Termination w.r.t. Q proof of /tmp/tmpyHULTR/OvConsOS_complete

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, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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:

PROPER(U71(X1, X2)) → PROPER(X1)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X1)
U71¹(mark(X1), X2) → U71¹(X1, X2)
ACTIVE(U32(X)) → ACTIVE(X)
PROPER(U52(X1, X2)) → U52¹(proper(X1), proper(X2))
ACTIVE(cons(X1, X2)) → CONS(active(X1), X2)
PROPER(U11(X1, X2)) → U11¹(proper(X1), proper(X2))
PROPER(U31(X1, X2)) → PROPER(X1)
ACTIVE(U52(tt, V2)) → U53¹(isNatList(V2))
ACTIVE(isNatIList(cons(V1, V2))) → ISNATKIND(V1)
ACTIVE(length(cons(N, L))) → U71¹(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
ACTIVE(length(cons(N, L))) → ISNATILISTKIND(L)
ACTIVE(U61(tt, V1, V2)) → U62¹(isNat(V1), V2)
ACTIVE(isNatIListKind(take(V1, V2))) → ISNATILISTKIND(V2)
ACTIVE(U21(X1, X2)) → U21¹(active(X1), X2)
U32¹(ok(X)) → U32¹(X)
U31¹(mark(X1), X2) → U31¹(X1, X2)
PROPER(U11(X1, X2)) → PROPER(X2)
ACTIVE(isNatList(cons(V1, V2))) → ISNATKIND(V1)
U62¹(ok(X1), ok(X2)) → U62¹(X1, X2)
ACTIVE(isNatList(take(V1, V2))) → ISNATILISTKIND(V2)
ACTIVE(take(s(M), cons(N, IL))) → U91¹(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
ACTIVE(isNatIListKind(take(V1, V2))) → AND(isNatKind(V1), isNatIListKind(V2))
PROPER(isNatList(X)) → ISNATLIST(proper(X))
PROPER(U53(X)) → PROPER(X)
ACTIVE(length(cons(N, L))) → AND(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N)))
ACTIVE(U63(X)) → ACTIVE(X)
U71¹(ok(X1), ok(X2)) → U71¹(X1, X2)
ACTIVE(isNatIListKind(cons(V1, V2))) → ISNATILISTKIND(V2)
S(ok(X)) → S(X)
CONS(mark(X1), X2) → CONS(X1, X2)
ACTIVE(U71(X1, X2)) → U71¹(active(X1), X2)
ACTIVE(isNatList(take(V1, V2))) → U61¹(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
PROPER(isNatKind(X)) → ISNATKIND(proper(X))
PROPER(U41(X1, X2, X3)) → U41¹(proper(X1), proper(X2), proper(X3))
U12¹(ok(X)) → U12¹(X)
ACTIVE(U42(tt, V2)) → ISNATILIST(V2)
U43¹(ok(X)) → U43¹(X)
PROPER(and(X1, X2)) → AND(proper(X1), proper(X2))
ACTIVE(U51(tt, V1, V2)) → U52¹(isNat(V1), V2)
ACTIVE(U41(X1, X2, X3)) → U41¹(active(X1), X2, X3)
U63¹(ok(X)) → U63¹(X)
PROPER(cons(X1, X2)) → PROPER(X1)
ACTIVE(take(s(M), cons(N, IL))) → ISNATILISTKIND(IL)
U31¹(ok(X1), ok(X2)) → U31¹(X1, X2)
ACTIVE(U11(tt, V1)) → U12¹(isNatList(V1))
S(mark(X)) → S(X)
PROPER(cons(X1, X2)) → CONS(proper(X1), proper(X2))
PROPER(and(X1, X2)) → PROPER(X1)
ISNATLIST(ok(X)) → ISNATLIST(X)
ACTIVE(U43(X)) → U43¹(active(X))
U61¹(mark(X1), X2, X3) → U61¹(X1, X2, X3)
ACTIVE(isNatIList(cons(V1, V2))) → AND(isNatKind(V1), isNatIListKind(V2))
ACTIVE(take(s(M), cons(N, IL))) → AND(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
PROPER(U42(X1, X2)) → PROPER(X1)
ACTIVE(U62(X1, X2)) → ACTIVE(X1)
U51¹(ok(X1), ok(X2), ok(X3)) → U51¹(X1, X2, X3)
U12¹(mark(X)) → U12¹(X)
PROPER(U91(X1, X2, X3, X4)) → U91¹(proper(X1), proper(X2), proper(X3), proper(X4))
PROPER(U61(X1, X2, X3)) → U61¹(proper(X1), proper(X2), proper(X3))
ACTIVE(U41(tt, V1, V2)) → ISNAT(V1)
PROPER(U42(X1, X2)) → PROPER(X2)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X2)
U43¹(mark(X)) → U43¹(X)
ACTIVE(U91(tt, IL, M, N)) → TAKE(M, IL)
U61¹(ok(X1), ok(X2), ok(X3)) → U61¹(X1, X2, X3)
U53¹(ok(X)) → U53¹(X)
ACTIVE(zeros) → CONS(0, zeros)
PROPER(U63(X)) → PROPER(X)
ACTIVE(U63(X)) → U63¹(active(X))
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(U11(tt, V1)) → ISNATLIST(V1)
ACTIVE(take(s(M), cons(N, IL))) → AND(isNat(M), isNatKind(M))
ISNATILIST(ok(X)) → ISNATILIST(X)
ACTIVE(U51(tt, V1, V2)) → ISNAT(V1)
ACTIVE(U81(X)) → ACTIVE(X)
PROPER(U81(X)) → U81¹(proper(X))
PROPER(U21(X1, X2)) → PROPER(X1)
ACTIVE(take(s(M), cons(N, IL))) → AND(isNat(N), isNatKind(N))
U11¹(ok(X1), ok(X2)) → U11¹(X1, X2)
ACTIVE(U62(tt, V2)) → U63¹(isNatIList(V2))
PROPER(isNatIList(X)) → ISNATILIST(proper(X))
ACTIVE(take(0, IL)) → ISNATILIST(IL)
ACTIVE(U61(tt, V1, V2)) → ISNAT(V1)
PROPER(isNatIListKind(X)) → PROPER(X)
ACTIVE(isNatIList(V)) → ISNATILISTKIND(V)
ACTIVE(isNat(length(V1))) → U11¹(isNatIListKind(V1), V1)
PROPER(U41(X1, X2, X3)) → PROPER(X2)
ACTIVE(U62(X1, X2)) → U62¹(active(X1), X2)
ISNATKIND(ok(X)) → ISNATKIND(X)
U22¹(mark(X)) → U22¹(X)
PROPER(take(X1, X2)) → TAKE(proper(X1), proper(X2))
ACTIVE(take(X1, X2)) → ACTIVE(X2)
ACTIVE(U21(tt, V1)) → U22¹(isNat(V1))
ACTIVE(U41(tt, V1, V2)) → U42¹(isNat(V1), V2)
AND(ok(X1), ok(X2)) → AND(X1, X2)
ACTIVE(isNatIListKind(take(V1, V2))) → ISNATKIND(V1)
ACTIVE(isNatIListKind(cons(V1, V2))) → AND(isNatKind(V1), isNatIListKind(V2))
ACTIVE(take(s(M), cons(N, IL))) → ISNATILIST(IL)
U62¹(mark(X1), X2) → U62¹(X1, X2)
ACTIVE(U61(X1, X2, X3)) → ACTIVE(X1)
U81¹(mark(X)) → U81¹(X)
PROPER(U42(X1, X2)) → U42¹(proper(X1), proper(X2))
ACTIVE(U51(X1, X2, X3)) → ACTIVE(X1)
PROPER(U61(X1, X2, X3)) → PROPER(X1)
TOP(ok(X)) → TOP(active(X))
ACTIVE(U32(X)) → U32¹(active(X))
U52¹(ok(X1), ok(X2)) → U52¹(X1, X2)
TAKE(ok(X1), ok(X2)) → TAKE(X1, X2)
ISNAT(ok(X)) → ISNAT(X)
CONS(ok(X1), ok(X2)) → CONS(X1, X2)
ACTIVE(U61(X1, X2, X3)) → U61¹(active(X1), X2, X3)
ACTIVE(U41(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U22(X)) → ACTIVE(X)
ACTIVE(length(cons(N, L))) → ISNATLIST(L)
U81¹(ok(X)) → U81¹(X)
ACTIVE(isNat(s(V1))) → U21¹(isNatKind(V1), V1)
ACTIVE(length(cons(N, L))) → ISNAT(N)
PROPER(take(X1, X2)) → PROPER(X1)
PROPER(U62(X1, X2)) → U62¹(proper(X1), proper(X2))
ACTIVE(take(0, IL)) → ISNATILISTKIND(IL)
U22¹(ok(X)) → U22¹(X)
PROPER(U11(X1, X2)) → PROPER(X1)
ACTIVE(s(X)) → S(active(X))
ACTIVE(take(X1, X2)) → TAKE(X1, active(X2))
ACTIVE(U53(X)) → ACTIVE(X)
PROPER(U62(X1, X2)) → PROPER(X2)
ACTIVE(U52(X1, X2)) → U52¹(active(X1), X2)
ACTIVE(length(cons(N, L))) → AND(isNatList(L), isNatIListKind(L))
PROPER(isNatList(X)) → PROPER(X)
ACTIVE(U42(X1, X2)) → ACTIVE(X1)
ACTIVE(isNatKind(length(V1))) → ISNATILISTKIND(V1)
ACTIVE(isNatList(cons(V1, V2))) → U51¹(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
PROPER(length(X)) → LENGTH(proper(X))
ACTIVE(isNatList(take(V1, V2))) → ISNATKIND(V1)
U63¹(mark(X)) → U63¹(X)
ACTIVE(take(X1, X2)) → ACTIVE(X1)
ACTIVE(take(s(M), cons(N, IL))) → ISNAT(N)
U21¹(mark(X1), X2) → U21¹(X1, X2)
ACTIVE(U53(X)) → U53¹(active(X))
ACTIVE(take(0, IL)) → AND(isNatIList(IL), isNatIListKind(IL))
ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(isNatList(take(V1, V2))) → AND(isNatKind(V1), isNatIListKind(V2))
U21¹(ok(X1), ok(X2)) → U21¹(X1, X2)
ACTIVE(U91(X1, X2, X3, X4)) → U91¹(active(X1), X2, X3, X4)
ACTIVE(isNatIListKind(cons(V1, V2))) → ISNATKIND(V1)
ACTIVE(U81(X)) → U81¹(active(X))
ACTIVE(take(X1, X2)) → TAKE(active(X1), X2)
PROPER(length(X)) → PROPER(X)
PROPER(U63(X)) → U63¹(proper(X))
PROPER(U91(X1, X2, X3, X4)) → PROPER(X3)
ACTIVE(U71(tt, L)) → LENGTH(L)
PROPER(U61(X1, X2, X3)) → PROPER(X3)
ACTIVE(U21(tt, V1)) → ISNAT(V1)
PROPER(U32(X)) → U32¹(proper(X))
ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(take(0, IL)) → U81¹(and(isNatIList(IL), isNatIListKind(IL)))
ACTIVE(U31(tt, V)) → ISNATLIST(V)
ACTIVE(s(X)) → ACTIVE(X)
U32¹(mark(X)) → U32¹(X)
ACTIVE(U12(X)) → U12¹(active(X))
ACTIVE(isNatIList(cons(V1, V2))) → ISNATILISTKIND(V2)
TOP(mark(X)) → PROPER(X)
ACTIVE(and(X1, X2)) → AND(active(X1), X2)
PROPER(U31(X1, X2)) → U31¹(proper(X1), proper(X2))
TOP(ok(X)) → ACTIVE(X)
ACTIVE(length(X)) → ACTIVE(X)
ACTIVE(isNat(length(V1))) → ISNATILISTKIND(V1)
PROPER(U22(X)) → PROPER(X)
ACTIVE(length(cons(N, L))) → AND(isNat(N), isNatKind(N))
LENGTH(mark(X)) → LENGTH(X)
ACTIVE(take(s(M), cons(N, IL))) → ISNATKIND(N)
PROPER(isNatKind(X)) → PROPER(X)
ACTIVE(length(X)) → LENGTH(active(X))
ACTIVE(isNatList(cons(V1, V2))) → AND(isNatKind(V1), isNatIListKind(V2))
PROPER(U43(X)) → PROPER(X)
ACTIVE(take(s(M), cons(N, IL))) → AND(isNatIList(IL), isNatIListKind(IL))
PROPER(U61(X1, X2, X3)) → PROPER(X2)
AND(mark(X1), X2) → AND(X1, X2)
U41¹(ok(X1), ok(X2), ok(X3)) → U41¹(X1, X2, X3)
PROPER(isNatIListKind(X)) → ISNATILISTKIND(proper(X))
PROPER(U81(X)) → PROPER(X)
PROPER(U62(X1, X2)) → PROPER(X1)
ACTIVE(isNatList(cons(V1, V2))) → ISNATILISTKIND(V2)
PROPER(U51(X1, X2, X3)) → PROPER(X2)
ACTIVE(U71(tt, L)) → S(length(L))
PROPER(U21(X1, X2)) → U21¹(proper(X1), proper(X2))
U53¹(mark(X)) → U53¹(X)
ACTIVE(take(s(M), cons(N, IL))) → ISNATKIND(M)
ACTIVE(isNatIList(cons(V1, V2))) → U41¹(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
ACTIVE(U12(X)) → ACTIVE(X)
PROPER(U12(X)) → U12¹(proper(X))
ACTIVE(U21(X1, X2)) → ACTIVE(X1)
U41¹(mark(X1), X2, X3) → U41¹(X1, X2, X3)
U42¹(mark(X1), X2) → U42¹(X1, X2)
U42¹(ok(X1), ok(X2)) → U42¹(X1, X2)
ACTIVE(U22(X)) → U22¹(active(X))
ACTIVE(length(cons(N, L))) → ISNATKIND(N)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(U53(X)) → U53¹(proper(X))
ISNATILISTKIND(ok(X)) → ISNATILISTKIND(X)
ACTIVE(U42(X1, X2)) → U42¹(active(X1), X2)
U91¹(mark(X1), X2, X3, X4) → U91¹(X1, X2, X3, X4)
ACTIVE(U91(tt, IL, M, N)) → CONS(N, take(M, IL))
PROPER(U41(X1, X2, X3)) → PROPER(X1)
ACTIVE(U43(X)) → ACTIVE(X)
ACTIVE(U31(X1, X2)) → U31¹(active(X1), X2)
PROPER(U41(X1, X2, X3)) → PROPER(X3)
PROPER(s(X)) → S(proper(X))
ACTIVE(and(X1, X2)) → ACTIVE(X1)
ACTIVE(U42(tt, V2)) → U43¹(isNatIList(V2))
ACTIVE(U52(tt, V2)) → ISNATLIST(V2)
PROPER(U71(X1, X2)) → U71¹(proper(X1), proper(X2))
PROPER(and(X1, X2)) → PROPER(X2)
ACTIVE(U91(X1, X2, X3, X4)) → ACTIVE(X1)
PROPER(U51(X1, X2, X3)) → PROPER(X3)
ACTIVE(isNat(s(V1))) → ISNATKIND(V1)
PROPER(U52(X1, X2)) → PROPER(X2)
TAKE(mark(X1), X2) → TAKE(X1, X2)
ACTIVE(U62(tt, V2)) → ISNATILIST(V2)
U11¹(mark(X1), X2) → U11¹(X1, X2)
PROPER(isNat(X)) → PROPER(X)
LENGTH(ok(X)) → LENGTH(X)
PROPER(U52(X1, X2)) → PROPER(X1)
ACTIVE(U31(tt, V)) → U32¹(isNatList(V))
PROPER(U51(X1, X2, X3)) → PROPER(X1)
PROPER(U12(X)) → PROPER(X)
PROPER(isNatIList(X)) → PROPER(X)
ACTIVE(U51(X1, X2, X3)) → U51¹(active(X1), X2, X3)
ACTIVE(take(s(M), cons(N, IL))) → AND(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))))
PROPER(s(X)) → PROPER(X)
ACTIVE(isNatIList(V)) → U31¹(isNatIListKind(V), V)
PROPER(take(X1, X2)) → PROPER(X2)
ACTIVE(U11(X1, X2)) → U11¹(active(X1), X2)
PROPER(isNat(X)) → ISNAT(proper(X))
ACTIVE(isNatKind(s(V1))) → ISNATKIND(V1)
U52¹(mark(X1), X2) → U52¹(X1, X2)
PROPER(U32(X)) → PROPER(X)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X4)
ACTIVE(take(s(M), cons(N, IL))) → ISNAT(M)
ACTIVE(U71(X1, X2)) → ACTIVE(X1)
PROPER(U31(X1, X2)) → PROPER(X2)
U91¹(ok(X1), ok(X2), ok(X3), ok(X4)) → U91¹(X1, X2, X3, X4)
U51¹(mark(X1), X2, X3) → U51¹(X1, X2, X3)
PROPER(U71(X1, X2)) → PROPER(X2)
PROPER(U43(X)) → U43¹(proper(X))
TAKE(X1, mark(X2)) → TAKE(X1, X2)
ACTIVE(U52(X1, X2)) → ACTIVE(X1)
PROPER(U22(X)) → U22¹(proper(X))
TOP(mark(X)) → TOP(proper(X))
PROPER(U21(X1, X2)) → PROPER(X2)
PROPER(U51(X1, X2, X3)) → U51¹(proper(X1), proper(X2), proper(X3))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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:

PROPER(U71(X1, X2)) → PROPER(X1)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X1)
U71¹(mark(X1), X2) → U71¹(X1, X2)
ACTIVE(U32(X)) → ACTIVE(X)
PROPER(U52(X1, X2)) → U52¹(proper(X1), proper(X2))
ACTIVE(cons(X1, X2)) → CONS(active(X1), X2)
PROPER(U11(X1, X2)) → U11¹(proper(X1), proper(X2))
PROPER(U31(X1, X2)) → PROPER(X1)
ACTIVE(U52(tt, V2)) → U53¹(isNatList(V2))
ACTIVE(isNatIList(cons(V1, V2))) → ISNATKIND(V1)
ACTIVE(length(cons(N, L))) → U71¹(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
ACTIVE(length(cons(N, L))) → ISNATILISTKIND(L)
ACTIVE(U61(tt, V1, V2)) → U62¹(isNat(V1), V2)
ACTIVE(isNatIListKind(take(V1, V2))) → ISNATILISTKIND(V2)
ACTIVE(U21(X1, X2)) → U21¹(active(X1), X2)
U32¹(ok(X)) → U32¹(X)
U31¹(mark(X1), X2) → U31¹(X1, X2)
PROPER(U11(X1, X2)) → PROPER(X2)
ACTIVE(isNatList(cons(V1, V2))) → ISNATKIND(V1)
U62¹(ok(X1), ok(X2)) → U62¹(X1, X2)
ACTIVE(isNatList(take(V1, V2))) → ISNATILISTKIND(V2)
ACTIVE(take(s(M), cons(N, IL))) → U91¹(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
ACTIVE(isNatIListKind(take(V1, V2))) → AND(isNatKind(V1), isNatIListKind(V2))
PROPER(isNatList(X)) → ISNATLIST(proper(X))
PROPER(U53(X)) → PROPER(X)
ACTIVE(length(cons(N, L))) → AND(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N)))
ACTIVE(U63(X)) → ACTIVE(X)
U71¹(ok(X1), ok(X2)) → U71¹(X1, X2)
ACTIVE(isNatIListKind(cons(V1, V2))) → ISNATILISTKIND(V2)
S(ok(X)) → S(X)
CONS(mark(X1), X2) → CONS(X1, X2)
ACTIVE(U71(X1, X2)) → U71¹(active(X1), X2)
ACTIVE(isNatList(take(V1, V2))) → U61¹(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
PROPER(isNatKind(X)) → ISNATKIND(proper(X))
PROPER(U41(X1, X2, X3)) → U41¹(proper(X1), proper(X2), proper(X3))
U12¹(ok(X)) → U12¹(X)
ACTIVE(U42(tt, V2)) → ISNATILIST(V2)
U43¹(ok(X)) → U43¹(X)
PROPER(and(X1, X2)) → AND(proper(X1), proper(X2))
ACTIVE(U51(tt, V1, V2)) → U52¹(isNat(V1), V2)
ACTIVE(U41(X1, X2, X3)) → U41¹(active(X1), X2, X3)
U63¹(ok(X)) → U63¹(X)
PROPER(cons(X1, X2)) → PROPER(X1)
ACTIVE(take(s(M), cons(N, IL))) → ISNATILISTKIND(IL)
U31¹(ok(X1), ok(X2)) → U31¹(X1, X2)
ACTIVE(U11(tt, V1)) → U12¹(isNatList(V1))
S(mark(X)) → S(X)
PROPER(cons(X1, X2)) → CONS(proper(X1), proper(X2))
PROPER(and(X1, X2)) → PROPER(X1)
ISNATLIST(ok(X)) → ISNATLIST(X)
ACTIVE(U43(X)) → U43¹(active(X))
U61¹(mark(X1), X2, X3) → U61¹(X1, X2, X3)
ACTIVE(isNatIList(cons(V1, V2))) → AND(isNatKind(V1), isNatIListKind(V2))
ACTIVE(take(s(M), cons(N, IL))) → AND(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
PROPER(U42(X1, X2)) → PROPER(X1)
ACTIVE(U62(X1, X2)) → ACTIVE(X1)
U51¹(ok(X1), ok(X2), ok(X3)) → U51¹(X1, X2, X3)
U12¹(mark(X)) → U12¹(X)
PROPER(U91(X1, X2, X3, X4)) → U91¹(proper(X1), proper(X2), proper(X3), proper(X4))
PROPER(U61(X1, X2, X3)) → U61¹(proper(X1), proper(X2), proper(X3))
ACTIVE(U41(tt, V1, V2)) → ISNAT(V1)
PROPER(U42(X1, X2)) → PROPER(X2)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X2)
U43¹(mark(X)) → U43¹(X)
ACTIVE(U91(tt, IL, M, N)) → TAKE(M, IL)
U61¹(ok(X1), ok(X2), ok(X3)) → U61¹(X1, X2, X3)
U53¹(ok(X)) → U53¹(X)
ACTIVE(zeros) → CONS(0, zeros)
PROPER(U63(X)) → PROPER(X)
ACTIVE(U63(X)) → U63¹(active(X))
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(U11(tt, V1)) → ISNATLIST(V1)
ACTIVE(take(s(M), cons(N, IL))) → AND(isNat(M), isNatKind(M))
ISNATILIST(ok(X)) → ISNATILIST(X)
ACTIVE(U51(tt, V1, V2)) → ISNAT(V1)
ACTIVE(U81(X)) → ACTIVE(X)
PROPER(U81(X)) → U81¹(proper(X))
PROPER(U21(X1, X2)) → PROPER(X1)
ACTIVE(take(s(M), cons(N, IL))) → AND(isNat(N), isNatKind(N))
U11¹(ok(X1), ok(X2)) → U11¹(X1, X2)
ACTIVE(U62(tt, V2)) → U63¹(isNatIList(V2))
PROPER(isNatIList(X)) → ISNATILIST(proper(X))
ACTIVE(take(0, IL)) → ISNATILIST(IL)
ACTIVE(U61(tt, V1, V2)) → ISNAT(V1)
PROPER(isNatIListKind(X)) → PROPER(X)
ACTIVE(isNatIList(V)) → ISNATILISTKIND(V)
ACTIVE(isNat(length(V1))) → U11¹(isNatIListKind(V1), V1)
PROPER(U41(X1, X2, X3)) → PROPER(X2)
ACTIVE(U62(X1, X2)) → U62¹(active(X1), X2)
ISNATKIND(ok(X)) → ISNATKIND(X)
U22¹(mark(X)) → U22¹(X)
PROPER(take(X1, X2)) → TAKE(proper(X1), proper(X2))
ACTIVE(take(X1, X2)) → ACTIVE(X2)
ACTIVE(U21(tt, V1)) → U22¹(isNat(V1))
ACTIVE(U41(tt, V1, V2)) → U42¹(isNat(V1), V2)
AND(ok(X1), ok(X2)) → AND(X1, X2)
ACTIVE(isNatIListKind(take(V1, V2))) → ISNATKIND(V1)
ACTIVE(isNatIListKind(cons(V1, V2))) → AND(isNatKind(V1), isNatIListKind(V2))
ACTIVE(take(s(M), cons(N, IL))) → ISNATILIST(IL)
U62¹(mark(X1), X2) → U62¹(X1, X2)
ACTIVE(U61(X1, X2, X3)) → ACTIVE(X1)
U81¹(mark(X)) → U81¹(X)
PROPER(U42(X1, X2)) → U42¹(proper(X1), proper(X2))
ACTIVE(U51(X1, X2, X3)) → ACTIVE(X1)
PROPER(U61(X1, X2, X3)) → PROPER(X1)
TOP(ok(X)) → TOP(active(X))
ACTIVE(U32(X)) → U32¹(active(X))
U52¹(ok(X1), ok(X2)) → U52¹(X1, X2)
TAKE(ok(X1), ok(X2)) → TAKE(X1, X2)
ISNAT(ok(X)) → ISNAT(X)
CONS(ok(X1), ok(X2)) → CONS(X1, X2)
ACTIVE(U61(X1, X2, X3)) → U61¹(active(X1), X2, X3)
ACTIVE(U41(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U22(X)) → ACTIVE(X)
ACTIVE(length(cons(N, L))) → ISNATLIST(L)
U81¹(ok(X)) → U81¹(X)
ACTIVE(isNat(s(V1))) → U21¹(isNatKind(V1), V1)
ACTIVE(length(cons(N, L))) → ISNAT(N)
PROPER(take(X1, X2)) → PROPER(X1)
PROPER(U62(X1, X2)) → U62¹(proper(X1), proper(X2))
ACTIVE(take(0, IL)) → ISNATILISTKIND(IL)
U22¹(ok(X)) → U22¹(X)
PROPER(U11(X1, X2)) → PROPER(X1)
ACTIVE(s(X)) → S(active(X))
ACTIVE(take(X1, X2)) → TAKE(X1, active(X2))
ACTIVE(U53(X)) → ACTIVE(X)
PROPER(U62(X1, X2)) → PROPER(X2)
ACTIVE(U52(X1, X2)) → U52¹(active(X1), X2)
ACTIVE(length(cons(N, L))) → AND(isNatList(L), isNatIListKind(L))
PROPER(isNatList(X)) → PROPER(X)
ACTIVE(U42(X1, X2)) → ACTIVE(X1)
ACTIVE(isNatKind(length(V1))) → ISNATILISTKIND(V1)
ACTIVE(isNatList(cons(V1, V2))) → U51¹(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
PROPER(length(X)) → LENGTH(proper(X))
ACTIVE(isNatList(take(V1, V2))) → ISNATKIND(V1)
U63¹(mark(X)) → U63¹(X)
ACTIVE(take(X1, X2)) → ACTIVE(X1)
ACTIVE(take(s(M), cons(N, IL))) → ISNAT(N)
U21¹(mark(X1), X2) → U21¹(X1, X2)
ACTIVE(U53(X)) → U53¹(active(X))
ACTIVE(take(0, IL)) → AND(isNatIList(IL), isNatIListKind(IL))
ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(isNatList(take(V1, V2))) → AND(isNatKind(V1), isNatIListKind(V2))
U21¹(ok(X1), ok(X2)) → U21¹(X1, X2)
ACTIVE(U91(X1, X2, X3, X4)) → U91¹(active(X1), X2, X3, X4)
ACTIVE(isNatIListKind(cons(V1, V2))) → ISNATKIND(V1)
ACTIVE(U81(X)) → U81¹(active(X))
ACTIVE(take(X1, X2)) → TAKE(active(X1), X2)
PROPER(length(X)) → PROPER(X)
PROPER(U63(X)) → U63¹(proper(X))
PROPER(U91(X1, X2, X3, X4)) → PROPER(X3)
ACTIVE(U71(tt, L)) → LENGTH(L)
PROPER(U61(X1, X2, X3)) → PROPER(X3)
ACTIVE(U21(tt, V1)) → ISNAT(V1)
PROPER(U32(X)) → U32¹(proper(X))
ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(take(0, IL)) → U81¹(and(isNatIList(IL), isNatIListKind(IL)))
ACTIVE(U31(tt, V)) → ISNATLIST(V)
ACTIVE(s(X)) → ACTIVE(X)
U32¹(mark(X)) → U32¹(X)
ACTIVE(U12(X)) → U12¹(active(X))
ACTIVE(isNatIList(cons(V1, V2))) → ISNATILISTKIND(V2)
TOP(mark(X)) → PROPER(X)
ACTIVE(and(X1, X2)) → AND(active(X1), X2)
PROPER(U31(X1, X2)) → U31¹(proper(X1), proper(X2))
TOP(ok(X)) → ACTIVE(X)
ACTIVE(length(X)) → ACTIVE(X)
ACTIVE(isNat(length(V1))) → ISNATILISTKIND(V1)
PROPER(U22(X)) → PROPER(X)
ACTIVE(length(cons(N, L))) → AND(isNat(N), isNatKind(N))
LENGTH(mark(X)) → LENGTH(X)
ACTIVE(take(s(M), cons(N, IL))) → ISNATKIND(N)
PROPER(isNatKind(X)) → PROPER(X)
ACTIVE(length(X)) → LENGTH(active(X))
ACTIVE(isNatList(cons(V1, V2))) → AND(isNatKind(V1), isNatIListKind(V2))
PROPER(U43(X)) → PROPER(X)
ACTIVE(take(s(M), cons(N, IL))) → AND(isNatIList(IL), isNatIListKind(IL))
PROPER(U61(X1, X2, X3)) → PROPER(X2)
AND(mark(X1), X2) → AND(X1, X2)
U41¹(ok(X1), ok(X2), ok(X3)) → U41¹(X1, X2, X3)
PROPER(isNatIListKind(X)) → ISNATILISTKIND(proper(X))
PROPER(U81(X)) → PROPER(X)
PROPER(U62(X1, X2)) → PROPER(X1)
ACTIVE(isNatList(cons(V1, V2))) → ISNATILISTKIND(V2)
PROPER(U51(X1, X2, X3)) → PROPER(X2)
ACTIVE(U71(tt, L)) → S(length(L))
PROPER(U21(X1, X2)) → U21¹(proper(X1), proper(X2))
U53¹(mark(X)) → U53¹(X)
ACTIVE(take(s(M), cons(N, IL))) → ISNATKIND(M)
ACTIVE(isNatIList(cons(V1, V2))) → U41¹(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
ACTIVE(U12(X)) → ACTIVE(X)
PROPER(U12(X)) → U12¹(proper(X))
ACTIVE(U21(X1, X2)) → ACTIVE(X1)
U41¹(mark(X1), X2, X3) → U41¹(X1, X2, X3)
U42¹(mark(X1), X2) → U42¹(X1, X2)
U42¹(ok(X1), ok(X2)) → U42¹(X1, X2)
ACTIVE(U22(X)) → U22¹(active(X))
ACTIVE(length(cons(N, L))) → ISNATKIND(N)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(U53(X)) → U53¹(proper(X))
ISNATILISTKIND(ok(X)) → ISNATILISTKIND(X)
ACTIVE(U42(X1, X2)) → U42¹(active(X1), X2)
U91¹(mark(X1), X2, X3, X4) → U91¹(X1, X2, X3, X4)
ACTIVE(U91(tt, IL, M, N)) → CONS(N, take(M, IL))
PROPER(U41(X1, X2, X3)) → PROPER(X1)
ACTIVE(U43(X)) → ACTIVE(X)
ACTIVE(U31(X1, X2)) → U31¹(active(X1), X2)
PROPER(U41(X1, X2, X3)) → PROPER(X3)
PROPER(s(X)) → S(proper(X))
ACTIVE(and(X1, X2)) → ACTIVE(X1)
ACTIVE(U42(tt, V2)) → U43¹(isNatIList(V2))
ACTIVE(U52(tt, V2)) → ISNATLIST(V2)
PROPER(U71(X1, X2)) → U71¹(proper(X1), proper(X2))
PROPER(and(X1, X2)) → PROPER(X2)
ACTIVE(U91(X1, X2, X3, X4)) → ACTIVE(X1)
PROPER(U51(X1, X2, X3)) → PROPER(X3)
ACTIVE(isNat(s(V1))) → ISNATKIND(V1)
PROPER(U52(X1, X2)) → PROPER(X2)
TAKE(mark(X1), X2) → TAKE(X1, X2)
ACTIVE(U62(tt, V2)) → ISNATILIST(V2)
U11¹(mark(X1), X2) → U11¹(X1, X2)
PROPER(isNat(X)) → PROPER(X)
LENGTH(ok(X)) → LENGTH(X)
PROPER(U52(X1, X2)) → PROPER(X1)
ACTIVE(U31(tt, V)) → U32¹(isNatList(V))
PROPER(U51(X1, X2, X3)) → PROPER(X1)
PROPER(U12(X)) → PROPER(X)
PROPER(isNatIList(X)) → PROPER(X)
ACTIVE(U51(X1, X2, X3)) → U51¹(active(X1), X2, X3)
ACTIVE(take(s(M), cons(N, IL))) → AND(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))))
PROPER(s(X)) → PROPER(X)
ACTIVE(isNatIList(V)) → U31¹(isNatIListKind(V), V)
PROPER(take(X1, X2)) → PROPER(X2)
ACTIVE(U11(X1, X2)) → U11¹(active(X1), X2)
PROPER(isNat(X)) → ISNAT(proper(X))
ACTIVE(isNatKind(s(V1))) → ISNATKIND(V1)
U52¹(mark(X1), X2) → U52¹(X1, X2)
PROPER(U32(X)) → PROPER(X)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X4)
ACTIVE(take(s(M), cons(N, IL))) → ISNAT(M)
ACTIVE(U71(X1, X2)) → ACTIVE(X1)
PROPER(U31(X1, X2)) → PROPER(X2)
U91¹(ok(X1), ok(X2), ok(X3), ok(X4)) → U91¹(X1, X2, X3, X4)
U51¹(mark(X1), X2, X3) → U51¹(X1, X2, X3)
PROPER(U71(X1, X2)) → PROPER(X2)
PROPER(U43(X)) → U43¹(proper(X))
TAKE(X1, mark(X2)) → TAKE(X1, X2)
ACTIVE(U52(X1, X2)) → ACTIVE(X1)
PROPER(U22(X)) → U22¹(proper(X))
TOP(mark(X)) → TOP(proper(X))
PROPER(U21(X1, X2)) → PROPER(X2)
PROPER(U51(X1, X2, X3)) → U51¹(proper(X1), proper(X2), proper(X3))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 31 SCCs with 127 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ISNATKIND(ok(X)) → ISNATKIND(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ISNATKIND(ok(X)) → ISNATKIND(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:

ISNATKIND(ok(X)) → ISNATKIND(X)
The graph contains the following edges 1 > 1

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ISNATILISTKIND(ok(X)) → ISNATILISTKIND(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ISNATILISTKIND(ok(X)) → ISNATILISTKIND(X)

From the DPs we obtained the following set of size-change graphs:

ISNATILISTKIND(ok(X)) → ISNATILISTKIND(X)
The graph contains the following edges 1 > 1

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ISNATILIST(ok(X)) → ISNATILIST(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ISNATILIST(ok(X)) → ISNATILIST(X)

From the DPs we obtained the following set of size-change graphs:

ISNATILIST(ok(X)) → ISNATILIST(X)
The graph contains the following edges 1 > 1

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ISNAT(ok(X)) → ISNAT(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ISNAT(ok(X)) → ISNAT(X)

From the DPs we obtained the following set of size-change graphs:

ISNAT(ok(X)) → ISNAT(X)
The graph contains the following edges 1 > 1

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ISNATLIST(ok(X)) → ISNATLIST(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ISNATLIST(ok(X)) → ISNATLIST(X)

From the DPs we obtained the following set of size-change graphs:

ISNATLIST(ok(X)) → ISNATLIST(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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

AND(mark(X1), X2) → AND(X1, X2)
AND(ok(X1), ok(X2)) → AND(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

AND(mark(X1), X2) → AND(X1, X2)
AND(ok(X1), ok(X2)) → AND(X1, X2)

From the DPs we obtained the following set of size-change graphs:

AND(mark(X1), X2) → AND(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
AND(ok(X1), ok(X2)) → AND(X1, X2)
The graph contains the following edges 1 > 1, 2 > 2

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

TAKE(ok(X1), ok(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, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

TAKE(ok(X1), ok(X2)) → TAKE(X1, X2)
TAKE(mark(X1), X2) → TAKE(X1, X2)
TAKE(X1, mark(X2)) → TAKE(X1, X2)

From the DPs we obtained the following set of size-change graphs:

TAKE(ok(X1), ok(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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ 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:

U91¹(ok(X1), ok(X2), ok(X3), ok(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, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U91¹(ok(X1), ok(X2), ok(X3), ok(X4)) → U91¹(X1, X2, X3, X4)
U91¹(mark(X1), X2, X3, X4) → U91¹(X1, X2, X3, X4)

From the DPs we obtained the following set of size-change graphs:

U91¹(ok(X1), ok(X2), ok(X3), ok(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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ 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:

U81¹(mark(X)) → U81¹(X)
U81¹(ok(X)) → U81¹(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U81¹(ok(X)) → U81¹(X)
U81¹(mark(X)) → U81¹(X)

From the DPs we obtained the following set of size-change graphs:

U81¹(mark(X)) → U81¹(X)
The graph contains the following edges 1 > 1
U81¹(ok(X)) → U81¹(X)
The graph contains the following edges 1 > 1

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

LENGTH(ok(X)) → LENGTH(X)
LENGTH(mark(X)) → LENGTH(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

LENGTH(ok(X)) → LENGTH(X)
LENGTH(mark(X)) → LENGTH(X)

From the DPs we obtained the following set of size-change graphs:

LENGTH(ok(X)) → LENGTH(X)
The graph contains the following edges 1 > 1
LENGTH(mark(X)) → LENGTH(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

            ↳ 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:

S(ok(X)) → S(X)
S(mark(X)) → S(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

S(ok(X)) → S(X)
S(mark(X)) → S(X)

From the DPs we obtained the following set of size-change graphs:

S(ok(X)) → S(X)
The graph contains the following edges 1 > 1
S(mark(X)) → S(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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U71¹(mark(X1), X2) → U71¹(X1, X2)
U71¹(ok(X1), ok(X2)) → U71¹(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U71¹(mark(X1), X2) → U71¹(X1, X2)
U71¹(ok(X1), ok(X2)) → U71¹(X1, X2)

From the DPs we obtained the following set of size-change graphs:

U71¹(mark(X1), X2) → U71¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
U71¹(ok(X1), ok(X2)) → U71¹(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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U63¹(ok(X)) → U63¹(X)
U63¹(mark(X)) → U63¹(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U63¹(ok(X)) → U63¹(X)
U63¹(mark(X)) → U63¹(X)

From the DPs we obtained the following set of size-change graphs:

U63¹(ok(X)) → U63¹(X)
The graph contains the following edges 1 > 1
U63¹(mark(X)) → U63¹(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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U62¹(mark(X1), X2) → U62¹(X1, X2)
U62¹(ok(X1), ok(X2)) → U62¹(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U62¹(mark(X1), X2) → U62¹(X1, X2)
U62¹(ok(X1), ok(X2)) → U62¹(X1, X2)

From the DPs we obtained the following set of size-change graphs:

U62¹(mark(X1), X2) → U62¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
U62¹(ok(X1), ok(X2)) → U62¹(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

            ↳ 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:

U61¹(mark(X1), X2, X3) → U61¹(X1, X2, X3)
U61¹(ok(X1), ok(X2), ok(X3)) → U61¹(X1, X2, X3)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U61¹(mark(X1), X2, X3) → U61¹(X1, X2, X3)
U61¹(ok(X1), ok(X2), ok(X3)) → U61¹(X1, X2, X3)

From the DPs we obtained the following set of size-change graphs:

U61¹(mark(X1), X2, X3) → U61¹(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
U61¹(ok(X1), ok(X2), ok(X3)) → U61¹(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

          ↳ 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:

U53¹(mark(X)) → U53¹(X)
U53¹(ok(X)) → U53¹(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U53¹(mark(X)) → U53¹(X)
U53¹(ok(X)) → U53¹(X)

From the DPs we obtained the following set of size-change graphs:

U53¹(mark(X)) → U53¹(X)
The graph contains the following edges 1 > 1
U53¹(ok(X)) → U53¹(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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U52¹(mark(X1), X2) → U52¹(X1, X2)
U52¹(ok(X1), ok(X2)) → U52¹(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U52¹(mark(X1), X2) → U52¹(X1, X2)
U52¹(ok(X1), ok(X2)) → U52¹(X1, X2)

From the DPs we obtained the following set of size-change graphs:

U52¹(mark(X1), X2) → U52¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
U52¹(ok(X1), ok(X2)) → U52¹(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

            ↳ 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:

U51¹(mark(X1), X2, X3) → U51¹(X1, X2, X3)
U51¹(ok(X1), ok(X2), ok(X3)) → U51¹(X1, X2, X3)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U51¹(mark(X1), X2, X3) → U51¹(X1, X2, X3)
U51¹(ok(X1), ok(X2), ok(X3)) → U51¹(X1, X2, X3)

From the DPs we obtained the following set of size-change graphs:

U51¹(mark(X1), X2, X3) → U51¹(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
U51¹(ok(X1), ok(X2), ok(X3)) → U51¹(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

          ↳ 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:

U43¹(mark(X)) → U43¹(X)
U43¹(ok(X)) → U43¹(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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

            ↳ 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:

U43¹(ok(X)) → U43¹(X)
U43¹(mark(X)) → U43¹(X)

From the DPs we obtained the following set of size-change graphs:

U43¹(mark(X)) → U43¹(X)
The graph contains the following edges 1 > 1
U43¹(ok(X)) → U43¹(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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U42¹(mark(X1), X2) → U42¹(X1, X2)
U42¹(ok(X1), ok(X2)) → U42¹(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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

            ↳ 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:

U42¹(mark(X1), X2) → U42¹(X1, X2)
U42¹(ok(X1), ok(X2)) → U42¹(X1, X2)

From the DPs we obtained the following set of size-change graphs:

U42¹(mark(X1), X2) → U42¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
U42¹(ok(X1), ok(X2)) → U42¹(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

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U41¹(ok(X1), ok(X2), ok(X3)) → U41¹(X1, X2, X3)
U41¹(mark(X1), X2, X3) → U41¹(X1, X2, X3)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U41¹(ok(X1), ok(X2), ok(X3)) → U41¹(X1, X2, X3)
U41¹(mark(X1), X2, X3) → U41¹(X1, X2, X3)

From the DPs we obtained the following set of size-change graphs:

U41¹(ok(X1), ok(X2), ok(X3)) → U41¹(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 > 2, 3 > 3
U41¹(mark(X1), X2, X3) → U41¹(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

          ↳ 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:

U32¹(ok(X)) → U32¹(X)
U32¹(mark(X)) → U32¹(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U32¹(ok(X)) → U32¹(X)
U32¹(mark(X)) → U32¹(X)

From the DPs we obtained the following set of size-change graphs:

U32¹(ok(X)) → U32¹(X)
The graph contains the following edges 1 > 1
U32¹(mark(X)) → U32¹(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

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U31¹(mark(X1), X2) → U31¹(X1, X2)
U31¹(ok(X1), ok(X2)) → U31¹(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U31¹(mark(X1), X2) → U31¹(X1, X2)
U31¹(ok(X1), ok(X2)) → U31¹(X1, X2)

From the DPs we obtained the following set of size-change graphs:

U31¹(mark(X1), X2) → U31¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
U31¹(ok(X1), ok(X2)) → U31¹(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

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U22¹(mark(X)) → U22¹(X)
U22¹(ok(X)) → U22¹(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U22¹(mark(X)) → U22¹(X)
U22¹(ok(X)) → U22¹(X)

From the DPs we obtained the following set of size-change graphs:

U22¹(mark(X)) → U22¹(X)
The graph contains the following edges 1 > 1
U22¹(ok(X)) → U22¹(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

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U21¹(ok(X1), ok(X2)) → U21¹(X1, X2)
U21¹(mark(X1), X2) → U21¹(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U21¹(ok(X1), ok(X2)) → U21¹(X1, X2)
U21¹(mark(X1), X2) → U21¹(X1, X2)

From the DPs we obtained the following set of size-change graphs:

U21¹(ok(X1), ok(X2)) → U21¹(X1, X2)
The graph contains the following edges 1 > 1, 2 > 2
U21¹(mark(X1), X2) → U21¹(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

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U12¹(mark(X)) → U12¹(X)
U12¹(ok(X)) → U12¹(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U12¹(mark(X)) → U12¹(X)
U12¹(ok(X)) → U12¹(X)

From the DPs we obtained the following set of size-change graphs:

U12¹(mark(X)) → U12¹(X)
The graph contains the following edges 1 > 1
U12¹(ok(X)) → U12¹(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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U11¹(mark(X1), X2) → U11¹(X1, X2)
U11¹(ok(X1), ok(X2)) → U11¹(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U11¹(ok(X1), ok(X2)) → U11¹(X1, X2)
U11¹(mark(X1), X2) → U11¹(X1, X2)

From the DPs we obtained the following set of size-change graphs:

U11¹(mark(X1), X2) → U11¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
U11¹(ok(X1), ok(X2)) → U11¹(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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

CONS(mark(X1), X2) → CONS(X1, X2)
CONS(ok(X1), ok(X2)) → CONS(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

CONS(mark(X1), X2) → CONS(X1, X2)
CONS(ok(X1), ok(X2)) → CONS(X1, X2)

From the DPs we obtained the following set of size-change graphs:

CONS(mark(X1), X2) → CONS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
CONS(ok(X1), ok(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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

PROPER(U71(X1, X2)) → PROPER(X1)
PROPER(isNat(X)) → PROPER(X)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X1)
PROPER(U52(X1, X2)) → PROPER(X1)
PROPER(U51(X1, X2, X3)) → PROPER(X1)
PROPER(U42(X1, X2)) → PROPER(X2)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X2)
PROPER(U12(X)) → PROPER(X)
PROPER(isNatIList(X)) → PROPER(X)
PROPER(U62(X1, X2)) → PROPER(X2)
PROPER(isNatList(X)) → PROPER(X)
PROPER(U31(X1, X2)) → PROPER(X1)
PROPER(s(X)) → PROPER(X)
PROPER(U22(X)) → PROPER(X)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(take(X1, X2)) → PROPER(X2)
PROPER(U63(X)) → PROPER(X)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(isNatKind(X)) → PROPER(X)
PROPER(U61(X1, X2, X3)) → PROPER(X1)
PROPER(U41(X1, X2, X3)) → PROPER(X1)
PROPER(U32(X)) → PROPER(X)
PROPER(U21(X1, X2)) → PROPER(X1)
PROPER(and(X1, X2)) → PROPER(X1)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X4)
PROPER(U11(X1, X2)) → PROPER(X2)
PROPER(length(X)) → PROPER(X)
PROPER(U31(X1, X2)) → PROPER(X2)
PROPER(U41(X1, X2, X3)) → PROPER(X3)
PROPER(U43(X)) → PROPER(X)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X3)
PROPER(U61(X1, X2, X3)) → PROPER(X3)
PROPER(U71(X1, X2)) → PROPER(X2)
PROPER(U61(X1, X2, X3)) → PROPER(X2)
PROPER(and(X1, X2)) → PROPER(X2)
PROPER(U51(X1, X2, X3)) → PROPER(X3)
PROPER(isNatIListKind(X)) → PROPER(X)
PROPER(take(X1, X2)) → PROPER(X1)
PROPER(U53(X)) → PROPER(X)
PROPER(U52(X1, X2)) → PROPER(X2)
PROPER(U81(X)) → PROPER(X)
PROPER(U41(X1, X2, X3)) → PROPER(X2)
PROPER(U21(X1, X2)) → PROPER(X2)
PROPER(U62(X1, X2)) → PROPER(X1)
PROPER(U42(X1, X2)) → PROPER(X1)
PROPER(U11(X1, X2)) → PROPER(X1)
PROPER(U51(X1, X2, X3)) → PROPER(X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

PROPER(isNat(X)) → PROPER(X)
PROPER(U71(X1, X2)) → PROPER(X1)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X1)
PROPER(U52(X1, X2)) → PROPER(X1)
PROPER(U51(X1, X2, X3)) → PROPER(X1)
PROPER(U42(X1, X2)) → PROPER(X2)
PROPER(U12(X)) → PROPER(X)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X2)
PROPER(isNatIList(X)) → PROPER(X)
PROPER(U62(X1, X2)) → PROPER(X2)
PROPER(isNatList(X)) → PROPER(X)
PROPER(U31(X1, X2)) → PROPER(X1)
PROPER(s(X)) → PROPER(X)
PROPER(U22(X)) → PROPER(X)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(take(X1, X2)) → PROPER(X2)
PROPER(U63(X)) → PROPER(X)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(isNatKind(X)) → PROPER(X)
PROPER(U61(X1, X2, X3)) → PROPER(X1)
PROPER(U41(X1, X2, X3)) → PROPER(X1)
PROPER(U32(X)) → PROPER(X)
PROPER(U21(X1, X2)) → PROPER(X1)
PROPER(and(X1, X2)) → PROPER(X1)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X4)
PROPER(U11(X1, X2)) → PROPER(X2)
PROPER(U43(X)) → PROPER(X)
PROPER(U41(X1, X2, X3)) → PROPER(X3)
PROPER(U31(X1, X2)) → PROPER(X2)
PROPER(length(X)) → PROPER(X)
PROPER(U61(X1, X2, X3)) → PROPER(X3)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X3)
PROPER(U71(X1, X2)) → PROPER(X2)
PROPER(U61(X1, X2, X3)) → PROPER(X2)
PROPER(and(X1, X2)) → PROPER(X2)
PROPER(U51(X1, X2, X3)) → PROPER(X3)
PROPER(take(X1, X2)) → PROPER(X1)
PROPER(isNatIListKind(X)) → PROPER(X)
PROPER(U53(X)) → PROPER(X)
PROPER(U52(X1, X2)) → PROPER(X2)
PROPER(U41(X1, X2, X3)) → PROPER(X2)
PROPER(U81(X)) → PROPER(X)
PROPER(U21(X1, X2)) → PROPER(X2)
PROPER(U42(X1, X2)) → PROPER(X1)
PROPER(U62(X1, X2)) → PROPER(X1)
PROPER(U51(X1, X2, X3)) → PROPER(X2)
PROPER(U11(X1, X2)) → PROPER(X1)

From the DPs we obtained the following set of size-change graphs:

PROPER(U71(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(isNat(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(U91(X1, X2, X3, X4)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(U52(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(U51(X1, X2, X3)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(U42(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(U91(X1, X2, X3, X4)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(U12(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(isNatIList(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(U62(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(isNatList(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(U31(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(s(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(U22(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(cons(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(take(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(U63(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(cons(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(isNatKind(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(U61(X1, X2, X3)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(U41(X1, X2, X3)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(U32(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(U21(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(and(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(U91(X1, X2, X3, X4)) → PROPER(X4)
The graph contains the following edges 1 > 1
PROPER(U11(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(length(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(U31(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(U41(X1, X2, X3)) → PROPER(X3)
The graph contains the following edges 1 > 1
PROPER(U43(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(U91(X1, X2, X3, X4)) → PROPER(X3)
The graph contains the following edges 1 > 1
PROPER(U61(X1, X2, X3)) → PROPER(X3)
The graph contains the following edges 1 > 1
PROPER(U71(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(U61(X1, X2, X3)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(and(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(U51(X1, X2, X3)) → PROPER(X3)
The graph contains the following edges 1 > 1
PROPER(isNatIListKind(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(take(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(U53(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(U52(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(U81(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(U41(X1, X2, X3)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(U21(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(U62(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(U42(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(U11(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(U51(X1, X2, X3)) → PROPER(X2)
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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(U81(X)) → ACTIVE(X)
ACTIVE(take(X1, X2)) → ACTIVE(X2)
ACTIVE(U12(X)) → ACTIVE(X)
ACTIVE(U43(X)) → ACTIVE(X)
ACTIVE(U22(X)) → ACTIVE(X)
ACTIVE(U32(X)) → ACTIVE(X)
ACTIVE(U41(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U71(X1, X2)) → ACTIVE(X1)
ACTIVE(U21(X1, X2)) → ACTIVE(X1)
ACTIVE(and(X1, X2)) → ACTIVE(X1)
ACTIVE(U53(X)) → ACTIVE(X)
ACTIVE(length(X)) → ACTIVE(X)
ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(U91(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(U42(X1, X2)) → ACTIVE(X1)
ACTIVE(U52(X1, X2)) → ACTIVE(X1)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(U61(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → ACTIVE(X1)
ACTIVE(U63(X)) → ACTIVE(X)
ACTIVE(U51(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U62(X1, X2)) → ACTIVE(X1)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(U81(X)) → ACTIVE(X)
ACTIVE(take(X1, X2)) → ACTIVE(X2)
ACTIVE(U12(X)) → ACTIVE(X)
ACTIVE(U43(X)) → ACTIVE(X)
ACTIVE(U22(X)) → ACTIVE(X)
ACTIVE(U32(X)) → ACTIVE(X)
ACTIVE(U41(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U71(X1, X2)) → ACTIVE(X1)
ACTIVE(U21(X1, X2)) → ACTIVE(X1)
ACTIVE(and(X1, X2)) → ACTIVE(X1)
ACTIVE(U53(X)) → ACTIVE(X)
ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(length(X)) → ACTIVE(X)
ACTIVE(U91(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(U52(X1, X2)) → ACTIVE(X1)
ACTIVE(U42(X1, X2)) → ACTIVE(X1)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(U61(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → ACTIVE(X1)
ACTIVE(U63(X)) → ACTIVE(X)
ACTIVE(U51(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U62(X1, X2)) → ACTIVE(X1)

From the DPs we obtained the following set of size-change graphs:

ACTIVE(U31(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(U81(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
ACTIVE(take(X1, X2)) → ACTIVE(X2)
The graph contains the following edges 1 > 1
ACTIVE(U12(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
ACTIVE(U43(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
ACTIVE(U22(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
ACTIVE(U32(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
ACTIVE(U41(X1, X2, X3)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(U71(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(U21(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(and(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(U53(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
ACTIVE(length(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
ACTIVE(U11(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(U91(X1, X2, X3, X4)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(U42(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(U52(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(s(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
ACTIVE(U61(X1, X2, X3)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(take(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(U63(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
ACTIVE(U51(X1, X2, X3)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(U62(X1, X2)) → ACTIVE(X1)
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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

Q DP problem:
The TRS P consists of the following rules:

TOP(mark(X)) → TOP(proper(X))
TOP(ok(X)) → TOP(active(X))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the usable rules with reduction pair processor [15] with a polynomial ordering [25], all dependency pairs and the corresponding usable rules [17] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

No dependency pairs are removed.

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = 2·x₁
POL(U11(x₁, x₂)) = x₁ + 2·x₂
POL(U12(x₁)) = 2·x₁
POL(U21(x₁, x₂)) = x₁ + x₂
POL(U22(x₁)) = 2·x₁
POL(U31(x₁, x₂)) = x₁ + 2·x₂
POL(U32(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = 2·x₁ + x₂ + x₃
POL(U42(x₁, x₂)) = x₁ + x₂
POL(U43(x₁)) = x₁
POL(U51(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U52(x₁, x₂)) = x₁ + 2·x₂
POL(U53(x₁)) = x₁
POL(U61(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U62(x₁, x₂)) = x₁ + x₂
POL(U63(x₁)) = x₁
POL(U71(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U81(x₁)) = x₁
POL(U91(x₁, x₂, x₃, x₄)) = 2·x₁ + 2·x₂ + 2·x₃ + x₄
POL(active(x₁)) = 2·x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = 2·x₁
POL(isNatIListKind(x₁)) = x₁
POL(isNatKind(x₁)) = 2·x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 2·x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(ok(x₁)) = 2·x₁
POL(proper(x₁)) = x₁
POL(s(x₁)) = 2·x₁
POL(take(x₁, x₂)) = 2·x₁ + 2·x₂
POL(tt) = 0
POL(zeros) = 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

TOP(mark(X)) → TOP(proper(X))
TOP(ok(X)) → TOP(active(X))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule TOP(mark(X)) → TOP(proper(X)) at position [0] we obtained the following new rules:

TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(mark(tt)) → TOP(ok(tt))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(mark(nil)) → TOP(ok(nil))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(zeros)) → TOP(ok(zeros))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(0)) → TOP(ok(0))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(mark(tt)) → TOP(ok(tt))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(mark(nil)) → TOP(ok(nil))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(X)) → TOP(active(X))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(zeros)) → TOP(ok(zeros))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(0)) → TOP(ok(0))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule TOP(ok(X)) → TOP(active(X)) at position [0] we obtained the following new rules:

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(ok(U32(tt))) → TOP(mark(tt))
TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(U61(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(ok(U43(tt))) → TOP(mark(tt))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(isNatIListKind(zeros))) → TOP(mark(tt))
TOP(ok(U53(tt))) → TOP(mark(tt))
TOP(ok(isNatIListKind(nil))) → TOP(mark(tt))
TOP(ok(isNat(length(x0)))) → TOP(mark(U11(isNatIListKind(x0), x0)))
TOP(ok(zeros)) → TOP(mark(cons(0, zeros)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(ok(U63(tt))) → TOP(mark(tt))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(ok(isNatList(nil))) → TOP(mark(tt))
TOP(ok(isNat(0))) → TOP(mark(tt))
TOP(ok(U12(tt))) → TOP(mark(tt))
TOP(ok(isNatKind(0))) → TOP(mark(tt))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(ok(U71(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(U41(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(ok(U22(tt))) → TOP(mark(tt))
TOP(ok(U21(tt, x0))) → TOP(mark(U22(isNat(x0))))
TOP(ok(U81(tt))) → TOP(mark(nil))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U91(and(and(isNatIList(x2), isNatIListKind(x2)), and(and(isNat(x0), isNatKind(x0)), and(isNat(x1), isNatKind(x1)))), x2, x0, x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(isNatIList(x0))) → TOP(mark(U31(isNatIListKind(x0), x0)))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(ok(length(nil))) → TOP(mark(0))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U41(tt, x0, x1))) → TOP(mark(U42(isNat(x0), x1)))
TOP(ok(isNatIList(zeros))) → TOP(mark(tt))
TOP(ok(isNatKind(length(x0)))) → TOP(mark(isNatIListKind(x0)))
TOP(ok(take(0, x0))) → TOP(mark(U81(and(isNatIList(x0), isNatIListKind(x0)))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(ok(U31(tt, x0))) → TOP(mark(U32(isNatList(x0))))
TOP(ok(U62(tt, x0))) → TOP(mark(U63(isNatIList(x0))))
TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(U51(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNatKind(x0), 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(U61(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(ok(U43(tt))) → TOP(mark(tt))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(U53(tt))) → TOP(mark(tt))
TOP(ok(zeros)) → TOP(mark(cons(0, zeros)))
TOP(ok(isNat(length(x0)))) → TOP(mark(U11(isNatIListKind(x0), x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(ok(U63(tt))) → TOP(mark(tt))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(ok(isNatList(nil))) → TOP(mark(tt))
TOP(ok(U12(tt))) → TOP(mark(tt))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(U71(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(U41(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(mark(tt)) → TOP(ok(tt))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(ok(U22(tt))) → TOP(mark(tt))
TOP(ok(U21(tt, x0))) → TOP(mark(U22(isNat(x0))))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(nil)) → TOP(ok(nil))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U91(and(and(isNatIList(x2), isNatIListKind(x2)), and(and(isNat(x0), isNatKind(x0)), and(isNat(x1), isNatKind(x1)))), x2, x0, x1)))
TOP(mark(zeros)) → TOP(ok(zeros))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(isNatIList(x0))) → TOP(mark(U31(isNatIListKind(x0), x0)))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(isNatIList(zeros))) → TOP(mark(tt))
TOP(ok(take(0, x0))) → TOP(mark(U81(and(isNatIList(x0), isNatIListKind(x0)))))
TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(U51(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNatKind(x0), x0)))
TOP(ok(U32(tt))) → TOP(mark(tt))
TOP(ok(isNatIListKind(zeros))) → TOP(mark(tt))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(isNatIListKind(nil))) → TOP(mark(tt))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(ok(isNat(0))) → TOP(mark(tt))
TOP(ok(isNatKind(0))) → TOP(mark(tt))
TOP(mark(0)) → TOP(ok(0))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U81(tt))) → TOP(mark(nil))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(length(nil))) → TOP(mark(0))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U41(tt, x0, x1))) → TOP(mark(U42(isNat(x0), x1)))
TOP(ok(isNatKind(length(x0)))) → TOP(mark(isNatIListKind(x0)))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(ok(U31(tt, x0))) → TOP(mark(U32(isNatList(x0))))
TOP(ok(U62(tt, x0))) → TOP(mark(U63(isNatIList(x0))))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))

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 17 less nodes.

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(U61(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(zeros)) → TOP(mark(cons(0, zeros)))
TOP(ok(isNat(length(x0)))) → TOP(mark(U11(isNatIListKind(x0), x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(U71(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(U41(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U21(tt, x0))) → TOP(mark(U22(isNat(x0))))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U91(and(and(isNatIList(x2), isNatIListKind(x2)), and(and(isNat(x0), isNatKind(x0)), and(isNat(x1), isNatKind(x1)))), x2, x0, x1)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(mark(zeros)) → TOP(ok(zeros))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(isNatIList(x0))) → TOP(mark(U31(isNatIListKind(x0), x0)))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U41(tt, x0, x1))) → TOP(mark(U42(isNat(x0), x1)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(isNatKind(length(x0)))) → TOP(mark(isNatIListKind(x0)))
TOP(ok(take(0, x0))) → TOP(mark(U81(and(isNatIList(x0), isNatIListKind(x0)))))
TOP(ok(U31(tt, x0))) → TOP(mark(U32(isNatList(x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(ok(U62(tt, x0))) → TOP(mark(U63(isNatIList(x0))))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(U51(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNatKind(x0), x0)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))

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.

TOP(ok(zeros)) → TOP(mark(cons(0, zeros)))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(U41(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(ok(isNatIList(x0))) → TOP(mark(U31(isNatIListKind(x0), x0)))

The remaining pairs can at least be oriented weakly.

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(U61(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(isNat(length(x0)))) → TOP(mark(U11(isNatIListKind(x0), x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(U71(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U21(tt, x0))) → TOP(mark(U22(isNat(x0))))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U91(and(and(isNatIList(x2), isNatIListKind(x2)), and(and(isNat(x0), isNatKind(x0)), and(isNat(x1), isNatKind(x1)))), x2, x0, x1)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(mark(zeros)) → TOP(ok(zeros))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U41(tt, x0, x1))) → TOP(mark(U42(isNat(x0), x1)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(isNatKind(length(x0)))) → TOP(mark(isNatIListKind(x0)))
TOP(ok(take(0, x0))) → TOP(mark(U81(and(isNatIList(x0), isNatIListKind(x0)))))
TOP(ok(U31(tt, x0))) → TOP(mark(U32(isNatList(x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(ok(U62(tt, x0))) → TOP(mark(U63(isNatIList(x0))))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(U51(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNatKind(x0), x0)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U12(x₁)) = 0
POL(U21(x₁, x₂)) = 0
POL(U22(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 0
POL(U42(x₁, x₂)) = 0
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = 0
POL(U53(x₁)) = 0
POL(U61(x₁, x₂, x₃)) = 0
POL(U62(x₁, x₂)) = 0
POL(U63(x₁)) = 0
POL(U71(x₁, x₂)) = 0
POL(U81(x₁)) = 0
POL(U91(x₁, x₂, x₃, x₄)) = 0
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 1
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = x₁
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 0
POL(tt) = 0
POL(zeros) = 1

The following usable rules [17] were oriented:

take(ok(X1), ok(X2)) → ok(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
and(mark(X1), X2) → mark(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
proper(tt) → ok(tt)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(isNatList(X)) → isNatList(proper(X))
proper(U12(X)) → U12(proper(X))
proper(U22(X)) → U22(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U32(X)) → U32(proper(X))
proper(U43(X)) → U43(proper(X))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U53(X)) → U53(proper(X))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U43(mark(X)) → mark(U43(X))
U32(ok(X)) → ok(U32(X))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X)) → ok(U22(X))
U22(mark(X)) → mark(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(nil) → ok(nil)
proper(U81(X)) → U81(proper(X))
proper(length(X)) → length(proper(X))
proper(s(X)) → s(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(U61(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(isNat(length(x0)))) → TOP(mark(U11(isNatIListKind(x0), x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(U71(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U21(tt, x0))) → TOP(mark(U22(isNat(x0))))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U91(and(and(isNatIList(x2), isNatIListKind(x2)), and(and(isNat(x0), isNatKind(x0)), and(isNat(x1), isNatKind(x1)))), x2, x0, x1)))
TOP(mark(zeros)) → TOP(ok(zeros))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U41(tt, x0, x1))) → TOP(mark(U42(isNat(x0), x1)))
TOP(ok(isNatKind(length(x0)))) → TOP(mark(isNatIListKind(x0)))
TOP(ok(take(0, x0))) → TOP(mark(U81(and(isNatIList(x0), isNatIListKind(x0)))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(ok(U31(tt, x0))) → TOP(mark(U32(isNatList(x0))))
TOP(ok(U62(tt, x0))) → TOP(mark(U63(isNatIList(x0))))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(U51(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNatKind(x0), x0)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))

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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(U61(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(isNat(length(x0)))) → TOP(mark(U11(isNatIListKind(x0), x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(U71(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U21(tt, x0))) → TOP(mark(U22(isNat(x0))))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U91(and(and(isNatIList(x2), isNatIListKind(x2)), and(and(isNat(x0), isNatKind(x0)), and(isNat(x1), isNatKind(x1)))), x2, x0, x1)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U41(tt, x0, x1))) → TOP(mark(U42(isNat(x0), x1)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(isNatKind(length(x0)))) → TOP(mark(isNatIListKind(x0)))
TOP(ok(take(0, x0))) → TOP(mark(U81(and(isNatIList(x0), isNatIListKind(x0)))))
TOP(ok(U31(tt, x0))) → TOP(mark(U32(isNatList(x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(ok(U62(tt, x0))) → TOP(mark(U63(isNatIList(x0))))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(U51(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNatKind(x0), x0)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))

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.

TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))

The remaining pairs can at least be oriented weakly.

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(U61(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(isNat(length(x0)))) → TOP(mark(U11(isNatIListKind(x0), x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(U71(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U21(tt, x0))) → TOP(mark(U22(isNat(x0))))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U91(and(and(isNatIList(x2), isNatIListKind(x2)), and(and(isNat(x0), isNatKind(x0)), and(isNat(x1), isNatKind(x1)))), x2, x0, x1)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U41(tt, x0, x1))) → TOP(mark(U42(isNat(x0), x1)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(isNatKind(length(x0)))) → TOP(mark(isNatIListKind(x0)))
TOP(ok(take(0, x0))) → TOP(mark(U81(and(isNatIList(x0), isNatIListKind(x0)))))
TOP(ok(U31(tt, x0))) → TOP(mark(U32(isNatList(x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(ok(U62(tt, x0))) → TOP(mark(U63(isNatIList(x0))))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(U51(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNatKind(x0), x0)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁, x₂)) = 1
POL(U12(x₁)) = 1
POL(U21(x₁, x₂)) = 1
POL(U22(x₁)) = 1
POL(U31(x₁, x₂)) = 1
POL(U32(x₁)) = 1
POL(U41(x₁, x₂, x₃)) = 1
POL(U42(x₁, x₂)) = 1
POL(U43(x₁)) = 1
POL(U51(x₁, x₂, x₃)) = 1
POL(U52(x₁, x₂)) = 1
POL(U53(x₁)) = 1
POL(U61(x₁, x₂, x₃)) = 1
POL(U62(x₁, x₂)) = 1
POL(U63(x₁)) = 1
POL(U71(x₁, x₂)) = 1
POL(U81(x₁)) = 1
POL(U91(x₁, x₂, x₃, x₄)) = 1
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = 1
POL(cons(x₁, x₂)) = 1
POL(isNat(x₁)) = 1
POL(isNatIList(x₁)) = 0
POL(isNatIListKind(x₁)) = 1
POL(isNatKind(x₁)) = 1
POL(isNatList(x₁)) = 1
POL(length(x₁)) = 1
POL(mark(x₁)) = 1
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(s(x₁)) = 1
POL(take(x₁, x₂)) = 1
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

take(ok(X1), ok(X2)) → ok(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
and(mark(X1), X2) → mark(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U43(mark(X)) → mark(U43(X))
U32(ok(X)) → ok(U32(X))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X)) → ok(U22(X))
U22(mark(X)) → mark(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(U61(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(isNat(length(x0)))) → TOP(mark(U11(isNatIListKind(x0), x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(U71(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U21(tt, x0))) → TOP(mark(U22(isNat(x0))))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U91(and(and(isNatIList(x2), isNatIListKind(x2)), and(and(isNat(x0), isNatKind(x0)), and(isNat(x1), isNatKind(x1)))), x2, x0, x1)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U41(tt, x0, x1))) → TOP(mark(U42(isNat(x0), x1)))
TOP(ok(isNatKind(length(x0)))) → TOP(mark(isNatIListKind(x0)))
TOP(ok(take(0, x0))) → TOP(mark(U81(and(isNatIList(x0), isNatIListKind(x0)))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(ok(U31(tt, x0))) → TOP(mark(U32(isNatList(x0))))
TOP(ok(U62(tt, x0))) → TOP(mark(U63(isNatIList(x0))))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(U51(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNatKind(x0), x0)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))

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.

TOP(ok(isNat(length(x0)))) → TOP(mark(U11(isNatIListKind(x0), x0)))
TOP(ok(U21(tt, x0))) → TOP(mark(U22(isNat(x0))))
TOP(ok(U41(tt, x0, x1))) → TOP(mark(U42(isNat(x0), x1)))
TOP(ok(isNatKind(length(x0)))) → TOP(mark(isNatIListKind(x0)))
TOP(ok(U31(tt, x0))) → TOP(mark(U32(isNatList(x0))))
TOP(ok(U62(tt, x0))) → TOP(mark(U63(isNatIList(x0))))
TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(U51(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNatKind(x0), x0)))

The remaining pairs can at least be oriented weakly.

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(U61(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(U71(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U91(and(and(isNatIList(x2), isNatIListKind(x2)), and(and(isNat(x0), isNatKind(x0)), and(isNat(x1), isNatKind(x1)))), x2, x0, x1)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(take(0, x0))) → TOP(mark(U81(and(isNatIList(x0), isNatIListKind(x0)))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U12(x₁)) = 0
POL(U21(x₁, x₂)) = 1
POL(U22(x₁)) = 0
POL(U31(x₁, x₂)) = 1
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 1 + x₃
POL(U42(x₁, x₂)) = x₂
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = 0
POL(U53(x₁)) = 0
POL(U61(x₁, x₂, x₃)) = 1
POL(U62(x₁, x₂)) = 1
POL(U63(x₁)) = 0
POL(U71(x₁, x₂)) = 1
POL(U81(x₁)) = 0
POL(U91(x₁, x₂, x₃, x₄)) = 0
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 1 + x₁
POL(isNatIList(x₁)) = 0
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 1
POL(isNatList(x₁)) = 1
POL(length(x₁)) = 1
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = x₁
POL(s(x₁)) = 1
POL(take(x₁, x₂)) = 0
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

take(ok(X1), ok(X2)) → ok(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
and(mark(X1), X2) → mark(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
proper(tt) → ok(tt)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(isNatList(X)) → isNatList(proper(X))
proper(U12(X)) → U12(proper(X))
proper(U22(X)) → U22(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U32(X)) → U32(proper(X))
proper(U43(X)) → U43(proper(X))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U53(X)) → U53(proper(X))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U43(mark(X)) → mark(U43(X))
U32(ok(X)) → ok(U32(X))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X)) → ok(U22(X))
U22(mark(X)) → mark(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(nil) → ok(nil)
proper(U81(X)) → U81(proper(X))
proper(length(X)) → length(proper(X))
proper(s(X)) → s(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(U61(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(U71(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U91(and(and(isNatIList(x2), isNatIListKind(x2)), and(and(isNat(x0), isNatKind(x0)), and(isNat(x1), isNatKind(x1)))), x2, x0, x1)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(take(0, x0))) → TOP(mark(U81(and(isNatIList(x0), isNatIListKind(x0)))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))

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.

TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))

The remaining pairs can at least be oriented weakly.

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(U61(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(U71(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U91(and(and(isNatIList(x2), isNatIListKind(x2)), and(and(isNat(x0), isNatKind(x0)), and(isNat(x1), isNatKind(x1)))), x2, x0, x1)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(take(0, x0))) → TOP(mark(U81(and(isNatIList(x0), isNatIListKind(x0)))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁, x₂)) = 1
POL(U12(x₁)) = 1
POL(U21(x₁, x₂)) = 1
POL(U22(x₁)) = 1
POL(U31(x₁, x₂)) = 1
POL(U32(x₁)) = 1
POL(U41(x₁, x₂, x₃)) = 1
POL(U42(x₁, x₂)) = 1
POL(U43(x₁)) = 1
POL(U51(x₁, x₂, x₃)) = 1
POL(U52(x₁, x₂)) = 1
POL(U53(x₁)) = 1
POL(U61(x₁, x₂, x₃)) = 1
POL(U62(x₁, x₂)) = 1
POL(U63(x₁)) = 1
POL(U71(x₁, x₂)) = 1
POL(U81(x₁)) = 1
POL(U91(x₁, x₂, x₃, x₄)) = 1
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = 1
POL(cons(x₁, x₂)) = 1
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatIListKind(x₁)) = 1
POL(isNatKind(x₁)) = 1
POL(isNatList(x₁)) = 1
POL(length(x₁)) = 1
POL(mark(x₁)) = 1
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(s(x₁)) = 1
POL(take(x₁, x₂)) = 1
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

take(ok(X1), ok(X2)) → ok(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
and(mark(X1), X2) → mark(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U43(mark(X)) → mark(U43(X))
U32(ok(X)) → ok(U32(X))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X)) → ok(U22(X))
U22(mark(X)) → mark(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(U61(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(U71(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U91(and(and(isNatIList(x2), isNatIListKind(x2)), and(and(isNat(x0), isNatKind(x0)), and(isNat(x1), isNatKind(x1)))), x2, x0, x1)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(take(0, x0))) → TOP(mark(U81(and(isNatIList(x0), isNatIListKind(x0)))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))

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.

TOP(ok(U71(tt, x0))) → TOP(mark(s(length(x0))))

The remaining pairs can at least be oriented weakly.

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(U61(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U91(and(and(isNatIList(x2), isNatIListKind(x2)), and(and(isNat(x0), isNatKind(x0)), and(isNat(x1), isNatKind(x1)))), x2, x0, x1)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(take(0, x0))) → TOP(mark(U81(and(isNatIList(x0), isNatIListKind(x0)))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U12(x₁)) = 0
POL(U21(x₁, x₂)) = 0
POL(U22(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 0
POL(U42(x₁, x₂)) = 0
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = 0
POL(U53(x₁)) = 0
POL(U61(x₁, x₂, x₃)) = 0
POL(U62(x₁, x₂)) = 0
POL(U63(x₁)) = 0
POL(U71(x₁, x₂)) = 1
POL(U81(x₁)) = 0
POL(U91(x₁, x₂, x₃, x₄)) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = 0
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 1
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 1
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = x₁
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 0
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

take(ok(X1), ok(X2)) → ok(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
and(mark(X1), X2) → mark(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
proper(tt) → ok(tt)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(isNatList(X)) → isNatList(proper(X))
proper(U12(X)) → U12(proper(X))
proper(U22(X)) → U22(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U32(X)) → U32(proper(X))
proper(U43(X)) → U43(proper(X))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U53(X)) → U53(proper(X))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U43(mark(X)) → mark(U43(X))
U32(ok(X)) → ok(U32(X))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X)) → ok(U22(X))
U22(mark(X)) → mark(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(nil) → ok(nil)
proper(U81(X)) → U81(proper(X))
proper(length(X)) → length(proper(X))
proper(s(X)) → s(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ 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:

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(U61(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U91(and(and(isNatIList(x2), isNatIListKind(x2)), and(and(isNat(x0), isNatKind(x0)), and(isNat(x1), isNatKind(x1)))), x2, x0, x1)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(take(0, x0))) → TOP(mark(U81(and(isNatIList(x0), isNatIListKind(x0)))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))

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.

TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(U61(and(isNatKind(x0), isNatIListKind(x1)), x0, x1)))

The remaining pairs can at least be oriented weakly.

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U91(and(and(isNatIList(x2), isNatIListKind(x2)), and(and(isNat(x0), isNatKind(x0)), and(isNat(x1), isNatKind(x1)))), x2, x0, x1)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(take(0, x0))) → TOP(mark(U81(and(isNatIList(x0), isNatIListKind(x0)))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U12(x₁)) = 0
POL(U21(x₁, x₂)) = 0
POL(U22(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 0
POL(U42(x₁, x₂)) = 0
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = 0
POL(U53(x₁)) = 0
POL(U61(x₁, x₂, x₃)) = 0
POL(U62(x₁, x₂)) = 0
POL(U63(x₁)) = 0
POL(U71(x₁, x₂)) = 0
POL(U81(x₁)) = 0
POL(U91(x₁, x₂, x₃, x₄)) = 0
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = 1
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = x₁
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 0
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

take(ok(X1), ok(X2)) → ok(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
and(mark(X1), X2) → mark(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
proper(tt) → ok(tt)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(isNatList(X)) → isNatList(proper(X))
proper(U12(X)) → U12(proper(X))
proper(U22(X)) → U22(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U32(X)) → U32(proper(X))
proper(U43(X)) → U43(proper(X))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U53(X)) → U53(proper(X))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U43(mark(X)) → mark(U43(X))
U32(ok(X)) → ok(U32(X))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X)) → ok(U22(X))
U22(mark(X)) → mark(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(nil) → ok(nil)
proper(U81(X)) → U81(proper(X))
proper(length(X)) → length(proper(X))
proper(s(X)) → s(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ 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:

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U91(and(and(isNatIList(x2), isNatIListKind(x2)), and(and(isNat(x0), isNatKind(x0)), and(isNat(x1), isNatKind(x1)))), x2, x0, x1)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(take(0, x0))) → TOP(mark(U81(and(isNatIList(x0), isNatIListKind(x0)))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))

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.

TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))

The remaining pairs can at least be oriented weakly.

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U91(and(and(isNatIList(x2), isNatIListKind(x2)), and(and(isNat(x0), isNatKind(x0)), and(isNat(x1), isNatKind(x1)))), x2, x0, x1)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(take(0, x0))) → TOP(mark(U81(and(isNatIList(x0), isNatIListKind(x0)))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁, x₂)) = 1
POL(U12(x₁)) = 1
POL(U21(x₁, x₂)) = 1
POL(U22(x₁)) = 1
POL(U31(x₁, x₂)) = 1
POL(U32(x₁)) = 1
POL(U41(x₁, x₂, x₃)) = 1
POL(U42(x₁, x₂)) = 1
POL(U43(x₁)) = 1
POL(U51(x₁, x₂, x₃)) = 1
POL(U52(x₁, x₂)) = 1
POL(U53(x₁)) = 1
POL(U61(x₁, x₂, x₃)) = 1
POL(U62(x₁, x₂)) = 1
POL(U63(x₁)) = 1
POL(U71(x₁, x₂)) = 1
POL(U81(x₁)) = 1
POL(U91(x₁, x₂, x₃, x₄)) = 1
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = 1
POL(cons(x₁, x₂)) = 1
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatIListKind(x₁)) = 1
POL(isNatKind(x₁)) = 1
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 1
POL(mark(x₁)) = 1
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(s(x₁)) = 1
POL(take(x₁, x₂)) = 1
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

take(ok(X1), ok(X2)) → ok(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
and(mark(X1), X2) → mark(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U43(mark(X)) → mark(U43(X))
U32(ok(X)) → ok(U32(X))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X)) → ok(U22(X))
U22(mark(X)) → mark(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ 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:

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U91(and(and(isNatIList(x2), isNatIListKind(x2)), and(and(isNat(x0), isNatKind(x0)), and(isNat(x1), isNatKind(x1)))), x2, x0, x1)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(take(0, x0))) → TOP(mark(U81(and(isNatIList(x0), isNatIListKind(x0)))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))

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.

TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U91(and(and(isNatIList(x2), isNatIListKind(x2)), and(and(isNat(x0), isNatKind(x0)), and(isNat(x1), isNatKind(x1)))), x2, x0, x1)))
TOP(ok(take(0, x0))) → TOP(mark(U81(and(isNatIList(x0), isNatIListKind(x0)))))

The remaining pairs can at least be oriented weakly.

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U12(x₁)) = 0
POL(U21(x₁, x₂)) = 0
POL(U22(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 0
POL(U42(x₁, x₂)) = 0
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = 0
POL(U53(x₁)) = 0
POL(U61(x₁, x₂, x₃)) = 0
POL(U62(x₁, x₂)) = 0
POL(U63(x₁)) = 0
POL(U71(x₁, x₂)) = 0
POL(U81(x₁)) = 0
POL(U91(x₁, x₂, x₃, x₄)) = 1
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = 1
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = x₁
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 1 + x₂
POL(tt) = 0
POL(zeros) = 1

The following usable rules [17] were oriented:

active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U81(X)) → U81(active(X))
active(take(X1, X2)) → take(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U63(X)) → U63(active(X))
active(length(X)) → length(active(X))
active(s(X)) → s(active(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
and(mark(X1), X2) → mark(and(X1, X2))
active(and(X1, X2)) → and(active(X1), X2)
take(mark(X1), X2) → mark(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(isNatKind(0)) → mark(tt)
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(nil)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(length(nil)) → mark(0)
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(nil)) → mark(tt)
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
proper(tt) → ok(tt)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(isNatList(X)) → isNatList(proper(X))
proper(U12(X)) → U12(proper(X))
proper(U22(X)) → U22(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U32(X)) → U32(proper(X))
proper(U43(X)) → U43(proper(X))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U53(X)) → U53(proper(X))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U43(mark(X)) → mark(U43(X))
U32(ok(X)) → ok(U32(X))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X)) → ok(U22(X))
U22(mark(X)) → mark(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(nil) → ok(nil)
proper(U81(X)) → U81(proper(X))
proper(length(X)) → length(proper(X))
proper(s(X)) → s(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(zeros) → mark(cons(0, zeros))
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ 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:

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))

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.

TOP(ok(isNatIListKind(take(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))

The remaining pairs can at least be oriented weakly.

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U12(x₁)) = 0
POL(U21(x₁, x₂)) = x₂
POL(U22(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 0
POL(U42(x₁, x₂)) = 0
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = x₃
POL(U52(x₁, x₂)) = 0
POL(U53(x₁)) = 0
POL(U61(x₁, x₂, x₃)) = x₃
POL(U62(x₁, x₂)) = x₂
POL(U63(x₁)) = 0
POL(U71(x₁, x₂)) = x₂
POL(U81(x₁)) = 0
POL(U91(x₁, x₂, x₃, x₄)) = 1 + x₂
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = 1 + x₁
POL(isNatIListKind(x₁)) = x₁
POL(isNatKind(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = x₁
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = 1 + x₂
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U81(X)) → U81(active(X))
active(take(X1, X2)) → take(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U63(X)) → U63(active(X))
active(length(X)) → length(active(X))
active(s(X)) → s(active(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
and(mark(X1), X2) → mark(and(X1, X2))
active(and(X1, X2)) → and(active(X1), X2)
take(mark(X1), X2) → mark(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(isNatKind(0)) → mark(tt)
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(nil)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(length(nil)) → mark(0)
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(nil)) → mark(tt)
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
proper(tt) → ok(tt)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(isNatList(X)) → isNatList(proper(X))
proper(U12(X)) → U12(proper(X))
proper(U22(X)) → U22(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U32(X)) → U32(proper(X))
proper(U43(X)) → U43(proper(X))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U53(X)) → U53(proper(X))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U43(mark(X)) → mark(U43(X))
U32(ok(X)) → ok(U32(X))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X)) → ok(U22(X))
U22(mark(X)) → mark(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(nil) → ok(nil)
proper(U81(X)) → U81(proper(X))
proper(length(X)) → length(proper(X))
proper(s(X)) → s(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(zeros) → mark(cons(0, zeros))
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ 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:

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))

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.

TOP(ok(length(cons(x0, x1)))) → TOP(mark(U71(and(and(isNatList(x1), isNatIListKind(x1)), and(isNat(x0), isNatKind(x0))), x1)))

The remaining pairs can at least be oriented weakly.

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 1
POL(TOP(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U12(x₁)) = 0
POL(U21(x₁, x₂)) = 0
POL(U22(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 0
POL(U42(x₁, x₂)) = 0
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = 0
POL(U53(x₁)) = 0
POL(U61(x₁, x₂, x₃)) = 0
POL(U62(x₁, x₂)) = 0
POL(U63(x₁)) = 0
POL(U71(x₁, x₂)) = 1
POL(U81(x₁)) = 0
POL(U91(x₁, x₂, x₃, x₄)) = 1
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = 1
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 1 + x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = x₁
POL(s(x₁)) = 1
POL(take(x₁, x₂)) = x₁ + x₂
POL(tt) = 0
POL(zeros) = 1

The following usable rules [17] were oriented:

active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U81(X)) → U81(active(X))
active(take(X1, X2)) → take(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U63(X)) → U63(active(X))
active(length(X)) → length(active(X))
active(s(X)) → s(active(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
and(mark(X1), X2) → mark(and(X1, X2))
active(and(X1, X2)) → and(active(X1), X2)
take(mark(X1), X2) → mark(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(isNatKind(0)) → mark(tt)
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(nil)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(length(nil)) → mark(0)
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(nil)) → mark(tt)
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
proper(tt) → ok(tt)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(isNatList(X)) → isNatList(proper(X))
proper(U12(X)) → U12(proper(X))
proper(U22(X)) → U22(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U32(X)) → U32(proper(X))
proper(U43(X)) → U43(proper(X))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U53(X)) → U53(proper(X))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U43(mark(X)) → mark(U43(X))
U32(ok(X)) → ok(U32(X))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X)) → ok(U22(X))
U22(mark(X)) → mark(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(nil) → ok(nil)
proper(U81(X)) → U81(proper(X))
proper(length(X)) → length(proper(X))
proper(s(X)) → s(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(zeros) → mark(cons(0, zeros))
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ 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:

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))

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.

TOP(ok(U52(tt, x0))) → TOP(mark(U53(isNatList(x0))))

The remaining pairs can at least be oriented weakly.

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U12(x₁)) = 0
POL(U21(x₁, x₂)) = 0
POL(U22(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 0
POL(U42(x₁, x₂)) = 0
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 1
POL(U52(x₁, x₂)) = 1
POL(U53(x₁)) = 0
POL(U61(x₁, x₂, x₃)) = 0
POL(U62(x₁, x₂)) = 0
POL(U63(x₁)) = 0
POL(U71(x₁, x₂)) = 0
POL(U81(x₁)) = 0
POL(U91(x₁, x₂, x₃, x₄)) = 0
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = x₁
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 0
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

take(ok(X1), ok(X2)) → ok(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
and(mark(X1), X2) → mark(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
proper(tt) → ok(tt)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(isNatList(X)) → isNatList(proper(X))
proper(U12(X)) → U12(proper(X))
proper(U22(X)) → U22(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U32(X)) → U32(proper(X))
proper(U43(X)) → U43(proper(X))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U53(X)) → U53(proper(X))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U43(mark(X)) → mark(U43(X))
U32(ok(X)) → ok(U32(X))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X)) → ok(U22(X))
U22(mark(X)) → mark(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(nil) → ok(nil)
proper(U81(X)) → U81(proper(X))
proper(length(X)) → length(proper(X))
proper(s(X)) → s(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ 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:

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))

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.

TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x0), x1)))
TOP(ok(isNatIListKind(cons(x0, x1)))) → TOP(mark(and(isNatKind(x0), isNatIListKind(x1))))

The remaining pairs can at least be oriented weakly.

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U12(x₁)) = 0
POL(U21(x₁, x₂)) = 0
POL(U22(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 0
POL(U42(x₁, x₂)) = 0
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 1
POL(U52(x₁, x₂)) = 0
POL(U53(x₁)) = 1
POL(U61(x₁, x₂, x₃)) = 0
POL(U62(x₁, x₂)) = 0
POL(U63(x₁)) = 0
POL(U71(x₁, x₂)) = 0
POL(U81(x₁)) = 0
POL(U91(x₁, x₂, x₃, x₄)) = 1
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = 1 + x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatIListKind(x₁)) = x₁
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = x₁
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 0
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

take(ok(X1), ok(X2)) → ok(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
and(mark(X1), X2) → mark(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
proper(tt) → ok(tt)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(isNatList(X)) → isNatList(proper(X))
proper(U12(X)) → U12(proper(X))
proper(U22(X)) → U22(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U32(X)) → U32(proper(X))
proper(U43(X)) → U43(proper(X))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U53(X)) → U53(proper(X))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U43(mark(X)) → mark(U43(X))
U32(ok(X)) → ok(U32(X))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X)) → ok(U22(X))
U22(mark(X)) → mark(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(nil) → ok(nil)
proper(U81(X)) → U81(proper(X))
proper(length(X)) → length(proper(X))
proper(s(X)) → s(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ 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:

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))

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.

TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNatList(x0))))

The remaining pairs can at least be oriented weakly.

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁, x₂)) = 1
POL(U12(x₁)) = 0
POL(U21(x₁, x₂)) = 0
POL(U22(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 0
POL(U42(x₁, x₂)) = 0
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = 0
POL(U53(x₁)) = 0
POL(U61(x₁, x₂, x₃)) = 0
POL(U62(x₁, x₂)) = 0
POL(U63(x₁)) = 0
POL(U71(x₁, x₂)) = 0
POL(U81(x₁)) = 0
POL(U91(x₁, x₂, x₃, x₄)) = 0
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = x₁
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 0
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

take(ok(X1), ok(X2)) → ok(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
and(mark(X1), X2) → mark(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
proper(tt) → ok(tt)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(isNatList(X)) → isNatList(proper(X))
proper(U12(X)) → U12(proper(X))
proper(U22(X)) → U22(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U32(X)) → U32(proper(X))
proper(U43(X)) → U43(proper(X))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U53(X)) → U53(proper(X))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U43(mark(X)) → mark(U43(X))
U32(ok(X)) → ok(U32(X))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X)) → ok(U22(X))
U22(mark(X)) → mark(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(nil) → ok(nil)
proper(U81(X)) → U81(proper(X))
proper(length(X)) → length(proper(X))
proper(s(X)) → s(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ 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:

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))

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.

TOP(mark(isNatIListKind(x0))) → TOP(isNatIListKind(proper(x0)))

The remaining pairs can at least be oriented weakly.

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁, x₂)) = 1
POL(U12(x₁)) = 1
POL(U21(x₁, x₂)) = 1
POL(U22(x₁)) = 1
POL(U31(x₁, x₂)) = 1
POL(U32(x₁)) = 1
POL(U41(x₁, x₂, x₃)) = 1
POL(U42(x₁, x₂)) = 1
POL(U43(x₁)) = 1
POL(U51(x₁, x₂, x₃)) = 1
POL(U52(x₁, x₂)) = 1
POL(U53(x₁)) = 1
POL(U61(x₁, x₂, x₃)) = 1
POL(U62(x₁, x₂)) = 1
POL(U63(x₁)) = 1
POL(U71(x₁, x₂)) = 1
POL(U81(x₁)) = 1
POL(U91(x₁, x₂, x₃, x₄)) = 1
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = 1
POL(cons(x₁, x₂)) = 1
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 1
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 1
POL(mark(x₁)) = 1
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(s(x₁)) = 1
POL(take(x₁, x₂)) = 1
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

take(ok(X1), ok(X2)) → ok(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
and(mark(X1), X2) → mark(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U43(mark(X)) → mark(U43(X))
U32(ok(X)) → ok(U32(X))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X)) → ok(U22(X))
U22(mark(X)) → mark(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ 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:

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))

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.

TOP(ok(and(tt, x0))) → TOP(mark(x0))

The remaining pairs can at least be oriented weakly.

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U12(x₁)) = 0
POL(U21(x₁, x₂)) = 0
POL(U22(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 0
POL(U42(x₁, x₂)) = 0
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = 0
POL(U53(x₁)) = 0
POL(U61(x₁, x₂, x₃)) = 0
POL(U62(x₁, x₂)) = 0
POL(U63(x₁)) = 0
POL(U71(x₁, x₂)) = 0
POL(U81(x₁)) = 0
POL(U91(x₁, x₂, x₃, x₄)) = 0
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = 1 + x₂
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = x₁
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 0
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

take(ok(X1), ok(X2)) → ok(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
and(mark(X1), X2) → mark(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
proper(tt) → ok(tt)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(isNatList(X)) → isNatList(proper(X))
proper(U12(X)) → U12(proper(X))
proper(U22(X)) → U22(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U32(X)) → U32(proper(X))
proper(U43(X)) → U43(proper(X))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U53(X)) → U53(proper(X))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U43(mark(X)) → mark(U43(X))
U32(ok(X)) → ok(U32(X))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X)) → ok(U22(X))
U22(mark(X)) → mark(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(nil) → ok(nil)
proper(U81(X)) → U81(proper(X))
proper(length(X)) → length(proper(X))
proper(s(X)) → s(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ 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:

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))

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.

TOP(ok(U91(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))

The remaining pairs can at least be oriented weakly.

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U12(x₁)) = 0
POL(U21(x₁, x₂)) = 0
POL(U22(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 0
POL(U42(x₁, x₂)) = 0
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = 0
POL(U53(x₁)) = 0
POL(U61(x₁, x₂, x₃)) = 0
POL(U62(x₁, x₂)) = 0
POL(U63(x₁)) = 0
POL(U71(x₁, x₂)) = 0
POL(U81(x₁)) = 0
POL(U91(x₁, x₂, x₃, x₄)) = 1
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = 0
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 0
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

take(ok(X1), ok(X2)) → ok(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
and(mark(X1), X2) → mark(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U43(mark(X)) → mark(U43(X))
U32(ok(X)) → ok(U32(X))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X)) → ok(U22(X))
U22(mark(X)) → mark(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ 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:

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))

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.

TOP(ok(U42(tt, x0))) → TOP(mark(U43(isNatIList(x0))))

The remaining pairs can at least be oriented weakly.

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U12(x₁)) = 0
POL(U21(x₁, x₂)) = 0
POL(U22(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 0
POL(U42(x₁, x₂)) = 1
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = 0
POL(U53(x₁)) = 0
POL(U61(x₁, x₂, x₃)) = 0
POL(U62(x₁, x₂)) = 0
POL(U63(x₁)) = 0
POL(U71(x₁, x₂)) = 0
POL(U81(x₁)) = 0
POL(U91(x₁, x₂, x₃, x₄)) = 0
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = 0
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 0
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

take(ok(X1), ok(X2)) → ok(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
and(mark(X1), X2) → mark(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U43(mark(X)) → mark(U43(X))
U32(ok(X)) → ok(U32(X))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X)) → ok(U22(X))
U22(mark(X)) → mark(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ 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:

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))

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.

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x0), x1)))

The remaining pairs can at least be oriented weakly.

TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U12(x₁)) = 0
POL(U21(x₁, x₂)) = 0
POL(U22(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 0
POL(U42(x₁, x₂)) = 0
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = 0
POL(U53(x₁)) = 0
POL(U61(x₁, x₂, x₃)) = 1
POL(U62(x₁, x₂)) = 0
POL(U63(x₁)) = 0
POL(U71(x₁, x₂)) = 0
POL(U81(x₁)) = 0
POL(U91(x₁, x₂, x₃, x₄)) = 0
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = 0
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 0
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

take(ok(X1), ok(X2)) → ok(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
and(mark(X1), X2) → mark(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U43(mark(X)) → mark(U43(X))
U32(ok(X)) → ok(U32(X))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X)) → ok(U22(X))
U22(mark(X)) → mark(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))

↳ 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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ 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

Q DP problem:
The TRS P consists of the following rules:

TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U22(x0))) → TOP(U22(active(x0)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(U81(x0))) → TOP(U81(active(x0)))
TOP(mark(U91(x0, x1, x2, x3))) → TOP(U91(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U63(x0))) → TOP(U63(proper(x0)))
TOP(ok(U63(x0))) → TOP(U63(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U42(x0, x1))) → TOP(U42(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(U53(x0))) → TOP(U53(proper(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(isNatKind(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1))) → TOP(U42(active(x0), x1))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U81(x0))) → TOP(U81(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U53(x0))) → TOP(U53(active(x0)))
TOP(ok(U43(x0))) → TOP(U43(active(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(x0, x1))) → TOP(U71(active(x0), x1))
TOP(ok(U91(x0, x1, x2, x3))) → TOP(U91(active(x0), x1, x2, x3))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U22(x0))) → TOP(U22(proper(x0)))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(mark(U71(x0, x1))) → TOP(U71(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U43(x0))) → TOP(U43(proper(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U81(mark(X)) → mark(U81(X))
U81(ok(X)) → ok(U81(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U63(mark(X)) → mark(U63(X))
U63(ok(X)) → ok(U63(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U53(mark(X)) → mark(U53(X))
U53(ok(X)) → ok(U53(X))
U52(mark(X1), X2) → mark(U52(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U43(mark(X)) → mark(U43(X))
U43(ok(X)) → ok(U43(X))
U42(mark(X1), X2) → mark(U42(X1, X2))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U22(mark(X)) → mark(U22(X))
U22(ok(X)) → ok(U22(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNat(ok(X)) → ok(isNat(X))
isNatKind(ok(X)) → ok(isNatKind(X))
isNatList(ok(X)) → ok(isNatList(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.