACTIVE(U72(active(U21(mark(x0))), y1))
MARK(length(x0)) → ACTIVE(length(x0))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(length(0)) → ACTIVE(length(active(0))) at position [0] we obtained the following new rules:
MARK(length(0)) → ACTIVE(length(0))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
MARK(length(0)) → ACTIVE(length(0))
MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(U72(nil, y1)) → ACTIVE(U72(active(nil), y1))
MARK(U72(U31(x0), y1)) → ACTIVE(U72(active(U31(mark(x0))), y1))
MARK(U71(isNat(x0), y1, y2)) → ACTIVE(U71(active(isNat(x0)), y1, y2))
MARK(length(U62(x0))) → ACTIVE(length(active(U62(mark(x0)))))
MARK(U71(y0, mark(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(length(x0), y1, y2)) → ACTIVE(U71(active(length(mark(x0))), y1, y2))
MARK(U72(U62(x0), y1)) → ACTIVE(U72(active(U62(mark(x0))), y1))
MARK(U71(U41(x0, x1), y1, y2)) → ACTIVE(U71(active(U41(mark(x0), x1)), y1, y2))
MARK(U72(length(x0), y1)) → ACTIVE(U72(active(length(mark(x0))), y1))
MARK(U72(0, y1)) → ACTIVE(U72(active(0), y1))
MARK(U71(y0, x1, mark(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(U21(x0), y1, y2)) → ACTIVE(U71(active(U21(mark(x0))), y1, y2))
MARK(U72(U71(x0, x1, x2), y1)) → ACTIVE(U72(active(U71(mark(x0), x1, x2)), y1))
MARK(U71(take(x0, x1), y1, y2)) → ACTIVE(U71(active(take(mark(x0), mark(x1))), y1, y2))
MARK(U71(y0, x1, active(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(U72(x0, x1), y1, y2)) → ACTIVE(U71(active(U72(mark(x0), x1)), y1, y2))
MARK(length(U41(x0, x1))) → ACTIVE(length(active(U41(mark(x0), x1))))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(U71(cons(x0, x1), y1, y2)) → ACTIVE(U71(active(cons(mark(x0), x1)), y1, y2))
MARK(length(U52(x0))) → ACTIVE(length(active(U52(mark(x0)))))
MARK(length(U61(x0, x1))) → ACTIVE(length(active(U61(mark(x0), x1))))
MARK(U72(U52(x0), y1)) → ACTIVE(U72(active(U52(mark(x0))), y1))
MARK(length(U93(x0, x1, x2, x3))) → ACTIVE(length(active(U93(mark(x0), x1, x2, x3))))
MARK(U71(U81(x0), y1, y2)) → ACTIVE(U71(active(U81(mark(x0))), y1, y2))
MARK(length(U31(x0))) → ACTIVE(length(active(U31(mark(x0)))))
MARK(U72(U81(x0), y1)) → ACTIVE(U72(active(U81(mark(x0))), y1))
MARK(length(U51(x0, x1))) → ACTIVE(length(active(U51(mark(x0), x1))))
MARK(U72(take(x0, x1), y1)) → ACTIVE(U72(active(take(mark(x0), mark(x1))), y1))
MARK(U71(U42(x0), y1, y2)) → ACTIVE(U71(active(U42(mark(x0))), y1, y2))
MARK(length(U11(x0))) → ACTIVE(length(active(U11(mark(x0)))))
MARK(U71(U52(x0), y1, y2)) → ACTIVE(U71(active(U52(mark(x0))), y1, y2))
MARK(length(U21(x0))) → ACTIVE(length(active(U21(mark(x0)))))
MARK(U72(isNatList(x0), y1)) → ACTIVE(U72(active(isNatList(x0)), y1))
MARK(U72(s(x0), y1)) → ACTIVE(U72(active(s(mark(x0))), y1))
MARK(U71(tt, y1, y2)) → ACTIVE(U71(active(tt), y1, y2))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(U71(U51(x0, x1), y1, y2)) → ACTIVE(U71(active(U51(mark(x0), x1)), y1, y2))
MARK(U71(0, y1, y2)) → ACTIVE(U71(active(0), y1, y2))
MARK(U72(y0, mark(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U71(U92(x0, x1, x2, x3), y1, y2)) → ACTIVE(U71(active(U92(mark(x0), x1, x2, x3)), y1, y2))
MARK(U72(U72(x0, x1), y1)) → ACTIVE(U72(active(U72(mark(x0), x1)), y1))
MARK(length(U72(x0, x1))) → ACTIVE(length(active(U72(mark(x0), x1))))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(U71(s(x0), y1, y2)) → ACTIVE(U71(active(s(mark(x0))), y1, y2))
MARK(s(X)) → MARK(X)
MARK(length(U91(x0, x1, x2, x3))) → ACTIVE(length(active(U91(mark(x0), x1, x2, x3))))
MARK(length(U81(x0))) → ACTIVE(length(active(U81(mark(x0)))))
MARK(U71(U62(x0), y1, y2)) → ACTIVE(U71(active(U62(mark(x0))), y1, y2))
MARK(U71(U61(x0, x1), y1, y2)) → ACTIVE(U71(active(U61(mark(x0), x1)), y1, y2))
MARK(U71(x0, x1, x2)) → ACTIVE(U71(x0, x1, x2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U71(zeros, y1, y2)) → ACTIVE(U71(active(zeros), y1, y2))
MARK(U71(U11(x0), y1, y2)) → ACTIVE(U71(active(U11(mark(x0))), y1, y2))
MARK(U71(y0, active(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U72(U91(x0, x1, x2, x3), y1)) → ACTIVE(U72(active(U91(mark(x0), x1, x2, x3)), y1))
MARK(length(U42(x0))) → ACTIVE(length(active(U42(mark(x0)))))
MARK(U72(x0, x1)) → ACTIVE(U72(x0, x1))
MARK(U71(U31(x0), y1, y2)) → ACTIVE(U71(active(U31(mark(x0))), y1, y2))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(length(U92(x0, x1, x2, x3))) → ACTIVE(length(active(U92(mark(x0), x1, x2, x3))))
MARK(U71(U91(x0, x1, x2, x3), y1, y2)) → ACTIVE(U71(active(U91(mark(x0), x1, x2, x3)), y1, y2))
MARK(length(U71(x0, x1, x2))) → ACTIVE(length(active(U71(mark(x0), x1, x2))))
MARK(U71(isNatIList(x0), y1, y2)) → ACTIVE(U71(active(isNatIList(x0)), y1, y2))
MARK(U71(U93(x0, x1, x2, x3), y1, y2)) → ACTIVE(U71(active(U93(mark(x0), x1, x2, x3)), y1, y2))
MARK(U72(U11(x0), y1)) → ACTIVE(U72(active(U11(mark(x0))), y1))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(U72(isNat(x0), y1)) → ACTIVE(U72(active(isNat(x0)), y1))
MARK(U71(nil, y1, y2)) → ACTIVE(U71(active(nil), y1, y2))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U72(zeros, y1)) → ACTIVE(U72(active(zeros), y1))
MARK(U71(isNatList(x0), y1, y2)) → ACTIVE(U71(active(isNatList(x0)), y1, y2))
MARK(U72(y0, active(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U71(U71(x0, x1, x2), y1, y2)) → ACTIVE(U71(active(U71(mark(x0), x1, x2)), y1, y2))
MARK(U72(U21(x0), y1)) → ACTIVE(U72(active(U21(mark(x0))), y1))
MARK(length(x0)) → ACTIVE(length(x0))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(U71(isNat(x0), y1, y2)) → ACTIVE(U71(active(isNat(x0)), y1, y2))
MARK(U72(U31(x0), y1)) → ACTIVE(U72(active(U31(mark(x0))), y1))
MARK(U72(nil, y1)) → ACTIVE(U72(active(nil), y1))
MARK(length(U62(x0))) → ACTIVE(length(active(U62(mark(x0)))))
MARK(U71(y0, mark(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(length(x0), y1, y2)) → ACTIVE(U71(active(length(mark(x0))), y1, y2))
MARK(U71(U41(x0, x1), y1, y2)) → ACTIVE(U71(active(U41(mark(x0), x1)), y1, y2))
MARK(U72(U62(x0), y1)) → ACTIVE(U72(active(U62(mark(x0))), y1))
MARK(U72(0, y1)) → ACTIVE(U72(active(0), y1))
MARK(U72(length(x0), y1)) → ACTIVE(U72(active(length(mark(x0))), y1))
MARK(U71(y0, x1, mark(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(U21(x0), y1, y2)) → ACTIVE(U71(active(U21(mark(x0))), y1, y2))
MARK(U71(take(x0, x1), y1, y2)) → ACTIVE(U71(active(take(mark(x0), mark(x1))), y1, y2))
MARK(U72(U71(x0, x1, x2), y1)) → ACTIVE(U72(active(U71(mark(x0), x1, x2)), y1))
MARK(U71(y0, x1, active(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(length(U41(x0, x1))) → ACTIVE(length(active(U41(mark(x0), x1))))
MARK(U71(U72(x0, x1), y1, y2)) → ACTIVE(U71(active(U72(mark(x0), x1)), y1, y2))
MARK(length(U52(x0))) → ACTIVE(length(active(U52(mark(x0)))))
MARK(U71(cons(x0, x1), y1, y2)) → ACTIVE(U71(active(cons(mark(x0), x1)), y1, y2))
MARK(length(U61(x0, x1))) → ACTIVE(length(active(U61(mark(x0), x1))))
MARK(length(U93(x0, x1, x2, x3))) → ACTIVE(length(active(U93(mark(x0), x1, x2, x3))))
MARK(U72(U52(x0), y1)) → ACTIVE(U72(active(U52(mark(x0))), y1))
MARK(U71(U81(x0), y1, y2)) → ACTIVE(U71(active(U81(mark(x0))), y1, y2))
MARK(length(U31(x0))) → ACTIVE(length(active(U31(mark(x0)))))
MARK(length(U51(x0, x1))) → ACTIVE(length(active(U51(mark(x0), x1))))
MARK(U72(U81(x0), y1)) → ACTIVE(U72(active(U81(mark(x0))), y1))
MARK(U71(U42(x0), y1, y2)) → ACTIVE(U71(active(U42(mark(x0))), y1, y2))
MARK(U72(take(x0, x1), y1)) → ACTIVE(U72(active(take(mark(x0), mark(x1))), y1))
MARK(length(U11(x0))) → ACTIVE(length(active(U11(mark(x0)))))
MARK(U71(U52(x0), y1, y2)) → ACTIVE(U71(active(U52(mark(x0))), y1, y2))
MARK(length(U21(x0))) → ACTIVE(length(active(U21(mark(x0)))))
MARK(U72(s(x0), y1)) → ACTIVE(U72(active(s(mark(x0))), y1))
MARK(U72(isNatList(x0), y1)) → ACTIVE(U72(active(isNatList(x0)), y1))
MARK(U71(tt, y1, y2)) → ACTIVE(U71(active(tt), y1, y2))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(U71(0, y1, y2)) → ACTIVE(U71(active(0), y1, y2))
MARK(U71(U51(x0, x1), y1, y2)) → ACTIVE(U71(active(U51(mark(x0), x1)), y1, y2))
MARK(U72(y0, mark(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U71(U92(x0, x1, x2, x3), y1, y2)) → ACTIVE(U71(active(U92(mark(x0), x1, x2, x3)), y1, y2))
MARK(U72(U72(x0, x1), y1)) → ACTIVE(U72(active(U72(mark(x0), x1)), y1))
MARK(length(U72(x0, x1))) → ACTIVE(length(active(U72(mark(x0), x1))))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
MARK(U71(s(x0), y1, y2)) → ACTIVE(U71(active(s(mark(x0))), y1, y2))
MARK(length(U91(x0, x1, x2, x3))) → ACTIVE(length(active(U91(mark(x0), x1, x2, x3))))
MARK(length(U81(x0))) → ACTIVE(length(active(U81(mark(x0)))))
MARK(U71(U61(x0, x1), y1, y2)) → ACTIVE(U71(active(U61(mark(x0), x1)), y1, y2))
MARK(U71(U62(x0), y1, y2)) → ACTIVE(U71(active(U62(mark(x0))), y1, y2))
MARK(U71(x0, x1, x2)) → ACTIVE(U71(x0, x1, x2))
MARK(U71(zeros, y1, y2)) → ACTIVE(U71(active(zeros), y1, y2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U71(y0, active(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(U11(x0), y1, y2)) → ACTIVE(U71(active(U11(mark(x0))), y1, y2))
MARK(length(U42(x0))) → ACTIVE(length(active(U42(mark(x0)))))
MARK(U72(U91(x0, x1, x2, x3), y1)) → ACTIVE(U72(active(U91(mark(x0), x1, x2, x3)), y1))
MARK(U72(x0, x1)) → ACTIVE(U72(x0, x1))
MARK(U71(U31(x0), y1, y2)) → ACTIVE(U71(active(U31(mark(x0))), y1, y2))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(length(U92(x0, x1, x2, x3))) → ACTIVE(length(active(U92(mark(x0), x1, x2, x3))))
MARK(U71(U91(x0, x1, x2, x3), y1, y2)) → ACTIVE(U71(active(U91(mark(x0), x1, x2, x3)), y1, y2))
MARK(length(U71(x0, x1, x2))) → ACTIVE(length(active(U71(mark(x0), x1, x2))))
MARK(U71(U93(x0, x1, x2, x3), y1, y2)) → ACTIVE(U71(active(U93(mark(x0), x1, x2, x3)), y1, y2))
MARK(U71(isNatIList(x0), y1, y2)) → ACTIVE(U71(active(isNatIList(x0)), y1, y2))
MARK(U72(U11(x0), y1)) → ACTIVE(U72(active(U11(mark(x0))), y1))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(U72(isNat(x0), y1)) → ACTIVE(U72(active(isNat(x0)), y1))
MARK(U71(nil, y1, y2)) → ACTIVE(U71(active(nil), y1, y2))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
MARK(U72(zeros, y1)) → ACTIVE(U72(active(zeros), y1))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U71(isNatList(x0), y1, y2)) → ACTIVE(U71(active(isNatList(x0)), y1, y2))
MARK(U72(y0, active(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U71(U71(x0, x1, x2), y1, y2)) → ACTIVE(U71(active(U71(mark(x0), x1, x2)), y1, y2))
MARK(U72(U21(x0), y1)) → ACTIVE(U72(active(U21(mark(x0))), y1))
MARK(length(x0)) → ACTIVE(length(x0))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
MARK(U71(isNat(x0), y1, y2)) → ACTIVE(U71(active(isNat(x0)), y1, y2))
MARK(U71(U41(x0, x1), y1, y2)) → ACTIVE(U71(active(U41(mark(x0), x1)), y1, y2))
MARK(U71(U21(x0), y1, y2)) → ACTIVE(U71(active(U21(mark(x0))), y1, y2))
MARK(U71(take(x0, x1), y1, y2)) → ACTIVE(U71(active(take(mark(x0), mark(x1))), y1, y2))
MARK(U71(cons(x0, x1), y1, y2)) → ACTIVE(U71(active(cons(mark(x0), x1)), y1, y2))
MARK(U71(U81(x0), y1, y2)) → ACTIVE(U71(active(U81(mark(x0))), y1, y2))
MARK(U71(U42(x0), y1, y2)) → ACTIVE(U71(active(U42(mark(x0))), y1, y2))
MARK(U71(U52(x0), y1, y2)) → ACTIVE(U71(active(U52(mark(x0))), y1, y2))
MARK(U71(tt, y1, y2)) → ACTIVE(U71(active(tt), y1, y2))
MARK(U71(0, y1, y2)) → ACTIVE(U71(active(0), y1, y2))
MARK(U71(U51(x0, x1), y1, y2)) → ACTIVE(U71(active(U51(mark(x0), x1)), y1, y2))
MARK(U71(U92(x0, x1, x2, x3), y1, y2)) → ACTIVE(U71(active(U92(mark(x0), x1, x2, x3)), y1, y2))
MARK(U71(U61(x0, x1), y1, y2)) → ACTIVE(U71(active(U61(mark(x0), x1)), y1, y2))
MARK(U71(U62(x0), y1, y2)) → ACTIVE(U71(active(U62(mark(x0))), y1, y2))
MARK(U71(zeros, y1, y2)) → ACTIVE(U71(active(zeros), y1, y2))
MARK(U71(U11(x0), y1, y2)) → ACTIVE(U71(active(U11(mark(x0))), y1, y2))
MARK(U71(U31(x0), y1, y2)) → ACTIVE(U71(active(U31(mark(x0))), y1, y2))
MARK(U71(U91(x0, x1, x2, x3), y1, y2)) → ACTIVE(U71(active(U91(mark(x0), x1, x2, x3)), y1, y2))
MARK(U71(U93(x0, x1, x2, x3), y1, y2)) → ACTIVE(U71(active(U93(mark(x0), x1, x2, x3)), y1, y2))
MARK(U71(isNatIList(x0), y1, y2)) → ACTIVE(U71(active(isNatIList(x0)), y1, y2))
MARK(U71(nil, y1, y2)) → ACTIVE(U71(active(nil), y1, y2))
The remaining pairs can at least be oriented weakly.
MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(U72(U31(x0), y1)) → ACTIVE(U72(active(U31(mark(x0))), y1))
MARK(U72(nil, y1)) → ACTIVE(U72(active(nil), y1))
MARK(length(U62(x0))) → ACTIVE(length(active(U62(mark(x0)))))
MARK(U71(y0, mark(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(length(x0), y1, y2)) → ACTIVE(U71(active(length(mark(x0))), y1, y2))
MARK(U72(U62(x0), y1)) → ACTIVE(U72(active(U62(mark(x0))), y1))
MARK(U72(0, y1)) → ACTIVE(U72(active(0), y1))
MARK(U72(length(x0), y1)) → ACTIVE(U72(active(length(mark(x0))), y1))
MARK(U71(y0, x1, mark(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U72(U71(x0, x1, x2), y1)) → ACTIVE(U72(active(U71(mark(x0), x1, x2)), y1))
MARK(U71(y0, x1, active(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(length(U41(x0, x1))) → ACTIVE(length(active(U41(mark(x0), x1))))
MARK(U71(U72(x0, x1), y1, y2)) → ACTIVE(U71(active(U72(mark(x0), x1)), y1, y2))
MARK(length(U52(x0))) → ACTIVE(length(active(U52(mark(x0)))))
MARK(length(U61(x0, x1))) → ACTIVE(length(active(U61(mark(x0), x1))))
MARK(length(U93(x0, x1, x2, x3))) → ACTIVE(length(active(U93(mark(x0), x1, x2, x3))))
MARK(U72(U52(x0), y1)) → ACTIVE(U72(active(U52(mark(x0))), y1))
MARK(length(U31(x0))) → ACTIVE(length(active(U31(mark(x0)))))
MARK(length(U51(x0, x1))) → ACTIVE(length(active(U51(mark(x0), x1))))
MARK(U72(U81(x0), y1)) → ACTIVE(U72(active(U81(mark(x0))), y1))
MARK(U72(take(x0, x1), y1)) → ACTIVE(U72(active(take(mark(x0), mark(x1))), y1))
MARK(length(U11(x0))) → ACTIVE(length(active(U11(mark(x0)))))
MARK(length(U21(x0))) → ACTIVE(length(active(U21(mark(x0)))))
MARK(U72(s(x0), y1)) → ACTIVE(U72(active(s(mark(x0))), y1))
MARK(U72(isNatList(x0), y1)) → ACTIVE(U72(active(isNatList(x0)), y1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(U72(y0, mark(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U72(U72(x0, x1), y1)) → ACTIVE(U72(active(U72(mark(x0), x1)), y1))
MARK(length(U72(x0, x1))) → ACTIVE(length(active(U72(mark(x0), x1))))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
MARK(U71(s(x0), y1, y2)) → ACTIVE(U71(active(s(mark(x0))), y1, y2))
MARK(length(U91(x0, x1, x2, x3))) → ACTIVE(length(active(U91(mark(x0), x1, x2, x3))))
MARK(length(U81(x0))) → ACTIVE(length(active(U81(mark(x0)))))
MARK(U71(x0, x1, x2)) → ACTIVE(U71(x0, x1, x2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U71(y0, active(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(length(U42(x0))) → ACTIVE(length(active(U42(mark(x0)))))
MARK(U72(U91(x0, x1, x2, x3), y1)) → ACTIVE(U72(active(U91(mark(x0), x1, x2, x3)), y1))
MARK(U72(x0, x1)) → ACTIVE(U72(x0, x1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(length(U92(x0, x1, x2, x3))) → ACTIVE(length(active(U92(mark(x0), x1, x2, x3))))
MARK(length(U71(x0, x1, x2))) → ACTIVE(length(active(U71(mark(x0), x1, x2))))
MARK(U72(U11(x0), y1)) → ACTIVE(U72(active(U11(mark(x0))), y1))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(U72(isNat(x0), y1)) → ACTIVE(U72(active(isNat(x0)), y1))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
MARK(U72(zeros, y1)) → ACTIVE(U72(active(zeros), y1))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U71(isNatList(x0), y1, y2)) → ACTIVE(U71(active(isNatList(x0)), y1, y2))
MARK(U72(y0, active(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U71(U71(x0, x1, x2), y1, y2)) → ACTIVE(U71(active(U71(mark(x0), x1, x2)), y1, y2))
MARK(U72(U21(x0), y1)) → ACTIVE(U72(active(U21(mark(x0))), y1))
MARK(length(x0)) → ACTIVE(length(x0))
Used ordering: Polynomial interpretation [25]:
POL(0) = 1
POL(ACTIVE(x1)) = 0
POL(MARK(x1)) = x1
POL(U11(x1)) = 1
POL(U21(x1)) = 1 + x1
POL(U31(x1)) = 1
POL(U41(x1, x2)) = 1
POL(U42(x1)) = 1
POL(U51(x1, x2)) = 1
POL(U52(x1)) = 1
POL(U61(x1, x2)) = 1 + x2
POL(U62(x1)) = 1
POL(U71(x1, x2, x3)) = x1
POL(U72(x1, x2)) = 0
POL(U81(x1)) = 1 + x1
POL(U91(x1, x2, x3, x4)) = 1
POL(U92(x1, x2, x3, x4)) = 1 + x1 + x2 + x3 + x4
POL(U93(x1, x2, x3, x4)) = 1 + x2 + x4
POL(active(x1)) = x1
POL(cons(x1, x2)) = 1 + x2
POL(isNat(x1)) = 1 + x1
POL(isNatIList(x1)) = 1
POL(isNatList(x1)) = 0
POL(length(x1)) = 0
POL(mark(x1)) = x1
POL(nil) = 1
POL(s(x1)) = x1
POL(take(x1, x2)) = 1 + x2
POL(tt) = 1
POL(zeros) = 1
The following usable rules [17] were oriented:
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(U72(U31(x0), y1)) → ACTIVE(U72(active(U31(mark(x0))), y1))
MARK(U72(nil, y1)) → ACTIVE(U72(active(nil), y1))
MARK(length(U62(x0))) → ACTIVE(length(active(U62(mark(x0)))))
MARK(U71(y0, mark(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(length(x0), y1, y2)) → ACTIVE(U71(active(length(mark(x0))), y1, y2))
MARK(U72(U62(x0), y1)) → ACTIVE(U72(active(U62(mark(x0))), y1))
MARK(U72(0, y1)) → ACTIVE(U72(active(0), y1))
MARK(U72(length(x0), y1)) → ACTIVE(U72(active(length(mark(x0))), y1))
MARK(U71(y0, x1, mark(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U72(U71(x0, x1, x2), y1)) → ACTIVE(U72(active(U71(mark(x0), x1, x2)), y1))
MARK(U71(y0, x1, active(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(U72(x0, x1), y1, y2)) → ACTIVE(U71(active(U72(mark(x0), x1)), y1, y2))
MARK(length(U41(x0, x1))) → ACTIVE(length(active(U41(mark(x0), x1))))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(length(U52(x0))) → ACTIVE(length(active(U52(mark(x0)))))
MARK(length(U61(x0, x1))) → ACTIVE(length(active(U61(mark(x0), x1))))
MARK(U72(U52(x0), y1)) → ACTIVE(U72(active(U52(mark(x0))), y1))
MARK(length(U93(x0, x1, x2, x3))) → ACTIVE(length(active(U93(mark(x0), x1, x2, x3))))
MARK(length(U31(x0))) → ACTIVE(length(active(U31(mark(x0)))))
MARK(U72(U81(x0), y1)) → ACTIVE(U72(active(U81(mark(x0))), y1))
MARK(length(U51(x0, x1))) → ACTIVE(length(active(U51(mark(x0), x1))))
MARK(U72(take(x0, x1), y1)) → ACTIVE(U72(active(take(mark(x0), mark(x1))), y1))
MARK(length(U11(x0))) → ACTIVE(length(active(U11(mark(x0)))))
MARK(length(U21(x0))) → ACTIVE(length(active(U21(mark(x0)))))
MARK(U72(s(x0), y1)) → ACTIVE(U72(active(s(mark(x0))), y1))
MARK(U72(isNatList(x0), y1)) → ACTIVE(U72(active(isNatList(x0)), y1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(U72(y0, mark(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U72(U72(x0, x1), y1)) → ACTIVE(U72(active(U72(mark(x0), x1)), y1))
MARK(length(U72(x0, x1))) → ACTIVE(length(active(U72(mark(x0), x1))))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
MARK(U71(s(x0), y1, y2)) → ACTIVE(U71(active(s(mark(x0))), y1, y2))
MARK(length(U91(x0, x1, x2, x3))) → ACTIVE(length(active(U91(mark(x0), x1, x2, x3))))
MARK(length(U81(x0))) → ACTIVE(length(active(U81(mark(x0)))))
MARK(U71(x0, x1, x2)) → ACTIVE(U71(x0, x1, x2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U71(y0, active(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(length(U42(x0))) → ACTIVE(length(active(U42(mark(x0)))))
MARK(U72(U91(x0, x1, x2, x3), y1)) → ACTIVE(U72(active(U91(mark(x0), x1, x2, x3)), y1))
MARK(U72(x0, x1)) → ACTIVE(U72(x0, x1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(length(U92(x0, x1, x2, x3))) → ACTIVE(length(active(U92(mark(x0), x1, x2, x3))))
MARK(length(U71(x0, x1, x2))) → ACTIVE(length(active(U71(mark(x0), x1, x2))))
MARK(U72(U11(x0), y1)) → ACTIVE(U72(active(U11(mark(x0))), y1))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(U72(isNat(x0), y1)) → ACTIVE(U72(active(isNat(x0)), y1))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U72(zeros, y1)) → ACTIVE(U72(active(zeros), y1))
MARK(U71(isNatList(x0), y1, y2)) → ACTIVE(U71(active(isNatList(x0)), y1, y2))
MARK(U72(y0, active(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U71(U71(x0, x1, x2), y1, y2)) → ACTIVE(U71(active(U71(mark(x0), x1, x2)), y1, y2))
MARK(U72(U21(x0), y1)) → ACTIVE(U72(active(U21(mark(x0))), y1))
MARK(length(x0)) → ACTIVE(length(x0))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
MARK(U72(U62(x0), y1)) → ACTIVE(U72(active(U62(mark(x0))), y1))
MARK(U72(U81(x0), y1)) → ACTIVE(U72(active(U81(mark(x0))), y1))
MARK(U72(take(x0, x1), y1)) → ACTIVE(U72(active(take(mark(x0), mark(x1))), y1))
The remaining pairs can at least be oriented weakly.
MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(U72(U31(x0), y1)) → ACTIVE(U72(active(U31(mark(x0))), y1))
MARK(U72(nil, y1)) → ACTIVE(U72(active(nil), y1))
MARK(length(U62(x0))) → ACTIVE(length(active(U62(mark(x0)))))
MARK(U71(y0, mark(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(length(x0), y1, y2)) → ACTIVE(U71(active(length(mark(x0))), y1, y2))
MARK(U72(0, y1)) → ACTIVE(U72(active(0), y1))
MARK(U72(length(x0), y1)) → ACTIVE(U72(active(length(mark(x0))), y1))
MARK(U71(y0, x1, mark(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U72(U71(x0, x1, x2), y1)) → ACTIVE(U72(active(U71(mark(x0), x1, x2)), y1))
MARK(U71(y0, x1, active(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(U72(x0, x1), y1, y2)) → ACTIVE(U71(active(U72(mark(x0), x1)), y1, y2))
MARK(length(U41(x0, x1))) → ACTIVE(length(active(U41(mark(x0), x1))))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(length(U52(x0))) → ACTIVE(length(active(U52(mark(x0)))))
MARK(length(U61(x0, x1))) → ACTIVE(length(active(U61(mark(x0), x1))))
MARK(U72(U52(x0), y1)) → ACTIVE(U72(active(U52(mark(x0))), y1))
MARK(length(U93(x0, x1, x2, x3))) → ACTIVE(length(active(U93(mark(x0), x1, x2, x3))))
MARK(length(U31(x0))) → ACTIVE(length(active(U31(mark(x0)))))
MARK(length(U51(x0, x1))) → ACTIVE(length(active(U51(mark(x0), x1))))
MARK(length(U11(x0))) → ACTIVE(length(active(U11(mark(x0)))))
MARK(length(U21(x0))) → ACTIVE(length(active(U21(mark(x0)))))
MARK(U72(s(x0), y1)) → ACTIVE(U72(active(s(mark(x0))), y1))
MARK(U72(isNatList(x0), y1)) → ACTIVE(U72(active(isNatList(x0)), y1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(U72(y0, mark(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U72(U72(x0, x1), y1)) → ACTIVE(U72(active(U72(mark(x0), x1)), y1))
MARK(length(U72(x0, x1))) → ACTIVE(length(active(U72(mark(x0), x1))))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
MARK(U71(s(x0), y1, y2)) → ACTIVE(U71(active(s(mark(x0))), y1, y2))
MARK(length(U91(x0, x1, x2, x3))) → ACTIVE(length(active(U91(mark(x0), x1, x2, x3))))
MARK(length(U81(x0))) → ACTIVE(length(active(U81(mark(x0)))))
MARK(U71(x0, x1, x2)) → ACTIVE(U71(x0, x1, x2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U71(y0, active(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(length(U42(x0))) → ACTIVE(length(active(U42(mark(x0)))))
MARK(U72(U91(x0, x1, x2, x3), y1)) → ACTIVE(U72(active(U91(mark(x0), x1, x2, x3)), y1))
MARK(U72(x0, x1)) → ACTIVE(U72(x0, x1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(length(U92(x0, x1, x2, x3))) → ACTIVE(length(active(U92(mark(x0), x1, x2, x3))))
MARK(length(U71(x0, x1, x2))) → ACTIVE(length(active(U71(mark(x0), x1, x2))))
MARK(U72(U11(x0), y1)) → ACTIVE(U72(active(U11(mark(x0))), y1))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(U72(isNat(x0), y1)) → ACTIVE(U72(active(isNat(x0)), y1))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U72(zeros, y1)) → ACTIVE(U72(active(zeros), y1))
MARK(U71(isNatList(x0), y1, y2)) → ACTIVE(U71(active(isNatList(x0)), y1, y2))
MARK(U72(y0, active(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U71(U71(x0, x1, x2), y1, y2)) → ACTIVE(U71(active(U71(mark(x0), x1, x2)), y1, y2))
MARK(U72(U21(x0), y1)) → ACTIVE(U72(active(U21(mark(x0))), y1))
MARK(length(x0)) → ACTIVE(length(x0))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVE(x1)) = 0
POL(MARK(x1)) = x1
POL(U11(x1)) = x1
POL(U21(x1)) = x1
POL(U31(x1)) = 0
POL(U41(x1, x2)) = 0
POL(U42(x1)) = 0
POL(U51(x1, x2)) = 0
POL(U52(x1)) = 0
POL(U61(x1, x2)) = 0
POL(U62(x1)) = 1 + x1
POL(U71(x1, x2, x3)) = 0
POL(U72(x1, x2)) = x1
POL(U81(x1)) = 1
POL(U91(x1, x2, x3, x4)) = 0
POL(U92(x1, x2, x3, x4)) = 0
POL(U93(x1, x2, x3, x4)) = 0
POL(active(x1)) = x1
POL(cons(x1, x2)) = 0
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 0
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = 1 + x2
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(U72(U31(x0), y1)) → ACTIVE(U72(active(U31(mark(x0))), y1))
MARK(U72(nil, y1)) → ACTIVE(U72(active(nil), y1))
MARK(length(U62(x0))) → ACTIVE(length(active(U62(mark(x0)))))
MARK(U71(y0, mark(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(length(x0), y1, y2)) → ACTIVE(U71(active(length(mark(x0))), y1, y2))
MARK(U72(0, y1)) → ACTIVE(U72(active(0), y1))
MARK(U72(length(x0), y1)) → ACTIVE(U72(active(length(mark(x0))), y1))
MARK(U71(y0, x1, mark(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U72(U71(x0, x1, x2), y1)) → ACTIVE(U72(active(U71(mark(x0), x1, x2)), y1))
MARK(U71(y0, x1, active(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(length(U41(x0, x1))) → ACTIVE(length(active(U41(mark(x0), x1))))
MARK(U71(U72(x0, x1), y1, y2)) → ACTIVE(U71(active(U72(mark(x0), x1)), y1, y2))
MARK(length(U52(x0))) → ACTIVE(length(active(U52(mark(x0)))))
MARK(length(U61(x0, x1))) → ACTIVE(length(active(U61(mark(x0), x1))))
MARK(length(U93(x0, x1, x2, x3))) → ACTIVE(length(active(U93(mark(x0), x1, x2, x3))))
MARK(U72(U52(x0), y1)) → ACTIVE(U72(active(U52(mark(x0))), y1))
MARK(length(U31(x0))) → ACTIVE(length(active(U31(mark(x0)))))
MARK(length(U51(x0, x1))) → ACTIVE(length(active(U51(mark(x0), x1))))
MARK(length(U11(x0))) → ACTIVE(length(active(U11(mark(x0)))))
MARK(length(U21(x0))) → ACTIVE(length(active(U21(mark(x0)))))
MARK(U72(isNatList(x0), y1)) → ACTIVE(U72(active(isNatList(x0)), y1))
MARK(U72(s(x0), y1)) → ACTIVE(U72(active(s(mark(x0))), y1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(U72(y0, mark(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U72(U72(x0, x1), y1)) → ACTIVE(U72(active(U72(mark(x0), x1)), y1))
MARK(length(U72(x0, x1))) → ACTIVE(length(active(U72(mark(x0), x1))))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
MARK(U71(s(x0), y1, y2)) → ACTIVE(U71(active(s(mark(x0))), y1, y2))
MARK(length(U91(x0, x1, x2, x3))) → ACTIVE(length(active(U91(mark(x0), x1, x2, x3))))
MARK(length(U81(x0))) → ACTIVE(length(active(U81(mark(x0)))))
MARK(U71(x0, x1, x2)) → ACTIVE(U71(x0, x1, x2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U71(y0, active(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(length(U42(x0))) → ACTIVE(length(active(U42(mark(x0)))))
MARK(U72(U91(x0, x1, x2, x3), y1)) → ACTIVE(U72(active(U91(mark(x0), x1, x2, x3)), y1))
MARK(U72(x0, x1)) → ACTIVE(U72(x0, x1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(length(U92(x0, x1, x2, x3))) → ACTIVE(length(active(U92(mark(x0), x1, x2, x3))))
MARK(length(U71(x0, x1, x2))) → ACTIVE(length(active(U71(mark(x0), x1, x2))))
MARK(U72(U11(x0), y1)) → ACTIVE(U72(active(U11(mark(x0))), y1))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(U72(isNat(x0), y1)) → ACTIVE(U72(active(isNat(x0)), y1))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
MARK(U72(zeros, y1)) → ACTIVE(U72(active(zeros), y1))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U71(isNatList(x0), y1, y2)) → ACTIVE(U71(active(isNatList(x0)), y1, y2))
MARK(U72(y0, active(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U71(U71(x0, x1, x2), y1, y2)) → ACTIVE(U71(active(U71(mark(x0), x1, x2)), y1, y2))
MARK(U72(U21(x0), y1)) → ACTIVE(U72(active(U21(mark(x0))), y1))
MARK(length(x0)) → ACTIVE(length(x0))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
MARK(U72(U21(x0), y1)) → ACTIVE(U72(active(U21(mark(x0))), y1))
The remaining pairs can at least be oriented weakly.
MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(U72(U31(x0), y1)) → ACTIVE(U72(active(U31(mark(x0))), y1))
MARK(U72(nil, y1)) → ACTIVE(U72(active(nil), y1))
MARK(length(U62(x0))) → ACTIVE(length(active(U62(mark(x0)))))
MARK(U71(y0, mark(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(length(x0), y1, y2)) → ACTIVE(U71(active(length(mark(x0))), y1, y2))
MARK(U72(0, y1)) → ACTIVE(U72(active(0), y1))
MARK(U72(length(x0), y1)) → ACTIVE(U72(active(length(mark(x0))), y1))
MARK(U71(y0, x1, mark(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U72(U71(x0, x1, x2), y1)) → ACTIVE(U72(active(U71(mark(x0), x1, x2)), y1))
MARK(U71(y0, x1, active(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(length(U41(x0, x1))) → ACTIVE(length(active(U41(mark(x0), x1))))
MARK(U71(U72(x0, x1), y1, y2)) → ACTIVE(U71(active(U72(mark(x0), x1)), y1, y2))
MARK(length(U52(x0))) → ACTIVE(length(active(U52(mark(x0)))))
MARK(length(U61(x0, x1))) → ACTIVE(length(active(U61(mark(x0), x1))))
MARK(length(U93(x0, x1, x2, x3))) → ACTIVE(length(active(U93(mark(x0), x1, x2, x3))))
MARK(U72(U52(x0), y1)) → ACTIVE(U72(active(U52(mark(x0))), y1))
MARK(length(U31(x0))) → ACTIVE(length(active(U31(mark(x0)))))
MARK(length(U51(x0, x1))) → ACTIVE(length(active(U51(mark(x0), x1))))
MARK(length(U11(x0))) → ACTIVE(length(active(U11(mark(x0)))))
MARK(length(U21(x0))) → ACTIVE(length(active(U21(mark(x0)))))
MARK(U72(isNatList(x0), y1)) → ACTIVE(U72(active(isNatList(x0)), y1))
MARK(U72(s(x0), y1)) → ACTIVE(U72(active(s(mark(x0))), y1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(U72(y0, mark(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U72(U72(x0, x1), y1)) → ACTIVE(U72(active(U72(mark(x0), x1)), y1))
MARK(length(U72(x0, x1))) → ACTIVE(length(active(U72(mark(x0), x1))))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
MARK(U71(s(x0), y1, y2)) → ACTIVE(U71(active(s(mark(x0))), y1, y2))
MARK(length(U91(x0, x1, x2, x3))) → ACTIVE(length(active(U91(mark(x0), x1, x2, x3))))
MARK(length(U81(x0))) → ACTIVE(length(active(U81(mark(x0)))))
MARK(U71(x0, x1, x2)) → ACTIVE(U71(x0, x1, x2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U71(y0, active(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(length(U42(x0))) → ACTIVE(length(active(U42(mark(x0)))))
MARK(U72(U91(x0, x1, x2, x3), y1)) → ACTIVE(U72(active(U91(mark(x0), x1, x2, x3)), y1))
MARK(U72(x0, x1)) → ACTIVE(U72(x0, x1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(length(U92(x0, x1, x2, x3))) → ACTIVE(length(active(U92(mark(x0), x1, x2, x3))))
MARK(length(U71(x0, x1, x2))) → ACTIVE(length(active(U71(mark(x0), x1, x2))))
MARK(U72(U11(x0), y1)) → ACTIVE(U72(active(U11(mark(x0))), y1))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(U72(isNat(x0), y1)) → ACTIVE(U72(active(isNat(x0)), y1))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
MARK(U72(zeros, y1)) → ACTIVE(U72(active(zeros), y1))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U71(isNatList(x0), y1, y2)) → ACTIVE(U71(active(isNatList(x0)), y1, y2))
MARK(U72(y0, active(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U71(U71(x0, x1, x2), y1, y2)) → ACTIVE(U71(active(U71(mark(x0), x1, x2)), y1, y2))
MARK(length(x0)) → ACTIVE(length(x0))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVE(x1)) = 0
POL(MARK(x1)) = x1
POL(U11(x1)) = x1
POL(U21(x1)) = 1
POL(U31(x1)) = 0
POL(U41(x1, x2)) = 0
POL(U42(x1)) = 0
POL(U51(x1, x2)) = 0
POL(U52(x1)) = x1
POL(U61(x1, x2)) = 0
POL(U62(x1)) = 0
POL(U71(x1, x2, x3)) = 0
POL(U72(x1, x2)) = x1
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 0
POL(U92(x1, x2, x3, x4)) = 0
POL(U93(x1, x2, x3, x4)) = 0
POL(active(x1)) = x1
POL(cons(x1, x2)) = 0
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 0
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = 0
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(U72(U31(x0), y1)) → ACTIVE(U72(active(U31(mark(x0))), y1))
MARK(U72(nil, y1)) → ACTIVE(U72(active(nil), y1))
MARK(length(U62(x0))) → ACTIVE(length(active(U62(mark(x0)))))
MARK(U71(y0, mark(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(length(x0), y1, y2)) → ACTIVE(U71(active(length(mark(x0))), y1, y2))
MARK(U72(0, y1)) → ACTIVE(U72(active(0), y1))
MARK(U72(length(x0), y1)) → ACTIVE(U72(active(length(mark(x0))), y1))
MARK(U71(y0, x1, mark(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U72(U71(x0, x1, x2), y1)) → ACTIVE(U72(active(U71(mark(x0), x1, x2)), y1))
MARK(U71(y0, x1, active(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(U72(x0, x1), y1, y2)) → ACTIVE(U71(active(U72(mark(x0), x1)), y1, y2))
MARK(length(U41(x0, x1))) → ACTIVE(length(active(U41(mark(x0), x1))))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(length(U52(x0))) → ACTIVE(length(active(U52(mark(x0)))))
MARK(length(U61(x0, x1))) → ACTIVE(length(active(U61(mark(x0), x1))))
MARK(U72(U52(x0), y1)) → ACTIVE(U72(active(U52(mark(x0))), y1))
MARK(length(U93(x0, x1, x2, x3))) → ACTIVE(length(active(U93(mark(x0), x1, x2, x3))))
MARK(length(U31(x0))) → ACTIVE(length(active(U31(mark(x0)))))
MARK(length(U51(x0, x1))) → ACTIVE(length(active(U51(mark(x0), x1))))
MARK(length(U11(x0))) → ACTIVE(length(active(U11(mark(x0)))))
MARK(length(U21(x0))) → ACTIVE(length(active(U21(mark(x0)))))
MARK(U72(s(x0), y1)) → ACTIVE(U72(active(s(mark(x0))), y1))
MARK(U72(isNatList(x0), y1)) → ACTIVE(U72(active(isNatList(x0)), y1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(U72(y0, mark(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U72(U72(x0, x1), y1)) → ACTIVE(U72(active(U72(mark(x0), x1)), y1))
MARK(length(U72(x0, x1))) → ACTIVE(length(active(U72(mark(x0), x1))))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
MARK(U71(s(x0), y1, y2)) → ACTIVE(U71(active(s(mark(x0))), y1, y2))
MARK(length(U91(x0, x1, x2, x3))) → ACTIVE(length(active(U91(mark(x0), x1, x2, x3))))
MARK(length(U81(x0))) → ACTIVE(length(active(U81(mark(x0)))))
MARK(U71(x0, x1, x2)) → ACTIVE(U71(x0, x1, x2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U71(y0, active(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(length(U42(x0))) → ACTIVE(length(active(U42(mark(x0)))))
MARK(U72(U91(x0, x1, x2, x3), y1)) → ACTIVE(U72(active(U91(mark(x0), x1, x2, x3)), y1))
MARK(U72(x0, x1)) → ACTIVE(U72(x0, x1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(length(U92(x0, x1, x2, x3))) → ACTIVE(length(active(U92(mark(x0), x1, x2, x3))))
MARK(length(U71(x0, x1, x2))) → ACTIVE(length(active(U71(mark(x0), x1, x2))))
MARK(U72(U11(x0), y1)) → ACTIVE(U72(active(U11(mark(x0))), y1))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(U72(isNat(x0), y1)) → ACTIVE(U72(active(isNat(x0)), y1))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U72(zeros, y1)) → ACTIVE(U72(active(zeros), y1))
MARK(U71(isNatList(x0), y1, y2)) → ACTIVE(U71(active(isNatList(x0)), y1, y2))
MARK(U72(y0, active(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U71(U71(x0, x1, x2), y1, y2)) → ACTIVE(U71(active(U71(mark(x0), x1, x2)), y1, y2))
MARK(length(x0)) → ACTIVE(length(x0))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
MARK(U72(U31(x0), y1)) → ACTIVE(U72(active(U31(mark(x0))), y1))
MARK(U72(nil, y1)) → ACTIVE(U72(active(nil), y1))
MARK(U72(0, y1)) → ACTIVE(U72(active(0), y1))
MARK(U72(U52(x0), y1)) → ACTIVE(U72(active(U52(mark(x0))), y1))
MARK(U72(isNatList(x0), y1)) → ACTIVE(U72(active(isNatList(x0)), y1))
MARK(U72(U91(x0, x1, x2, x3), y1)) → ACTIVE(U72(active(U91(mark(x0), x1, x2, x3)), y1))
MARK(U72(U11(x0), y1)) → ACTIVE(U72(active(U11(mark(x0))), y1))
MARK(U72(zeros, y1)) → ACTIVE(U72(active(zeros), y1))
The remaining pairs can at least be oriented weakly.
MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(length(U62(x0))) → ACTIVE(length(active(U62(mark(x0)))))
MARK(U71(y0, mark(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(length(x0), y1, y2)) → ACTIVE(U71(active(length(mark(x0))), y1, y2))
MARK(U72(length(x0), y1)) → ACTIVE(U72(active(length(mark(x0))), y1))
MARK(U71(y0, x1, mark(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U72(U71(x0, x1, x2), y1)) → ACTIVE(U72(active(U71(mark(x0), x1, x2)), y1))
MARK(U71(y0, x1, active(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(U72(x0, x1), y1, y2)) → ACTIVE(U71(active(U72(mark(x0), x1)), y1, y2))
MARK(length(U41(x0, x1))) → ACTIVE(length(active(U41(mark(x0), x1))))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(length(U52(x0))) → ACTIVE(length(active(U52(mark(x0)))))
MARK(length(U61(x0, x1))) → ACTIVE(length(active(U61(mark(x0), x1))))
MARK(length(U93(x0, x1, x2, x3))) → ACTIVE(length(active(U93(mark(x0), x1, x2, x3))))
MARK(length(U31(x0))) → ACTIVE(length(active(U31(mark(x0)))))
MARK(length(U51(x0, x1))) → ACTIVE(length(active(U51(mark(x0), x1))))
MARK(length(U11(x0))) → ACTIVE(length(active(U11(mark(x0)))))
MARK(length(U21(x0))) → ACTIVE(length(active(U21(mark(x0)))))
MARK(U72(s(x0), y1)) → ACTIVE(U72(active(s(mark(x0))), y1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(U72(y0, mark(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U72(U72(x0, x1), y1)) → ACTIVE(U72(active(U72(mark(x0), x1)), y1))
MARK(length(U72(x0, x1))) → ACTIVE(length(active(U72(mark(x0), x1))))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
MARK(U71(s(x0), y1, y2)) → ACTIVE(U71(active(s(mark(x0))), y1, y2))
MARK(length(U91(x0, x1, x2, x3))) → ACTIVE(length(active(U91(mark(x0), x1, x2, x3))))
MARK(length(U81(x0))) → ACTIVE(length(active(U81(mark(x0)))))
MARK(U71(x0, x1, x2)) → ACTIVE(U71(x0, x1, x2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U71(y0, active(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(length(U42(x0))) → ACTIVE(length(active(U42(mark(x0)))))
MARK(U72(x0, x1)) → ACTIVE(U72(x0, x1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(length(U92(x0, x1, x2, x3))) → ACTIVE(length(active(U92(mark(x0), x1, x2, x3))))
MARK(length(U71(x0, x1, x2))) → ACTIVE(length(active(U71(mark(x0), x1, x2))))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(U72(isNat(x0), y1)) → ACTIVE(U72(active(isNat(x0)), y1))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U71(isNatList(x0), y1, y2)) → ACTIVE(U71(active(isNatList(x0)), y1, y2))
MARK(U72(y0, active(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U71(U71(x0, x1, x2), y1, y2)) → ACTIVE(U71(active(U71(mark(x0), x1, x2)), y1, y2))
MARK(length(x0)) → ACTIVE(length(x0))
Used ordering: Polynomial interpretation [25]:
POL(0) = 1
POL(ACTIVE(x1)) = 0
POL(MARK(x1)) = x1
POL(U11(x1)) = 1
POL(U21(x1)) = 0
POL(U31(x1)) = 1
POL(U41(x1, x2)) = 0
POL(U42(x1)) = 0
POL(U51(x1, x2)) = 0
POL(U52(x1)) = 1 + x1
POL(U61(x1, x2)) = 0
POL(U62(x1)) = 0
POL(U71(x1, x2, x3)) = 0
POL(U72(x1, x2)) = x1
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 1
POL(U92(x1, x2, x3, x4)) = 0
POL(U93(x1, x2, x3, x4)) = 0
POL(active(x1)) = x1
POL(cons(x1, x2)) = 0
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatList(x1)) = 1
POL(length(x1)) = 0
POL(mark(x1)) = x1
POL(nil) = 1
POL(s(x1)) = x1
POL(take(x1, x2)) = 0
POL(tt) = 0
POL(zeros) = 1
The following usable rules [17] were oriented:
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(length(U62(x0))) → ACTIVE(length(active(U62(mark(x0)))))
MARK(U71(s(x0), y1, y2)) → ACTIVE(U71(active(s(mark(x0))), y1, y2))
MARK(s(X)) → MARK(X)
MARK(U71(y0, mark(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(length(x0), y1, y2)) → ACTIVE(U71(active(length(mark(x0))), y1, y2))
MARK(length(U91(x0, x1, x2, x3))) → ACTIVE(length(active(U91(mark(x0), x1, x2, x3))))
MARK(length(U81(x0))) → ACTIVE(length(active(U81(mark(x0)))))
MARK(U72(length(x0), y1)) → ACTIVE(U72(active(length(mark(x0))), y1))
MARK(U71(x0, x1, x2)) → ACTIVE(U71(x0, x1, x2))
MARK(U71(y0, x1, mark(x2))) → ACTIVE(U71(mark(y0), x1, x2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U71(y0, active(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(length(U42(x0))) → ACTIVE(length(active(U42(mark(x0)))))
MARK(U72(x0, x1)) → ACTIVE(U72(x0, x1))
MARK(U72(U71(x0, x1, x2), y1)) → ACTIVE(U72(active(U71(mark(x0), x1, x2)), y1))
MARK(U71(y0, x1, active(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(length(U41(x0, x1))) → ACTIVE(length(active(U41(mark(x0), x1))))
MARK(U71(U72(x0, x1), y1, y2)) → ACTIVE(U71(active(U72(mark(x0), x1)), y1, y2))
MARK(length(U52(x0))) → ACTIVE(length(active(U52(mark(x0)))))
MARK(length(U61(x0, x1))) → ACTIVE(length(active(U61(mark(x0), x1))))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(length(U92(x0, x1, x2, x3))) → ACTIVE(length(active(U92(mark(x0), x1, x2, x3))))
MARK(length(U71(x0, x1, x2))) → ACTIVE(length(active(U71(mark(x0), x1, x2))))
MARK(length(U93(x0, x1, x2, x3))) → ACTIVE(length(active(U93(mark(x0), x1, x2, x3))))
MARK(length(U31(x0))) → ACTIVE(length(active(U31(mark(x0)))))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(U51(x0, x1))) → ACTIVE(length(active(U51(mark(x0), x1))))
MARK(length(U11(x0))) → ACTIVE(length(active(U11(mark(x0)))))
MARK(length(U21(x0))) → ACTIVE(length(active(U21(mark(x0)))))
MARK(U72(isNat(x0), y1)) → ACTIVE(U72(active(isNat(x0)), y1))
MARK(U72(s(x0), y1)) → ACTIVE(U72(active(s(mark(x0))), y1))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U71(isNatList(x0), y1, y2)) → ACTIVE(U71(active(isNatList(x0)), y1, y2))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(U72(y0, active(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(U72(y0, mark(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U71(U71(x0, x1, x2), y1, y2)) → ACTIVE(U71(active(U71(mark(x0), x1, x2)), y1, y2))
MARK(U72(U72(x0, x1), y1)) → ACTIVE(U72(active(U72(mark(x0), x1)), y1))
MARK(length(U72(x0, x1))) → ACTIVE(length(active(U72(mark(x0), x1))))
MARK(length(x0)) → ACTIVE(length(x0))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
MARK(U71(length(x0), y1, y2)) → ACTIVE(U71(active(length(mark(x0))), y1, y2))
MARK(U71(U72(x0, x1), y1, y2)) → ACTIVE(U71(active(U72(mark(x0), x1)), y1, y2))
MARK(U71(U71(x0, x1, x2), y1, y2)) → ACTIVE(U71(active(U71(mark(x0), x1, x2)), y1, y2))
The remaining pairs can at least be oriented weakly.
MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(length(U62(x0))) → ACTIVE(length(active(U62(mark(x0)))))
MARK(U71(s(x0), y1, y2)) → ACTIVE(U71(active(s(mark(x0))), y1, y2))
MARK(s(X)) → MARK(X)
MARK(U71(y0, mark(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(length(U91(x0, x1, x2, x3))) → ACTIVE(length(active(U91(mark(x0), x1, x2, x3))))
MARK(length(U81(x0))) → ACTIVE(length(active(U81(mark(x0)))))
MARK(U72(length(x0), y1)) → ACTIVE(U72(active(length(mark(x0))), y1))
MARK(U71(x0, x1, x2)) → ACTIVE(U71(x0, x1, x2))
MARK(U71(y0, x1, mark(x2))) → ACTIVE(U71(mark(y0), x1, x2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U71(y0, active(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(length(U42(x0))) → ACTIVE(length(active(U42(mark(x0)))))
MARK(U72(x0, x1)) → ACTIVE(U72(x0, x1))
MARK(U72(U71(x0, x1, x2), y1)) → ACTIVE(U72(active(U71(mark(x0), x1, x2)), y1))
MARK(U71(y0, x1, active(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(length(U41(x0, x1))) → ACTIVE(length(active(U41(mark(x0), x1))))
MARK(length(U52(x0))) → ACTIVE(length(active(U52(mark(x0)))))
MARK(length(U61(x0, x1))) → ACTIVE(length(active(U61(mark(x0), x1))))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(length(U92(x0, x1, x2, x3))) → ACTIVE(length(active(U92(mark(x0), x1, x2, x3))))
MARK(length(U71(x0, x1, x2))) → ACTIVE(length(active(U71(mark(x0), x1, x2))))
MARK(length(U93(x0, x1, x2, x3))) → ACTIVE(length(active(U93(mark(x0), x1, x2, x3))))
MARK(length(U31(x0))) → ACTIVE(length(active(U31(mark(x0)))))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(U51(x0, x1))) → ACTIVE(length(active(U51(mark(x0), x1))))
MARK(length(U11(x0))) → ACTIVE(length(active(U11(mark(x0)))))
MARK(length(U21(x0))) → ACTIVE(length(active(U21(mark(x0)))))
MARK(U72(isNat(x0), y1)) → ACTIVE(U72(active(isNat(x0)), y1))
MARK(U72(s(x0), y1)) → ACTIVE(U72(active(s(mark(x0))), y1))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U71(isNatList(x0), y1, y2)) → ACTIVE(U71(active(isNatList(x0)), y1, y2))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(U72(y0, active(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(U72(y0, mark(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U72(U72(x0, x1), y1)) → ACTIVE(U72(active(U72(mark(x0), x1)), y1))
MARK(length(U72(x0, x1))) → ACTIVE(length(active(U72(mark(x0), x1))))
MARK(length(x0)) → ACTIVE(length(x0))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVE(x1)) = 1
POL(MARK(x1)) = x1
POL(U11(x1)) = 0
POL(U21(x1)) = 0
POL(U31(x1)) = 0
POL(U41(x1, x2)) = 0
POL(U42(x1)) = 0
POL(U51(x1, x2)) = 0
POL(U52(x1)) = 0
POL(U61(x1, x2)) = 0
POL(U62(x1)) = 0
POL(U71(x1, x2, x3)) = 1 + x1
POL(U72(x1, x2)) = 1
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 0
POL(U92(x1, x2, x3, x4)) = 0
POL(U93(x1, x2, x3, x4)) = 0
POL(active(x1)) = x1
POL(cons(x1, x2)) = 0
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 1
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = x1
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(length(U62(x0))) → ACTIVE(length(active(U62(mark(x0)))))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(U71(y0, mark(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(s(X)) → MARK(X)
MARK(U71(s(x0), y1, y2)) → ACTIVE(U71(active(s(mark(x0))), y1, y2))
MARK(length(U91(x0, x1, x2, x3))) → ACTIVE(length(active(U91(mark(x0), x1, x2, x3))))
MARK(length(U81(x0))) → ACTIVE(length(active(U81(mark(x0)))))
MARK(U72(length(x0), y1)) → ACTIVE(U72(active(length(mark(x0))), y1))
MARK(U71(y0, x1, mark(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(x0, x1, x2)) → ACTIVE(U71(x0, x1, x2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U71(y0, active(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(length(U42(x0))) → ACTIVE(length(active(U42(mark(x0)))))
MARK(U72(x0, x1)) → ACTIVE(U72(x0, x1))
MARK(U72(U71(x0, x1, x2), y1)) → ACTIVE(U72(active(U71(mark(x0), x1, x2)), y1))
MARK(U71(y0, x1, active(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(length(U41(x0, x1))) → ACTIVE(length(active(U41(mark(x0), x1))))
MARK(length(U52(x0))) → ACTIVE(length(active(U52(mark(x0)))))
MARK(length(U61(x0, x1))) → ACTIVE(length(active(U61(mark(x0), x1))))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(length(U92(x0, x1, x2, x3))) → ACTIVE(length(active(U92(mark(x0), x1, x2, x3))))
MARK(length(U93(x0, x1, x2, x3))) → ACTIVE(length(active(U93(mark(x0), x1, x2, x3))))
MARK(length(U71(x0, x1, x2))) → ACTIVE(length(active(U71(mark(x0), x1, x2))))
MARK(length(U31(x0))) → ACTIVE(length(active(U31(mark(x0)))))
MARK(length(U51(x0, x1))) → ACTIVE(length(active(U51(mark(x0), x1))))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(U11(x0))) → ACTIVE(length(active(U11(mark(x0)))))
MARK(length(U21(x0))) → ACTIVE(length(active(U21(mark(x0)))))
MARK(U72(s(x0), y1)) → ACTIVE(U72(active(s(mark(x0))), y1))
MARK(U72(isNat(x0), y1)) → ACTIVE(U72(active(isNat(x0)), y1))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(U71(isNatList(x0), y1, y2)) → ACTIVE(U71(active(isNatList(x0)), y1, y2))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(U72(y0, active(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U72(y0, mark(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U72(U72(x0, x1), y1)) → ACTIVE(U72(active(U72(mark(x0), x1)), y1))
MARK(length(U72(x0, x1))) → ACTIVE(length(active(U72(mark(x0), x1))))
MARK(length(x0)) → ACTIVE(length(x0))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
MARK(U72(length(x0), y1)) → ACTIVE(U72(active(length(mark(x0))), y1))
MARK(U72(U71(x0, x1, x2), y1)) → ACTIVE(U72(active(U71(mark(x0), x1, x2)), y1))
MARK(U72(U72(x0, x1), y1)) → ACTIVE(U72(active(U72(mark(x0), x1)), y1))
The remaining pairs can at least be oriented weakly.
MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(length(U62(x0))) → ACTIVE(length(active(U62(mark(x0)))))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(U71(y0, mark(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(s(X)) → MARK(X)
MARK(U71(s(x0), y1, y2)) → ACTIVE(U71(active(s(mark(x0))), y1, y2))
MARK(length(U91(x0, x1, x2, x3))) → ACTIVE(length(active(U91(mark(x0), x1, x2, x3))))
MARK(length(U81(x0))) → ACTIVE(length(active(U81(mark(x0)))))
MARK(U71(y0, x1, mark(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(x0, x1, x2)) → ACTIVE(U71(x0, x1, x2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U71(y0, active(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(length(U42(x0))) → ACTIVE(length(active(U42(mark(x0)))))
MARK(U72(x0, x1)) → ACTIVE(U72(x0, x1))
MARK(U71(y0, x1, active(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(length(U41(x0, x1))) → ACTIVE(length(active(U41(mark(x0), x1))))
MARK(length(U52(x0))) → ACTIVE(length(active(U52(mark(x0)))))
MARK(length(U61(x0, x1))) → ACTIVE(length(active(U61(mark(x0), x1))))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(length(U92(x0, x1, x2, x3))) → ACTIVE(length(active(U92(mark(x0), x1, x2, x3))))
MARK(length(U93(x0, x1, x2, x3))) → ACTIVE(length(active(U93(mark(x0), x1, x2, x3))))
MARK(length(U71(x0, x1, x2))) → ACTIVE(length(active(U71(mark(x0), x1, x2))))
MARK(length(U31(x0))) → ACTIVE(length(active(U31(mark(x0)))))
MARK(length(U51(x0, x1))) → ACTIVE(length(active(U51(mark(x0), x1))))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(U11(x0))) → ACTIVE(length(active(U11(mark(x0)))))
MARK(length(U21(x0))) → ACTIVE(length(active(U21(mark(x0)))))
MARK(U72(s(x0), y1)) → ACTIVE(U72(active(s(mark(x0))), y1))
MARK(U72(isNat(x0), y1)) → ACTIVE(U72(active(isNat(x0)), y1))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(U71(isNatList(x0), y1, y2)) → ACTIVE(U71(active(isNatList(x0)), y1, y2))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(U72(y0, active(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U72(y0, mark(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(length(U72(x0, x1))) → ACTIVE(length(active(U72(mark(x0), x1))))
MARK(length(x0)) → ACTIVE(length(x0))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVE(x1)) = 1
POL(MARK(x1)) = x1
POL(U11(x1)) = 0
POL(U21(x1)) = 0
POL(U31(x1)) = 0
POL(U41(x1, x2)) = 0
POL(U42(x1)) = 0
POL(U51(x1, x2)) = 0
POL(U52(x1)) = 0
POL(U61(x1, x2)) = 0
POL(U62(x1)) = 0
POL(U71(x1, x2, x3)) = 1
POL(U72(x1, x2)) = 1 + x1
POL(U81(x1)) = 0
POL(U91(x1, x2, x3, x4)) = 0
POL(U92(x1, x2, x3, x4)) = 0
POL(U93(x1, x2, x3, x4)) = 0
POL(active(x1)) = x1
POL(cons(x1, x2)) = 0
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 1
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = 0
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(length(U62(x0))) → ACTIVE(length(active(U62(mark(x0)))))
MARK(U71(s(x0), y1, y2)) → ACTIVE(U71(active(s(mark(x0))), y1, y2))
MARK(s(X)) → MARK(X)
MARK(U71(y0, mark(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(length(U91(x0, x1, x2, x3))) → ACTIVE(length(active(U91(mark(x0), x1, x2, x3))))
MARK(length(U81(x0))) → ACTIVE(length(active(U81(mark(x0)))))
MARK(U71(x0, x1, x2)) → ACTIVE(U71(x0, x1, x2))
MARK(U71(y0, x1, mark(x2))) → ACTIVE(U71(mark(y0), x1, x2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U71(y0, active(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(length(U42(x0))) → ACTIVE(length(active(U42(mark(x0)))))
MARK(U72(x0, x1)) → ACTIVE(U72(x0, x1))
MARK(U71(y0, x1, active(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(length(U41(x0, x1))) → ACTIVE(length(active(U41(mark(x0), x1))))
MARK(length(U52(x0))) → ACTIVE(length(active(U52(mark(x0)))))
MARK(length(U61(x0, x1))) → ACTIVE(length(active(U61(mark(x0), x1))))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(length(U92(x0, x1, x2, x3))) → ACTIVE(length(active(U92(mark(x0), x1, x2, x3))))
MARK(length(U93(x0, x1, x2, x3))) → ACTIVE(length(active(U93(mark(x0), x1, x2, x3))))
MARK(length(U71(x0, x1, x2))) → ACTIVE(length(active(U71(mark(x0), x1, x2))))
MARK(length(U31(x0))) → ACTIVE(length(active(U31(mark(x0)))))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(U51(x0, x1))) → ACTIVE(length(active(U51(mark(x0), x1))))
MARK(length(U11(x0))) → ACTIVE(length(active(U11(mark(x0)))))
MARK(length(U21(x0))) → ACTIVE(length(active(U21(mark(x0)))))
MARK(U72(isNat(x0), y1)) → ACTIVE(U72(active(isNat(x0)), y1))
MARK(U72(s(x0), y1)) → ACTIVE(U72(active(s(mark(x0))), y1))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U71(isNatList(x0), y1, y2)) → ACTIVE(U71(active(isNatList(x0)), y1, y2))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(U72(y0, active(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(U72(y0, mark(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(length(U72(x0, x1))) → ACTIVE(length(active(U72(mark(x0), x1))))
MARK(length(x0)) → ACTIVE(length(x0))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
MARK(U72(s(x0), y1)) → ACTIVE(U72(active(s(mark(x0))), y1))
The remaining pairs can at least be oriented weakly.
MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(length(U62(x0))) → ACTIVE(length(active(U62(mark(x0)))))
MARK(U71(s(x0), y1, y2)) → ACTIVE(U71(active(s(mark(x0))), y1, y2))
MARK(s(X)) → MARK(X)
MARK(U71(y0, mark(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(length(U91(x0, x1, x2, x3))) → ACTIVE(length(active(U91(mark(x0), x1, x2, x3))))
MARK(length(U81(x0))) → ACTIVE(length(active(U81(mark(x0)))))
MARK(U71(x0, x1, x2)) → ACTIVE(U71(x0, x1, x2))
MARK(U71(y0, x1, mark(x2))) → ACTIVE(U71(mark(y0), x1, x2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U71(y0, active(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(length(U42(x0))) → ACTIVE(length(active(U42(mark(x0)))))
MARK(U72(x0, x1)) → ACTIVE(U72(x0, x1))
MARK(U71(y0, x1, active(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(length(U41(x0, x1))) → ACTIVE(length(active(U41(mark(x0), x1))))
MARK(length(U52(x0))) → ACTIVE(length(active(U52(mark(x0)))))
MARK(length(U61(x0, x1))) → ACTIVE(length(active(U61(mark(x0), x1))))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(length(U92(x0, x1, x2, x3))) → ACTIVE(length(active(U92(mark(x0), x1, x2, x3))))
MARK(length(U93(x0, x1, x2, x3))) → ACTIVE(length(active(U93(mark(x0), x1, x2, x3))))
MARK(length(U71(x0, x1, x2))) → ACTIVE(length(active(U71(mark(x0), x1, x2))))
MARK(length(U31(x0))) → ACTIVE(length(active(U31(mark(x0)))))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(U51(x0, x1))) → ACTIVE(length(active(U51(mark(x0), x1))))
MARK(length(U11(x0))) → ACTIVE(length(active(U11(mark(x0)))))
MARK(length(U21(x0))) → ACTIVE(length(active(U21(mark(x0)))))
MARK(U72(isNat(x0), y1)) → ACTIVE(U72(active(isNat(x0)), y1))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(U71(isNatList(x0), y1, y2)) → ACTIVE(U71(active(isNatList(x0)), y1, y2))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(U72(y0, active(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(U72(y0, mark(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(length(U72(x0, x1))) → ACTIVE(length(active(U72(mark(x0), x1))))
MARK(length(x0)) → ACTIVE(length(x0))
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
| M( U92(x1, ..., x4) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
| M( take(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( U51(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( isNatList(x1) ) = | | + | | · | x1 |
| M( isNatIList(x1) ) = | | + | | · | x1 |
| M( U91(x1, ..., x4) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
| M( U93(x1, ..., x4) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
| M( U72(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( cons(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( U61(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( U41(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( U71(x1, ..., x3) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
Tuple symbols:
Matrix type:
We used a basic matrix type which is not further parametrizeable.
As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U62(active(X)) → U62(X)
U62(mark(X)) → U62(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U42(active(X)) → U42(X)
U42(mark(X)) → U42(X)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U52(active(X)) → U52(X)
U52(mark(X)) → U52(X)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
U11(active(X)) → U11(X)
U11(mark(X)) → U11(X)
U21(active(X)) → U21(X)
U21(mark(X)) → U21(X)
U31(active(X)) → U31(X)
U31(mark(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U81(active(X)) → U81(X)
U81(mark(X)) → U81(X)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
s(active(X)) → s(X)
s(mark(X)) → s(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(length(U62(x0))) → ACTIVE(length(active(U62(mark(x0)))))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(U71(y0, mark(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(s(X)) → MARK(X)
MARK(U71(s(x0), y1, y2)) → ACTIVE(U71(active(s(mark(x0))), y1, y2))
MARK(length(U91(x0, x1, x2, x3))) → ACTIVE(length(active(U91(mark(x0), x1, x2, x3))))
MARK(length(U81(x0))) → ACTIVE(length(active(U81(mark(x0)))))
MARK(U71(y0, x1, mark(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(U71(x0, x1, x2)) → ACTIVE(U71(x0, x1, x2))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U71(y0, active(x1), x2)) → ACTIVE(U71(mark(y0), x1, x2))
MARK(length(U42(x0))) → ACTIVE(length(active(U42(mark(x0)))))
MARK(U72(x0, x1)) → ACTIVE(U72(x0, x1))
MARK(U71(y0, x1, active(x2))) → ACTIVE(U71(mark(y0), x1, x2))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(length(U41(x0, x1))) → ACTIVE(length(active(U41(mark(x0), x1))))
MARK(length(U52(x0))) → ACTIVE(length(active(U52(mark(x0)))))
MARK(length(U61(x0, x1))) → ACTIVE(length(active(U61(mark(x0), x1))))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(length(U92(x0, x1, x2, x3))) → ACTIVE(length(active(U92(mark(x0), x1, x2, x3))))
MARK(length(U93(x0, x1, x2, x3))) → ACTIVE(length(active(U93(mark(x0), x1, x2, x3))))
MARK(length(U71(x0, x1, x2))) → ACTIVE(length(active(U71(mark(x0), x1, x2))))
MARK(length(U31(x0))) → ACTIVE(length(active(U31(mark(x0)))))
MARK(length(U51(x0, x1))) → ACTIVE(length(active(U51(mark(x0), x1))))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(length(U11(x0))) → ACTIVE(length(active(U11(mark(x0)))))
MARK(length(U21(x0))) → ACTIVE(length(active(U21(mark(x0)))))
MARK(U72(isNat(x0), y1)) → ACTIVE(U72(active(isNat(x0)), y1))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(U71(isNatList(x0), y1, y2)) → ACTIVE(U71(active(isNatList(x0)), y1, y2))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(U72(y0, active(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(U72(y0, mark(x1))) → ACTIVE(U72(mark(y0), x1))
MARK(length(U72(x0, x1))) → ACTIVE(length(active(U72(mark(x0), x1))))
MARK(length(x0)) → ACTIVE(length(x0))
The TRS R consists of the following rules:
active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.