Termination w.r.t. Q proof of /tmp/tmp5X6F_y/Ex4_DLMMU04

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(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

Q restricted rewrite system:
The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

ACTIVE(uTake2(tt, M, N, IL)) → MARK(cons(N, take(M, IL)))
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))
ACTIVE(isNat(length(L))) → MARK(isNatList(L))
CONS(X1, mark(X2)) → CONS(X1, X2)
ACTIVE(length(cons(N, L))) → AND(isNat(N), isNatList(L))
ACTIVE(isNatIList(zeros)) → MARK(tt)
MARK(length(X)) → MARK(X)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
ACTIVE(isNat(s(N))) → MARK(isNat(N))
ACTIVE(take(s(M), cons(N, IL))) → ISNAT(N)
MARK(take(X1, X2)) → TAKE(mark(X1), mark(X2))
ACTIVE(uLength(tt, L)) → LENGTH(L)
UTAKE2(active(X1), X2, X3, X4) → UTAKE2(X1, X2, X3, X4)
MARK(uTake2(X1, X2, X3, X4)) → ACTIVE(uTake2(mark(X1), X2, X3, X4))
ACTIVE(take(s(M), cons(N, IL))) → MARK(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
UTAKE1(mark(X)) → UTAKE1(X)
TAKE(active(X1), X2) → TAKE(X1, X2)
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
ACTIVE(take(s(M), cons(N, IL))) → AND(isNat(M), and(isNat(N), isNatIList(IL)))
MARK(uTake1(X)) → MARK(X)
ISNATILIST(mark(X)) → ISNATILIST(X)
ACTIVE(and(tt, T)) → MARK(T)
UTAKE2(X1, X2, active(X3), X4) → UTAKE2(X1, X2, X3, X4)
ULENGTH(active(X1), X2) → ULENGTH(X1, X2)
MARK(take(X1, X2)) → MARK(X2)
CONS(mark(X1), X2) → CONS(X1, X2)
ACTIVE(take(0, IL)) → UTAKE1(isNatIList(IL))
UTAKE2(X1, X2, X3, active(X4)) → UTAKE2(X1, X2, X3, X4)
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
UTAKE2(mark(X1), X2, X3, X4) → UTAKE2(X1, X2, X3, X4)
ACTIVE(isNat(0)) → MARK(tt)
UTAKE2(X1, mark(X2), X3, X4) → UTAKE2(X1, X2, X3, X4)
ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
CONS(X1, active(X2)) → CONS(X1, X2)
ULENGTH(mark(X1), X2) → ULENGTH(X1, X2)
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(take(s(M), cons(N, IL))) → AND(isNat(N), isNatIList(IL))
LENGTH(mark(X)) → LENGTH(X)
UTAKE2(X1, X2, mark(X3), X4) → UTAKE2(X1, X2, X3, X4)
ACTIVE(take(s(M), cons(N, IL))) → UTAKE2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)
LENGTH(active(X)) → LENGTH(X)
ACTIVE(isNatList(take(N, IL))) → ISNATILIST(IL)
S(mark(X)) → S(X)
ACTIVE(uTake2(tt, M, N, IL)) → CONS(N, take(M, IL))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ISNAT(mark(X)) → ISNAT(X)
ACTIVE(uLength(tt, L)) → S(length(L))
ACTIVE(isNatList(cons(N, L))) → ISNATLIST(L)
ACTIVE(isNat(s(N))) → ISNAT(N)
ACTIVE(isNatList(nil)) → MARK(tt)
AND(mark(X1), X2) → AND(X1, X2)
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
UTAKE2(X1, active(X2), X3, X4) → UTAKE2(X1, X2, X3, X4)
MARK(tt) → ACTIVE(tt)
MARK(isNat(X)) → ACTIVE(isNat(X))
TAKE(X1, active(X2)) → TAKE(X1, X2)
ACTIVE(take(0, IL)) → MARK(uTake1(isNatIList(IL)))
MARK(cons(X1, X2)) → MARK(X1)
UTAKE2(X1, X2, X3, mark(X4)) → UTAKE2(X1, X2, X3, X4)
AND(X1, mark(X2)) → AND(X1, X2)
MARK(uTake1(X)) → UTAKE1(mark(X))
ACTIVE(isNatList(take(N, IL))) → AND(isNat(N), isNatIList(IL))
MARK(uTake2(X1, X2, X3, X4)) → MARK(X1)
ACTIVE(isNatIList(IL)) → ISNATLIST(IL)
ACTIVE(zeros) → CONS(0, zeros)
ACTIVE(isNatList(take(N, IL))) → ISNAT(N)
MARK(take(X1, X2)) → MARK(X1)
ACTIVE(uTake2(tt, M, N, IL)) → TAKE(M, IL)
ACTIVE(isNatIList(cons(N, IL))) → ISNAT(N)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(uLength(X1, X2)) → ULENGTH(mark(X1), X2)
S(active(X)) → S(X)
MARK(cons(X1, X2)) → CONS(mark(X1), X2)
MARK(uTake1(X)) → ACTIVE(uTake1(mark(X)))
ACTIVE(isNatIList(cons(N, IL))) → ISNATILIST(IL)
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatList(cons(N, L))) → ISNAT(N)
AND(X1, active(X2)) → AND(X1, X2)
ACTIVE(isNatIList(cons(N, IL))) → AND(isNat(N), isNatIList(IL))
ACTIVE(take(0, IL)) → ISNATILIST(IL)
MARK(zeros) → ACTIVE(zeros)
MARK(length(X)) → LENGTH(mark(X))
ACTIVE(length(cons(N, L))) → ULENGTH(and(isNat(N), isNatList(L)), L)
MARK(uTake2(X1, X2, X3, X4)) → UTAKE2(mark(X1), X2, X3, X4)
TAKE(mark(X1), X2) → TAKE(X1, X2)
AND(active(X1), X2) → AND(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
ISNATLIST(active(X)) → ISNATLIST(X)
UTAKE1(active(X)) → UTAKE1(X)
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
ULENGTH(X1, active(X2)) → ULENGTH(X1, X2)
ACTIVE(isNat(length(L))) → ISNATLIST(L)
ISNATLIST(mark(X)) → ISNATLIST(X)
ACTIVE(take(s(M), cons(N, IL))) → ISNATILIST(IL)
ULENGTH(X1, mark(X2)) → ULENGTH(X1, X2)
MARK(and(X1, X2)) → MARK(X2)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(s(X)) → S(mark(X))
MARK(and(X1, X2)) → AND(mark(X1), mark(X2))
ACTIVE(length(cons(N, L))) → ISNATLIST(L)
ACTIVE(take(s(M), cons(N, IL))) → ISNAT(M)
ISNAT(active(X)) → ISNAT(X)
ACTIVE(isNatList(cons(N, L))) → AND(isNat(N), isNatList(L))
MARK(uLength(X1, X2)) → MARK(X1)
TAKE(X1, mark(X2)) → TAKE(X1, X2)
ACTIVE(length(cons(N, L))) → ISNAT(N)
ACTIVE(uTake1(tt)) → MARK(nil)
ACTIVE(zeros) → MARK(cons(0, zeros))
MARK(0) → ACTIVE(0)
ISNATILIST(active(X)) → ISNATILIST(X)
MARK(nil) → ACTIVE(nil)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

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

ACTIVE(uTake2(tt, M, N, IL)) → MARK(cons(N, take(M, IL)))
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))
ACTIVE(isNat(length(L))) → MARK(isNatList(L))
CONS(X1, mark(X2)) → CONS(X1, X2)
ACTIVE(length(cons(N, L))) → AND(isNat(N), isNatList(L))
ACTIVE(isNatIList(zeros)) → MARK(tt)
MARK(length(X)) → MARK(X)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
ACTIVE(isNat(s(N))) → MARK(isNat(N))
ACTIVE(take(s(M), cons(N, IL))) → ISNAT(N)
MARK(take(X1, X2)) → TAKE(mark(X1), mark(X2))
ACTIVE(uLength(tt, L)) → LENGTH(L)
UTAKE2(active(X1), X2, X3, X4) → UTAKE2(X1, X2, X3, X4)
MARK(uTake2(X1, X2, X3, X4)) → ACTIVE(uTake2(mark(X1), X2, X3, X4))
ACTIVE(take(s(M), cons(N, IL))) → MARK(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
UTAKE1(mark(X)) → UTAKE1(X)
TAKE(active(X1), X2) → TAKE(X1, X2)
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
ACTIVE(take(s(M), cons(N, IL))) → AND(isNat(M), and(isNat(N), isNatIList(IL)))
MARK(uTake1(X)) → MARK(X)
ISNATILIST(mark(X)) → ISNATILIST(X)
ACTIVE(and(tt, T)) → MARK(T)
UTAKE2(X1, X2, active(X3), X4) → UTAKE2(X1, X2, X3, X4)
ULENGTH(active(X1), X2) → ULENGTH(X1, X2)
MARK(take(X1, X2)) → MARK(X2)
CONS(mark(X1), X2) → CONS(X1, X2)
ACTIVE(take(0, IL)) → UTAKE1(isNatIList(IL))
UTAKE2(X1, X2, X3, active(X4)) → UTAKE2(X1, X2, X3, X4)
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
UTAKE2(mark(X1), X2, X3, X4) → UTAKE2(X1, X2, X3, X4)
ACTIVE(isNat(0)) → MARK(tt)
UTAKE2(X1, mark(X2), X3, X4) → UTAKE2(X1, X2, X3, X4)
ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
CONS(X1, active(X2)) → CONS(X1, X2)
ULENGTH(mark(X1), X2) → ULENGTH(X1, X2)
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(take(s(M), cons(N, IL))) → AND(isNat(N), isNatIList(IL))
LENGTH(mark(X)) → LENGTH(X)
UTAKE2(X1, X2, mark(X3), X4) → UTAKE2(X1, X2, X3, X4)
ACTIVE(take(s(M), cons(N, IL))) → UTAKE2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)
LENGTH(active(X)) → LENGTH(X)
ACTIVE(isNatList(take(N, IL))) → ISNATILIST(IL)
S(mark(X)) → S(X)
ACTIVE(uTake2(tt, M, N, IL)) → CONS(N, take(M, IL))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ISNAT(mark(X)) → ISNAT(X)
ACTIVE(uLength(tt, L)) → S(length(L))
ACTIVE(isNatList(cons(N, L))) → ISNATLIST(L)
ACTIVE(isNat(s(N))) → ISNAT(N)
ACTIVE(isNatList(nil)) → MARK(tt)
AND(mark(X1), X2) → AND(X1, X2)
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
UTAKE2(X1, active(X2), X3, X4) → UTAKE2(X1, X2, X3, X4)
MARK(tt) → ACTIVE(tt)
MARK(isNat(X)) → ACTIVE(isNat(X))
TAKE(X1, active(X2)) → TAKE(X1, X2)
ACTIVE(take(0, IL)) → MARK(uTake1(isNatIList(IL)))
MARK(cons(X1, X2)) → MARK(X1)
UTAKE2(X1, X2, X3, mark(X4)) → UTAKE2(X1, X2, X3, X4)
AND(X1, mark(X2)) → AND(X1, X2)
MARK(uTake1(X)) → UTAKE1(mark(X))
ACTIVE(isNatList(take(N, IL))) → AND(isNat(N), isNatIList(IL))
MARK(uTake2(X1, X2, X3, X4)) → MARK(X1)
ACTIVE(isNatIList(IL)) → ISNATLIST(IL)
ACTIVE(zeros) → CONS(0, zeros)
ACTIVE(isNatList(take(N, IL))) → ISNAT(N)
MARK(take(X1, X2)) → MARK(X1)
ACTIVE(uTake2(tt, M, N, IL)) → TAKE(M, IL)
ACTIVE(isNatIList(cons(N, IL))) → ISNAT(N)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(uLength(X1, X2)) → ULENGTH(mark(X1), X2)
S(active(X)) → S(X)
MARK(cons(X1, X2)) → CONS(mark(X1), X2)
MARK(uTake1(X)) → ACTIVE(uTake1(mark(X)))
ACTIVE(isNatIList(cons(N, IL))) → ISNATILIST(IL)
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatList(cons(N, L))) → ISNAT(N)
AND(X1, active(X2)) → AND(X1, X2)
ACTIVE(isNatIList(cons(N, IL))) → AND(isNat(N), isNatIList(IL))
ACTIVE(take(0, IL)) → ISNATILIST(IL)
MARK(zeros) → ACTIVE(zeros)
MARK(length(X)) → LENGTH(mark(X))
ACTIVE(length(cons(N, L))) → ULENGTH(and(isNat(N), isNatList(L)), L)
MARK(uTake2(X1, X2, X3, X4)) → UTAKE2(mark(X1), X2, X3, X4)
TAKE(mark(X1), X2) → TAKE(X1, X2)
AND(active(X1), X2) → AND(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
ISNATLIST(active(X)) → ISNATLIST(X)
UTAKE1(active(X)) → UTAKE1(X)
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
ULENGTH(X1, active(X2)) → ULENGTH(X1, X2)
ACTIVE(isNat(length(L))) → ISNATLIST(L)
ISNATLIST(mark(X)) → ISNATLIST(X)
ACTIVE(take(s(M), cons(N, IL))) → ISNATILIST(IL)
ULENGTH(X1, mark(X2)) → ULENGTH(X1, X2)
MARK(and(X1, X2)) → MARK(X2)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(s(X)) → S(mark(X))
MARK(and(X1, X2)) → AND(mark(X1), mark(X2))
ACTIVE(length(cons(N, L))) → ISNATLIST(L)
ACTIVE(take(s(M), cons(N, IL))) → ISNAT(M)
ISNAT(active(X)) → ISNAT(X)
ACTIVE(isNatList(cons(N, L))) → AND(isNat(N), isNatList(L))
MARK(uLength(X1, X2)) → MARK(X1)
TAKE(X1, mark(X2)) → TAKE(X1, X2)
ACTIVE(length(cons(N, L))) → ISNAT(N)
ACTIVE(uTake1(tt)) → MARK(nil)
ACTIVE(zeros) → MARK(cons(0, zeros))
MARK(0) → ACTIVE(0)
ISNATILIST(active(X)) → ISNATILIST(X)
MARK(nil) → ACTIVE(nil)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 12 SCCs with 44 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ULENGTH(active(X1), X2) → ULENGTH(X1, X2)
ULENGTH(mark(X1), X2) → ULENGTH(X1, X2)
ULENGTH(X1, mark(X2)) → ULENGTH(X1, X2)
ULENGTH(X1, active(X2)) → ULENGTH(X1, X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ULENGTH(active(X1), X2) → ULENGTH(X1, X2)
ULENGTH(mark(X1), X2) → ULENGTH(X1, X2)
ULENGTH(X1, mark(X2)) → ULENGTH(X1, X2)
ULENGTH(X1, active(X2)) → ULENGTH(X1, X2)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

UTAKE2(X1, active(X2), X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, mark(X2), X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, X2, X3, active(X4)) → UTAKE2(X1, X2, X3, X4)
UTAKE2(active(X1), X2, X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, X2, X3, mark(X4)) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, X2, mark(X3), X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(mark(X1), X2, X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, X2, active(X3), X4) → UTAKE2(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ 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:

UTAKE2(X1, active(X2), X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, mark(X2), X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(active(X1), X2, X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, X2, X3, active(X4)) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, X2, X3, mark(X4)) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, X2, mark(X3), X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(mark(X1), X2, X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, X2, active(X3), X4) → UTAKE2(X1, X2, X3, X4)

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

UTAKE2(X1, active(X2), X3, X4) → UTAKE2(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4
UTAKE2(X1, mark(X2), X3, X4) → UTAKE2(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4
UTAKE2(X1, X2, X3, active(X4)) → UTAKE2(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4
UTAKE2(active(X1), X2, X3, X4) → UTAKE2(X1, X2, X3, X4)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4
UTAKE2(X1, X2, X3, mark(X4)) → UTAKE2(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4
UTAKE2(X1, X2, mark(X3), X4) → UTAKE2(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4
UTAKE2(mark(X1), X2, X3, X4) → UTAKE2(X1, X2, X3, X4)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4
UTAKE2(X1, X2, active(X3), X4) → UTAKE2(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

UTAKE1(mark(X)) → UTAKE1(X)
UTAKE1(active(X)) → UTAKE1(X)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ 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:

UTAKE1(mark(X)) → UTAKE1(X)
UTAKE1(active(X)) → UTAKE1(X)

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

UTAKE1(mark(X)) → UTAKE1(X)
The graph contains the following edges 1 > 1
UTAKE1(active(X)) → UTAKE1(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

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

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

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ 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:

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

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ 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:

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

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

LENGTH(mark(X)) → LENGTH(X)
LENGTH(active(X)) → LENGTH(X)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ 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:

LENGTH(mark(X)) → LENGTH(X)
LENGTH(active(X)) → LENGTH(X)

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

S(mark(X)) → S(X)
S(active(X)) → S(X)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ 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:

S(active(X)) → S(X)
S(mark(X)) → S(X)

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ISNAT(active(X)) → ISNAT(X)
ISNAT(mark(X)) → ISNAT(X)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ 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:

ISNAT(active(X)) → ISNAT(X)
ISNAT(mark(X)) → ISNAT(X)

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ISNATLIST(mark(X)) → ISNATLIST(X)
ISNATLIST(active(X)) → ISNATLIST(X)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ 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:

ISNATLIST(mark(X)) → ISNATLIST(X)
ISNATLIST(active(X)) → ISNATLIST(X)

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

ISNATLIST(mark(X)) → ISNATLIST(X)
The graph contains the following edges 1 > 1
ISNATLIST(active(X)) → ISNATLIST(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

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

ISNATILIST(mark(X)) → ISNATILIST(X)
ISNATILIST(active(X)) → ISNATILIST(X)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ 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:

ISNATILIST(mark(X)) → ISNATILIST(X)
ISNATILIST(active(X)) → ISNATILIST(X)

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

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

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

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ 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:

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

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

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

ACTIVE(uTake2(tt, M, N, IL)) → MARK(cons(N, take(M, IL)))
MARK(take(X1, X2)) → MARK(X2)
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNat(length(L))) → MARK(isNatList(L))
ACTIVE(take(0, IL)) → MARK(uTake1(isNatIList(IL)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(length(X)) → MARK(X)
ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(uTake2(X1, X2, X3, X4)) → MARK(X1)
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(N))) → MARK(isNat(N))
MARK(take(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(uTake2(X1, X2, X3, X4)) → ACTIVE(uTake2(mark(X1), X2, X3, X4))
ACTIVE(take(s(M), cons(N, IL))) → MARK(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(uTake1(X)) → ACTIVE(uTake1(mark(X)))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(uLength(X1, X2)) → MARK(X1)
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(uTake1(X)) → MARK(X)
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
MARK(zeros) → ACTIVE(zeros)
ACTIVE(zeros) → MARK(cons(0, zeros))
ACTIVE(and(tt, T)) → MARK(T)
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(uTake1(X)) → ACTIVE(uTake1(mark(X)))

The remaining pairs can at least be oriented weakly.

ACTIVE(uTake2(tt, M, N, IL)) → MARK(cons(N, take(M, IL)))
MARK(take(X1, X2)) → MARK(X2)
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNat(length(L))) → MARK(isNatList(L))
ACTIVE(take(0, IL)) → MARK(uTake1(isNatIList(IL)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(length(X)) → MARK(X)
ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(uTake2(X1, X2, X3, X4)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(N))) → MARK(isNat(N))
MARK(take(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(uTake2(X1, X2, X3, X4)) → ACTIVE(uTake2(mark(X1), X2, X3, X4))
ACTIVE(take(s(M), cons(N, IL))) → MARK(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(uLength(X1, X2)) → MARK(X1)
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(uTake1(X)) → MARK(X)
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
MARK(zeros) → ACTIVE(zeros)
ACTIVE(zeros) → MARK(cons(0, zeros))
ACTIVE(and(tt, T)) → MARK(T)
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = 1
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = 1
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 1
POL(isNatIList(x₁)) = 1
POL(isNatList(x₁)) = 1
POL(length(x₁)) = 1
POL(mark(x₁)) = 0
POL(nil) = 0
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 1
POL(tt) = 0
POL(uLength(x₁, x₂)) = 1
POL(uTake1(x₁)) = 0
POL(uTake2(x₁, x₂, x₃, x₄)) = 1
POL(zeros) = 1

The following usable rules [17] were oriented:

uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

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

ACTIVE(uTake2(tt, M, N, IL)) → MARK(cons(N, take(M, IL)))
MARK(take(X1, X2)) → MARK(X2)
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNat(length(L))) → MARK(isNatList(L))
ACTIVE(take(0, IL)) → MARK(uTake1(isNatIList(IL)))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(length(X)) → MARK(X)
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
MARK(uTake2(X1, X2, X3, X4)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(N))) → MARK(isNat(N))
MARK(take(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(uTake2(X1, X2, X3, X4)) → ACTIVE(uTake2(mark(X1), X2, X3, X4))
ACTIVE(take(s(M), cons(N, IL))) → MARK(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(uLength(X1, X2)) → MARK(X1)
MARK(uTake1(X)) → MARK(X)
MARK(zeros) → ACTIVE(zeros)
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
ACTIVE(zeros) → MARK(cons(0, zeros))
ACTIVE(and(tt, T)) → MARK(T)
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(take(X1, X2)) → MARK(X2)
MARK(uTake2(X1, X2, X3, X4)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X1)
MARK(uTake1(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.

ACTIVE(uTake2(tt, M, N, IL)) → MARK(cons(N, take(M, IL)))
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNat(length(L))) → MARK(isNatList(L))
ACTIVE(take(0, IL)) → MARK(uTake1(isNatIList(IL)))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(length(X)) → MARK(X)
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(N))) → MARK(isNat(N))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(uTake2(X1, X2, X3, X4)) → ACTIVE(uTake2(mark(X1), X2, X3, X4))
ACTIVE(take(s(M), cons(N, IL))) → MARK(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(uLength(X1, X2)) → MARK(X1)
MARK(zeros) → ACTIVE(zeros)
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
ACTIVE(zeros) → MARK(cons(0, zeros))
ACTIVE(and(tt, T)) → MARK(T)
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = 1 + x₁ + x₂
POL(tt) = 0
POL(uLength(x₁, x₂)) = x₁ + x₂
POL(uTake1(x₁)) = 1 + x₁
POL(uTake2(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₂ + x₃ + x₄
POL(zeros) = 0

The following usable rules [17] were oriented:

uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
active(isNat(0)) → mark(tt)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
mark(tt) → active(tt)
active(isNatIList(zeros)) → mark(tt)
active(isNatList(nil)) → mark(tt)
active(uTake1(tt)) → mark(nil)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
mark(0) → active(0)
mark(nil) → active(nil)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

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

ACTIVE(uTake2(tt, M, N, IL)) → MARK(cons(N, take(M, IL)))
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNat(length(L))) → MARK(isNatList(L))
ACTIVE(take(0, IL)) → MARK(uTake1(isNatIList(IL)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(length(X)) → MARK(X)
ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(N))) → MARK(isNat(N))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(uTake2(X1, X2, X3, X4)) → ACTIVE(uTake2(mark(X1), X2, X3, X4))
ACTIVE(take(s(M), cons(N, IL))) → MARK(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(uLength(X1, X2)) → MARK(X1)
MARK(zeros) → ACTIVE(zeros)
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
ACTIVE(zeros) → MARK(cons(0, zeros))
ACTIVE(and(tt, T)) → MARK(T)
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

ACTIVE(take(0, IL)) → MARK(uTake1(isNatIList(IL)))

The remaining pairs can at least be oriented weakly.

ACTIVE(uTake2(tt, M, N, IL)) → MARK(cons(N, take(M, IL)))
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNat(length(L))) → MARK(isNatList(L))
MARK(cons(X1, X2)) → MARK(X1)
MARK(length(X)) → MARK(X)
ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(N))) → MARK(isNat(N))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(uTake2(X1, X2, X3, X4)) → ACTIVE(uTake2(mark(X1), X2, X3, X4))
ACTIVE(take(s(M), cons(N, IL))) → MARK(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(uLength(X1, X2)) → MARK(X1)
MARK(zeros) → ACTIVE(zeros)
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
ACTIVE(zeros) → MARK(cons(0, zeros))
ACTIVE(and(tt, T)) → MARK(T)
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = 1 + x₂
POL(tt) = 0
POL(uLength(x₁, x₂)) = x₁ + x₂
POL(uTake1(x₁)) = 0
POL(uTake2(x₁, x₂, x₃, x₄)) = 1 + x₃ + x₄
POL(zeros) = 0

The following usable rules [17] were oriented:

uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
active(isNat(0)) → mark(tt)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
mark(tt) → active(tt)
active(isNatIList(zeros)) → mark(tt)
active(isNatList(nil)) → mark(tt)
active(uTake1(tt)) → mark(nil)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
mark(0) → active(0)
mark(nil) → active(nil)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

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

ACTIVE(uTake2(tt, M, N, IL)) → MARK(cons(N, take(M, IL)))
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNat(length(L))) → MARK(isNatList(L))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(length(X)) → MARK(X)
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(N))) → MARK(isNat(N))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(uTake2(X1, X2, X3, X4)) → ACTIVE(uTake2(mark(X1), X2, X3, X4))
ACTIVE(take(s(M), cons(N, IL))) → MARK(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(uLength(X1, X2)) → MARK(X1)
MARK(zeros) → ACTIVE(zeros)
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
ACTIVE(zeros) → MARK(cons(0, zeros))
ACTIVE(and(tt, T)) → MARK(T)
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(length(X)) → MARK(X)
MARK(uLength(X1, X2)) → MARK(X1)

The remaining pairs can at least be oriented weakly.

ACTIVE(uTake2(tt, M, N, IL)) → MARK(cons(N, take(M, IL)))
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNat(length(L))) → MARK(isNatList(L))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(N))) → MARK(isNat(N))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(uTake2(X1, X2, X3, X4)) → ACTIVE(uTake2(mark(X1), X2, X3, X4))
ACTIVE(take(s(M), cons(N, IL))) → MARK(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(zeros) → ACTIVE(zeros)
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
ACTIVE(zeros) → MARK(cons(0, zeros))
ACTIVE(and(tt, T)) → MARK(T)
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 1 + x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = x₂
POL(tt) = 0
POL(uLength(x₁, x₂)) = 1 + x₁ + x₂
POL(uTake1(x₁)) = x₁
POL(uTake2(x₁, x₂, x₃, x₄)) = x₃ + x₄
POL(zeros) = 0

The following usable rules [17] were oriented:

uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
active(isNat(0)) → mark(tt)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
mark(tt) → active(tt)
active(isNatIList(zeros)) → mark(tt)
active(isNatList(nil)) → mark(tt)
active(uTake1(tt)) → mark(nil)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
mark(0) → active(0)
mark(nil) → active(nil)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

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

ACTIVE(uTake2(tt, M, N, IL)) → MARK(cons(N, take(M, IL)))
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNat(length(L))) → MARK(isNatList(L))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(N))) → MARK(isNat(N))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(uTake2(X1, X2, X3, X4)) → ACTIVE(uTake2(mark(X1), X2, X3, X4))
ACTIVE(take(s(M), cons(N, IL))) → MARK(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(zeros) → ACTIVE(zeros)
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
ACTIVE(zeros) → MARK(cons(0, zeros))
ACTIVE(and(tt, T)) → MARK(T)
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

ACTIVE(zeros) → MARK(cons(0, zeros))

The remaining pairs can at least be oriented weakly.

ACTIVE(uTake2(tt, M, N, IL)) → MARK(cons(N, take(M, IL)))
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNat(length(L))) → MARK(isNatList(L))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(N))) → MARK(isNat(N))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(uTake2(X1, X2, X3, X4)) → ACTIVE(uTake2(mark(X1), X2, X3, X4))
ACTIVE(take(s(M), cons(N, IL))) → MARK(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(zeros) → ACTIVE(zeros)
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
ACTIVE(and(tt, T)) → MARK(T)
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = x₁
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = x₂
POL(tt) = 0
POL(uLength(x₁, x₂)) = 0
POL(uTake1(x₁)) = x₁
POL(uTake2(x₁, x₂, x₃, x₄)) = x₃
POL(zeros) = 1

The following usable rules [17] were oriented:

uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
active(isNat(0)) → mark(tt)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
mark(tt) → active(tt)
active(isNatIList(zeros)) → mark(tt)
active(isNatList(nil)) → mark(tt)
active(uTake1(tt)) → mark(nil)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
mark(0) → active(0)
mark(nil) → active(nil)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

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

ACTIVE(uTake2(tt, M, N, IL)) → MARK(cons(N, take(M, IL)))
MARK(uTake2(X1, X2, X3, X4)) → ACTIVE(uTake2(mark(X1), X2, X3, X4))
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))
ACTIVE(take(s(M), cons(N, IL))) → MARK(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNat(length(L))) → MARK(isNatList(L))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
MARK(cons(X1, X2)) → MARK(X1)
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
MARK(zeros) → ACTIVE(zeros)
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(N))) → MARK(isNat(N))
ACTIVE(and(tt, T)) → MARK(T)
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

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

ACTIVE(uTake2(tt, M, N, IL)) → MARK(cons(N, take(M, IL)))
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))
MARK(uTake2(X1, X2, X3, X4)) → ACTIVE(uTake2(mark(X1), X2, X3, X4))
ACTIVE(take(s(M), cons(N, IL))) → MARK(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNat(length(L))) → MARK(isNatList(L))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(N))) → MARK(isNat(N))
ACTIVE(and(tt, T)) → MARK(T)
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

ACTIVE(take(s(M), cons(N, IL))) → MARK(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))

The remaining pairs can at least be oriented weakly.

ACTIVE(uTake2(tt, M, N, IL)) → MARK(cons(N, take(M, IL)))
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))
MARK(uTake2(X1, X2, X3, X4)) → ACTIVE(uTake2(mark(X1), X2, X3, X4))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNat(length(L))) → MARK(isNatList(L))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(N))) → MARK(isNat(N))
ACTIVE(and(tt, T)) → MARK(T)
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = x₁
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 1
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = 1 + x₂
POL(tt) = 0
POL(uLength(x₁, x₂)) = 0
POL(uTake1(x₁)) = 1 + x₁
POL(uTake2(x₁, x₂, x₃, x₄)) = x₃
POL(zeros) = 0

The following usable rules [17] were oriented:

uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
active(isNat(0)) → mark(tt)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
mark(tt) → active(tt)
active(isNatIList(zeros)) → mark(tt)
active(isNatList(nil)) → mark(tt)
active(uTake1(tt)) → mark(nil)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
mark(0) → active(0)
mark(nil) → active(nil)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

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

ACTIVE(uTake2(tt, M, N, IL)) → MARK(cons(N, take(M, IL)))
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))
MARK(uTake2(X1, X2, X3, X4)) → ACTIVE(uTake2(mark(X1), X2, X3, X4))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNat(length(L))) → MARK(isNatList(L))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(N))) → MARK(isNat(N))
ACTIVE(and(tt, T)) → MARK(T)
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

ACTIVE(uTake2(tt, M, N, IL)) → MARK(cons(N, take(M, IL)))

The remaining pairs can at least be oriented weakly.

ACTIVE(uLength(tt, L)) → MARK(s(length(L)))
MARK(uTake2(X1, X2, X3, X4)) → ACTIVE(uTake2(mark(X1), X2, X3, X4))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNat(length(L))) → MARK(isNatList(L))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(N))) → MARK(isNat(N))
ACTIVE(and(tt, T)) → MARK(T)
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = x₁
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = 1 + x₁ + x₂
POL(tt) = 0
POL(uLength(x₁, x₂)) = 0
POL(uTake1(x₁)) = x₁
POL(uTake2(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₃
POL(zeros) = 0

The following usable rules [17] were oriented:

uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
active(isNat(0)) → mark(tt)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
mark(0) → active(0)
mark(nil) → active(nil)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
mark(tt) → active(tt)
active(isNatIList(zeros)) → mark(tt)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
active(isNatList(nil)) → mark(tt)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
active(uTake1(tt)) → mark(nil)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

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

ACTIVE(uLength(tt, L)) → MARK(s(length(L)))
MARK(uTake2(X1, X2, X3, X4)) → ACTIVE(uTake2(mark(X1), X2, X3, X4))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNat(length(L))) → MARK(isNatList(L))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(N))) → MARK(isNat(N))
ACTIVE(and(tt, T)) → MARK(T)
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(uTake2(X1, X2, X3, X4)) → ACTIVE(uTake2(mark(X1), X2, X3, X4))
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))

The remaining pairs can at least be oriented weakly.

ACTIVE(uLength(tt, L)) → MARK(s(length(L)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNat(length(L))) → MARK(isNatList(L))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(N))) → MARK(isNat(N))
ACTIVE(and(tt, T)) → MARK(T)
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = 1
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = 1
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 1
POL(isNatIList(x₁)) = 1
POL(isNatList(x₁)) = 1
POL(length(x₁)) = 1
POL(mark(x₁)) = 0
POL(nil) = 0
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 0
POL(tt) = 0
POL(uLength(x₁, x₂)) = 1
POL(uTake1(x₁)) = 0
POL(uTake2(x₁, x₂, x₃, x₄)) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
length(active(X)) → length(X)
length(mark(X)) → length(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

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

ACTIVE(uLength(tt, L)) → MARK(s(length(L)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNat(length(L))) → MARK(isNatList(L))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(N))) → MARK(isNat(N))
ACTIVE(and(tt, T)) → MARK(T)
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(cons(X1, X2)) → MARK(X1)

The remaining pairs can at least be oriented weakly.

ACTIVE(uLength(tt, L)) → MARK(s(length(L)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNat(length(L))) → MARK(isNatList(L))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(N))) → MARK(isNat(N))
ACTIVE(and(tt, T)) → MARK(T)
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = 1 + x₁
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = 1 + x₂
POL(tt) = 0
POL(uLength(x₁, x₂)) = 0
POL(uTake1(x₁)) = 0
POL(uTake2(x₁, x₂, x₃, x₄)) = 1 + x₃
POL(zeros) = 1

The following usable rules [17] were oriented:

uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
active(isNat(0)) → mark(tt)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
mark(0) → active(0)
mark(nil) → active(nil)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
mark(tt) → active(tt)
active(isNatIList(zeros)) → mark(tt)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
active(isNatList(nil)) → mark(tt)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
active(uTake1(tt)) → mark(nil)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

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

ACTIVE(uLength(tt, L)) → MARK(s(length(L)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNat(length(L))) → MARK(isNatList(L))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(N))) → MARK(isNat(N))
ACTIVE(and(tt, T)) → MARK(T)
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

ACTIVE(isNat(length(L))) → MARK(isNatList(L))

The remaining pairs can at least be oriented weakly.

ACTIVE(uLength(tt, L)) → MARK(s(length(L)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(N))) → MARK(isNat(N))
ACTIVE(and(tt, T)) → MARK(T)
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 1 + x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = x₁ + x₂
POL(tt) = 0
POL(uLength(x₁, x₂)) = 1 + x₂
POL(uTake1(x₁)) = 0
POL(uTake2(x₁, x₂, x₃, x₄)) = x₂ + x₃ + x₄
POL(zeros) = 0

The following usable rules [17] were oriented:

uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
active(isNat(0)) → mark(tt)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
mark(0) → active(0)
mark(nil) → active(nil)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
mark(tt) → active(tt)
active(isNatIList(zeros)) → mark(tt)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
active(isNatList(nil)) → mark(tt)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
active(uTake1(tt)) → mark(nil)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

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

ACTIVE(uLength(tt, L)) → MARK(s(length(L)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(N))) → MARK(isNat(N))
ACTIVE(and(tt, T)) → MARK(T)
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))

The remaining pairs can at least be oriented weakly.

ACTIVE(uLength(tt, L)) → MARK(s(length(L)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(N))) → MARK(isNat(N))
ACTIVE(and(tt, T)) → MARK(T)
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = 1 + x₁ + x₂
POL(tt) = 0
POL(uLength(x₁, x₂)) = x₂
POL(uTake1(x₁)) = 1 + x₁
POL(uTake2(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₃ + x₄
POL(zeros) = 0

The following usable rules [17] were oriented:

uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
active(isNat(0)) → mark(tt)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
mark(0) → active(0)
mark(nil) → active(nil)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
mark(tt) → active(tt)
active(isNatIList(zeros)) → mark(tt)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
active(isNatList(nil)) → mark(tt)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
active(uTake1(tt)) → mark(nil)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

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

ACTIVE(uLength(tt, L)) → MARK(s(length(L)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(N))) → MARK(isNat(N))
ACTIVE(and(tt, T)) → MARK(T)
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))

The remaining pairs can at least be oriented weakly.

ACTIVE(uLength(tt, L)) → MARK(s(length(L)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(N))) → MARK(isNat(N))
ACTIVE(and(tt, T)) → MARK(T)
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = 1 + x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = 1 + x₁ + x₂
POL(tt) = 0
POL(uLength(x₁, x₂)) = x₂
POL(uTake1(x₁)) = x₁
POL(uTake2(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₃ + x₄
POL(zeros) = 0

The following usable rules [17] were oriented:

uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
active(isNat(0)) → mark(tt)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
mark(0) → active(0)
mark(nil) → active(nil)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
mark(tt) → active(tt)
active(isNatIList(zeros)) → mark(tt)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
active(isNatList(nil)) → mark(tt)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
active(uTake1(tt)) → mark(nil)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

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

ACTIVE(uLength(tt, L)) → MARK(s(length(L)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(N))) → MARK(isNat(N))
ACTIVE(and(tt, T)) → MARK(T)
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

ACTIVE(isNat(s(N))) → MARK(isNat(N))

The remaining pairs can at least be oriented weakly.

ACTIVE(uLength(tt, L)) → MARK(s(length(L)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(and(tt, T)) → MARK(T)
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( mark(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( take(x₁, x₂) ) =

/	0	\
\	1	/

/	1	1	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( 0 ) =

/	0	\
\	0	/

M( active(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	1	1	\
\	0	0	/

x₁

/	1	1	\
\	0	1	/

x₂

M( tt ) =

/	0	\
\	1	/

M( uTake2(x₁, ..., x₄) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	1	1	\
\	0	0	/

x₂

/	1	1	\
\	0	0	/

x₃

/	1	1	\
\	0	1	/

x₄

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( isNatIList(x₁) ) =

/	0	\
\	1	/

/	1	0	\
\	0	1	/

x₁

M( uTake1(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( s(x₁) ) =

/	0	\
\	1	/

/	1	1	\
\	0	0	/

x₁

M( length(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( isNat(x₁) ) =

/	0	\
\	1	/

/	1	1	\
\	1	1	/

x₁

M( nil ) =

/	0	\
\	1	/

M( uLength(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	1	1	\
\	0	0	/

x₂

Tuple symbols:

M( MARK(x₁) ) =

[

]

x₁

M( ACTIVE(x₁) ) =

[

]

x₁

Matrix type:
We used a basic matrix type which is not further parametrizeable.

As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:

uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
active(isNat(0)) → mark(tt)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
mark(0) → active(0)
mark(nil) → active(nil)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
mark(tt) → active(tt)
active(isNatIList(zeros)) → mark(tt)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
active(isNatList(nil)) → mark(tt)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
active(uTake1(tt)) → mark(nil)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

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

ACTIVE(uLength(tt, L)) → MARK(s(length(L)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(and(tt, T)) → MARK(T)
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(isNat(X)) → ACTIVE(isNat(X))

The remaining pairs can at least be oriented weakly.

ACTIVE(uLength(tt, L)) → MARK(s(length(L)))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(and(tt, T)) → MARK(T)
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = 1
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = 1
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 1
POL(isNatList(x₁)) = 1
POL(length(x₁)) = 1
POL(mark(x₁)) = 0
POL(nil) = 0
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 0
POL(tt) = 0
POL(uLength(x₁, x₂)) = 1
POL(uTake1(x₁)) = 0
POL(uTake2(x₁, x₂, x₃, x₄)) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

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

ACTIVE(uLength(tt, L)) → MARK(s(length(L)))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(and(tt, T)) → MARK(T)
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))

The remaining pairs can at least be oriented weakly.

ACTIVE(uLength(tt, L)) → MARK(s(length(L)))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(and(tt, T)) → MARK(T)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( mark(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	1	0	\
\	0	1	/

x₂

M( take(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	1	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( 0 ) =

/	1	\
\	0	/

M( active(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	1	1	/

x₂

M( tt ) =

/	1	\
\	0	/

M( uTake2(x₁, ..., x₄) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	1	0	/

x₂

/	0	0	\
\	0	0	/

x₃

/	1	0	\
\	1	1	/

x₄

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( isNatIList(x₁) ) =

/	1	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( uTake1(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	1	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( length(x₁) ) =

/	0	\
\	1	/

/	1	1	\
\	0	1	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( nil ) =

/	1	\
\	0	/

M( uLength(x₁, x₂) ) =

/	0	\
\	0	/

/	1	0	\
\	1	0	/

x₁

/	1	1	\
\	0	1	/

x₂

Tuple symbols:

M( MARK(x₁) ) =

[

]

x₁

M( ACTIVE(x₁) ) =

[

]

x₁

uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
active(isNat(0)) → mark(tt)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
mark(0) → active(0)
mark(nil) → active(nil)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
mark(tt) → active(tt)
active(isNatIList(zeros)) → mark(tt)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
active(isNatList(nil)) → mark(tt)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
active(uTake1(tt)) → mark(nil)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

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

ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))
MARK(s(X)) → MARK(X)
MARK(and(X1, X2)) → MARK(X1)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(and(tt, T)) → MARK(T)
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(length(X)) → ACTIVE(length(mark(X)))

The remaining pairs can at least be oriented weakly.

ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))
MARK(s(X)) → MARK(X)
MARK(and(X1, X2)) → MARK(X1)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(and(tt, T)) → MARK(T)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = 1
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = 1
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 1
POL(isNatList(x₁)) = 1
POL(length(x₁)) = 0
POL(mark(x₁)) = 0
POL(nil) = 0
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 0
POL(tt) = 0
POL(uLength(x₁, x₂)) = 1
POL(uTake1(x₁)) = 0
POL(uTake2(x₁, x₂, x₃, x₄)) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

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

ACTIVE(uLength(tt, L)) → MARK(s(length(L)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(and(X1, X2)) → MARK(X1)
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
ACTIVE(and(tt, T)) → MARK(T)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

ACTIVE(uLength(tt, L)) → MARK(s(length(L)))

The remaining pairs can at least be oriented weakly.

MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(and(X1, X2)) → MARK(X1)
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
ACTIVE(and(tt, T)) → MARK(T)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( mark(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( take(x₁, x₂) ) =

/	0	\
\	1	/

/	0	1	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( 0 ) =

/	0	\
\	0	/

M( active(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	1	\
\	0	1	/

x₂

M( tt ) =

/	0	\
\	1	/

M( uTake2(x₁, ..., x₄) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	1	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

/	1	1	\
\	0	1	/

x₄

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( isNatIList(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	1	/

x₁

M( uTake1(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( s(x₁) ) =

/	0	\
\	1	/

/	1	0	\
\	0	1	/

x₁

M( length(x₁) ) =

/	0	\
\	1	/

/	1	1	\
\	1	0	/

x₁

M( isNat(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	1	0	/

x₁

M( nil ) =

/	0	\
\	1	/

M( uLength(x₁, x₂) ) =

/	0	\
\	1	/

/	0	1	\
\	0	1	/

x₁

/	1	1	\
\	1	0	/

x₂

Tuple symbols:

M( MARK(x₁) ) =

[

]

x₁

M( ACTIVE(x₁) ) =

[

]

x₁

uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
active(isNat(0)) → mark(tt)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
mark(0) → active(0)
mark(nil) → active(nil)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
mark(tt) → active(tt)
active(isNatIList(zeros)) → mark(tt)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
active(isNatList(nil)) → mark(tt)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
active(uTake1(tt)) → mark(nil)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

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

ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
MARK(s(X)) → MARK(X)
MARK(and(X1, X2)) → MARK(X1)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(and(tt, T)) → MARK(T)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))

The remaining pairs can at least be oriented weakly.

ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
MARK(s(X)) → MARK(X)
MARK(and(X1, X2)) → MARK(X1)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
ACTIVE(and(tt, T)) → MARK(T)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = 1
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = 1
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 1
POL(isNatList(x₁)) = 1
POL(length(x₁)) = 0
POL(mark(x₁)) = 0
POL(nil) = 0
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 0
POL(tt) = 0
POL(uLength(x₁, x₂)) = 0
POL(uTake1(x₁)) = 0
POL(uTake2(x₁, x₂, x₃, x₄)) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

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

MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
ACTIVE(and(tt, T)) → MARK(T)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(s(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.

MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
ACTIVE(and(tt, T)) → MARK(T)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( mark(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	1	0	\
\	0	1	/

x₂

M( take(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	1	1	/

x₁

/	1	0	\
\	0	1	/

x₂

M( 0 ) =

/	1	\
\	0	/

M( active(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	1	1	/

x₂

M( tt ) =

/	1	\
\	0	/

M( uTake2(x₁, ..., x₄) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	1	1	/

x₂

/	0	0	\
\	0	0	/

x₃

/	1	0	\
\	1	1	/

x₄

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( isNatIList(x₁) ) =

/	1	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( uTake1(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	1	/

/	1	0	\
\	0	1	/

x₁

M( length(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( nil ) =

/	1	\
\	0	/

M( uLength(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

/	1	0	\
\	0	1	/

x₂

Tuple symbols:

M( MARK(x₁) ) =

[

]

x₁

M( ACTIVE(x₁) ) =

[

]

x₁

uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
active(isNat(0)) → mark(tt)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
mark(0) → active(0)
mark(nil) → active(nil)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
mark(tt) → active(tt)
active(isNatIList(zeros)) → mark(tt)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
active(isNatList(nil)) → mark(tt)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
active(uTake1(tt)) → mark(nil)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

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

ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
ACTIVE(and(tt, T)) → MARK(T)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2))) at position [0] we obtained the following new rules:

MARK(and(length(x0), y1)) → ACTIVE(and(active(length(mark(x0))), mark(y1)))
MARK(and(y0, cons(x0, x1))) → ACTIVE(and(mark(y0), active(cons(mark(x0), x1))))
MARK(and(tt, y1)) → ACTIVE(and(active(tt), mark(y1)))
MARK(and(y0, tt)) → ACTIVE(and(mark(y0), active(tt)))
MARK(and(y0, zeros)) → ACTIVE(and(mark(y0), active(zeros)))
MARK(and(zeros, y1)) → ACTIVE(and(active(zeros), mark(y1)))
MARK(and(isNat(x0), y1)) → ACTIVE(and(active(isNat(x0)), mark(y1)))
MARK(and(y0, uTake1(x0))) → ACTIVE(and(mark(y0), active(uTake1(mark(x0)))))
MARK(and(cons(x0, x1), y1)) → ACTIVE(and(active(cons(mark(x0), x1)), mark(y1)))
MARK(and(y0, and(x0, x1))) → ACTIVE(and(mark(y0), active(and(mark(x0), mark(x1)))))
MARK(and(y0, isNatList(x0))) → ACTIVE(and(mark(y0), active(isNatList(x0))))
MARK(and(y0, x1)) → ACTIVE(and(mark(y0), x1))
MARK(and(y0, isNat(x0))) → ACTIVE(and(mark(y0), active(isNat(x0))))
MARK(and(and(x0, x1), y1)) → ACTIVE(and(active(and(mark(x0), mark(x1))), mark(y1)))
MARK(and(y0, take(x0, x1))) → ACTIVE(and(mark(y0), active(take(mark(x0), mark(x1)))))
MARK(and(y0, nil)) → ACTIVE(and(mark(y0), active(nil)))
MARK(and(nil, y1)) → ACTIVE(and(active(nil), mark(y1)))
MARK(and(uLength(x0, x1), y1)) → ACTIVE(and(active(uLength(mark(x0), x1)), mark(y1)))
MARK(and(isNatIList(x0), y1)) → ACTIVE(and(active(isNatIList(x0)), mark(y1)))
MARK(and(s(x0), y1)) → ACTIVE(and(active(s(mark(x0))), mark(y1)))
MARK(and(uTake1(x0), y1)) → ACTIVE(and(active(uTake1(mark(x0))), mark(y1)))
MARK(and(y0, s(x0))) → ACTIVE(and(mark(y0), active(s(mark(x0)))))
MARK(and(y0, uLength(x0, x1))) → ACTIVE(and(mark(y0), active(uLength(mark(x0), x1))))
MARK(and(0, y1)) → ACTIVE(and(active(0), mark(y1)))
MARK(and(take(x0, x1), y1)) → ACTIVE(and(active(take(mark(x0), mark(x1))), mark(y1)))
MARK(and(y0, 0)) → ACTIVE(and(mark(y0), active(0)))
MARK(and(y0, length(x0))) → ACTIVE(and(mark(y0), active(length(mark(x0)))))
MARK(and(y0, uTake2(x0, x1, x2, x3))) → ACTIVE(and(mark(y0), active(uTake2(mark(x0), x1, x2, x3))))
MARK(and(y0, isNatIList(x0))) → ACTIVE(and(mark(y0), active(isNatIList(x0))))
MARK(and(x0, y1)) → ACTIVE(and(x0, mark(y1)))
MARK(and(isNatList(x0), y1)) → ACTIVE(and(active(isNatList(x0)), mark(y1)))
MARK(and(uTake2(x0, x1, x2, x3), y1)) → ACTIVE(and(active(uTake2(mark(x0), x1, x2, x3)), mark(y1)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

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

MARK(and(length(x0), y1)) → ACTIVE(and(active(length(mark(x0))), mark(y1)))
MARK(and(y0, cons(x0, x1))) → ACTIVE(and(mark(y0), active(cons(mark(x0), x1))))
MARK(and(y0, tt)) → ACTIVE(and(mark(y0), active(tt)))
MARK(and(tt, y1)) → ACTIVE(and(active(tt), mark(y1)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(and(y0, zeros)) → ACTIVE(and(mark(y0), active(zeros)))
MARK(and(zeros, y1)) → ACTIVE(and(active(zeros), mark(y1)))
MARK(and(isNat(x0), y1)) → ACTIVE(and(active(isNat(x0)), mark(y1)))
MARK(and(y0, uTake1(x0))) → ACTIVE(and(mark(y0), active(uTake1(mark(x0)))))
MARK(and(cons(x0, x1), y1)) → ACTIVE(and(active(cons(mark(x0), x1)), mark(y1)))
MARK(and(y0, and(x0, x1))) → ACTIVE(and(mark(y0), active(and(mark(x0), mark(x1)))))
MARK(and(y0, isNatList(x0))) → ACTIVE(and(mark(y0), active(isNatList(x0))))
MARK(and(y0, x1)) → ACTIVE(and(mark(y0), x1))
MARK(and(y0, isNat(x0))) → ACTIVE(and(mark(y0), active(isNat(x0))))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(and(x0, x1), y1)) → ACTIVE(and(active(and(mark(x0), mark(x1))), mark(y1)))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(y0, take(x0, x1))) → ACTIVE(and(mark(y0), active(take(mark(x0), mark(x1)))))
MARK(and(y0, nil)) → ACTIVE(and(mark(y0), active(nil)))
MARK(and(nil, y1)) → ACTIVE(and(active(nil), mark(y1)))
MARK(and(uLength(x0, x1), y1)) → ACTIVE(and(active(uLength(mark(x0), x1)), mark(y1)))
MARK(and(isNatIList(x0), y1)) → ACTIVE(and(active(isNatIList(x0)), mark(y1)))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)
MARK(and(s(x0), y1)) → ACTIVE(and(active(s(mark(x0))), mark(y1)))
MARK(and(uTake1(x0), y1)) → ACTIVE(and(active(uTake1(mark(x0))), mark(y1)))
MARK(and(y0, s(x0))) → ACTIVE(and(mark(y0), active(s(mark(x0)))))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(y0, uLength(x0, x1))) → ACTIVE(and(mark(y0), active(uLength(mark(x0), x1))))
MARK(and(y0, 0)) → ACTIVE(and(mark(y0), active(0)))
MARK(and(take(x0, x1), y1)) → ACTIVE(and(active(take(mark(x0), mark(x1))), mark(y1)))
MARK(and(0, y1)) → ACTIVE(and(active(0), mark(y1)))
MARK(and(y0, length(x0))) → ACTIVE(and(mark(y0), active(length(mark(x0)))))
MARK(and(y0, uTake2(x0, x1, x2, x3))) → ACTIVE(and(mark(y0), active(uTake2(mark(x0), x1, x2, x3))))
MARK(and(y0, isNatIList(x0))) → ACTIVE(and(mark(y0), active(isNatIList(x0))))
MARK(and(x0, y1)) → ACTIVE(and(x0, mark(y1)))
ACTIVE(and(tt, T)) → MARK(T)
MARK(and(isNatList(x0), y1)) → ACTIVE(and(active(isNatList(x0)), mark(y1)))
MARK(and(uTake2(x0, x1, x2, x3), y1)) → ACTIVE(and(active(uTake2(mark(x0), x1, x2, x3)), mark(y1)))

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(and(nil, y1)) → ACTIVE(and(active(nil), mark(y1)))
MARK(and(uTake1(x0), y1)) → ACTIVE(and(active(uTake1(mark(x0))), mark(y1)))
MARK(and(take(x0, x1), y1)) → ACTIVE(and(active(take(mark(x0), mark(x1))), mark(y1)))
MARK(and(0, y1)) → ACTIVE(and(active(0), mark(y1)))
MARK(and(uTake2(x0, x1, x2, x3), y1)) → ACTIVE(and(active(uTake2(mark(x0), x1, x2, x3)), mark(y1)))

The remaining pairs can at least be oriented weakly.

MARK(and(length(x0), y1)) → ACTIVE(and(active(length(mark(x0))), mark(y1)))
MARK(and(y0, cons(x0, x1))) → ACTIVE(and(mark(y0), active(cons(mark(x0), x1))))
MARK(and(y0, tt)) → ACTIVE(and(mark(y0), active(tt)))
MARK(and(tt, y1)) → ACTIVE(and(active(tt), mark(y1)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(and(y0, zeros)) → ACTIVE(and(mark(y0), active(zeros)))
MARK(and(zeros, y1)) → ACTIVE(and(active(zeros), mark(y1)))
MARK(and(isNat(x0), y1)) → ACTIVE(and(active(isNat(x0)), mark(y1)))
MARK(and(y0, uTake1(x0))) → ACTIVE(and(mark(y0), active(uTake1(mark(x0)))))
MARK(and(cons(x0, x1), y1)) → ACTIVE(and(active(cons(mark(x0), x1)), mark(y1)))
MARK(and(y0, and(x0, x1))) → ACTIVE(and(mark(y0), active(and(mark(x0), mark(x1)))))
MARK(and(y0, isNatList(x0))) → ACTIVE(and(mark(y0), active(isNatList(x0))))
MARK(and(y0, x1)) → ACTIVE(and(mark(y0), x1))
MARK(and(y0, isNat(x0))) → ACTIVE(and(mark(y0), active(isNat(x0))))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(and(x0, x1), y1)) → ACTIVE(and(active(and(mark(x0), mark(x1))), mark(y1)))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(y0, take(x0, x1))) → ACTIVE(and(mark(y0), active(take(mark(x0), mark(x1)))))
MARK(and(y0, nil)) → ACTIVE(and(mark(y0), active(nil)))
MARK(and(uLength(x0, x1), y1)) → ACTIVE(and(active(uLength(mark(x0), x1)), mark(y1)))
MARK(and(isNatIList(x0), y1)) → ACTIVE(and(active(isNatIList(x0)), mark(y1)))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)
MARK(and(s(x0), y1)) → ACTIVE(and(active(s(mark(x0))), mark(y1)))
MARK(and(y0, s(x0))) → ACTIVE(and(mark(y0), active(s(mark(x0)))))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(y0, uLength(x0, x1))) → ACTIVE(and(mark(y0), active(uLength(mark(x0), x1))))
MARK(and(y0, 0)) → ACTIVE(and(mark(y0), active(0)))
MARK(and(y0, length(x0))) → ACTIVE(and(mark(y0), active(length(mark(x0)))))
MARK(and(y0, uTake2(x0, x1, x2, x3))) → ACTIVE(and(mark(y0), active(uTake2(mark(x0), x1, x2, x3))))
MARK(and(y0, isNatIList(x0))) → ACTIVE(and(mark(y0), active(isNatIList(x0))))
MARK(and(x0, y1)) → ACTIVE(and(x0, mark(y1)))
ACTIVE(and(tt, T)) → MARK(T)
MARK(and(isNatList(x0), y1)) → ACTIVE(and(active(isNatList(x0)), mark(y1)))

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( mark(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	0	1	\
\	0	1	/

x₂

M( take(x₁, x₂) ) =

/	0	\
\	1	/

/	0	0	\
\	0	1	/

x₁

/	0	1	\
\	0	1	/

x₂

M( 0 ) =

/	0	\
\	1	/

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( active(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( tt ) =

/	0	\
\	0	/

M( uTake2(x₁, ..., x₄) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

/	0	0	\
\	0	0	/

x₄

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( isNatIList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( uTake1(x₁) ) =

/	0	\
\	1	/

/	0	1	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( length(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( uLength(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	0	1	\
\	0	0	/

x₂

M( nil ) =

/	0	\
\	1	/

Tuple symbols:

M( MARK(x₁) ) =

[

]

x₁

M( ACTIVE(x₁) ) =

[

]

x₁

uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
active(isNat(0)) → mark(tt)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
mark(tt) → active(tt)
active(isNatIList(zeros)) → mark(tt)
active(isNatList(nil)) → mark(tt)
active(uTake1(tt)) → mark(nil)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
mark(0) → active(0)
mark(nil) → active(nil)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

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

MARK(and(length(x0), y1)) → ACTIVE(and(active(length(mark(x0))), mark(y1)))
MARK(and(y0, cons(x0, x1))) → ACTIVE(and(mark(y0), active(cons(mark(x0), x1))))
MARK(and(tt, y1)) → ACTIVE(and(active(tt), mark(y1)))
MARK(and(y0, tt)) → ACTIVE(and(mark(y0), active(tt)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(and(isNat(x0), y1)) → ACTIVE(and(active(isNat(x0)), mark(y1)))
MARK(and(zeros, y1)) → ACTIVE(and(active(zeros), mark(y1)))
MARK(and(y0, zeros)) → ACTIVE(and(mark(y0), active(zeros)))
MARK(and(y0, uTake1(x0))) → ACTIVE(and(mark(y0), active(uTake1(mark(x0)))))
MARK(and(y0, isNatList(x0))) → ACTIVE(and(mark(y0), active(isNatList(x0))))
MARK(and(y0, and(x0, x1))) → ACTIVE(and(mark(y0), active(and(mark(x0), mark(x1)))))
MARK(and(cons(x0, x1), y1)) → ACTIVE(and(active(cons(mark(x0), x1)), mark(y1)))
MARK(and(y0, x1)) → ACTIVE(and(mark(y0), x1))
MARK(and(y0, isNat(x0))) → ACTIVE(and(mark(y0), active(isNat(x0))))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(and(x0, x1), y1)) → ACTIVE(and(active(and(mark(x0), mark(x1))), mark(y1)))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(y0, take(x0, x1))) → ACTIVE(and(mark(y0), active(take(mark(x0), mark(x1)))))
MARK(and(y0, nil)) → ACTIVE(and(mark(y0), active(nil)))
MARK(and(uLength(x0, x1), y1)) → ACTIVE(and(active(uLength(mark(x0), x1)), mark(y1)))
MARK(and(isNatIList(x0), y1)) → ACTIVE(and(active(isNatIList(x0)), mark(y1)))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)
MARK(and(s(x0), y1)) → ACTIVE(and(active(s(mark(x0))), mark(y1)))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(y0, s(x0))) → ACTIVE(and(mark(y0), active(s(mark(x0)))))
MARK(and(y0, uLength(x0, x1))) → ACTIVE(and(mark(y0), active(uLength(mark(x0), x1))))
MARK(and(y0, 0)) → ACTIVE(and(mark(y0), active(0)))
MARK(and(y0, length(x0))) → ACTIVE(and(mark(y0), active(length(mark(x0)))))
MARK(and(y0, uTake2(x0, x1, x2, x3))) → ACTIVE(and(mark(y0), active(uTake2(mark(x0), x1, x2, x3))))
MARK(and(y0, isNatIList(x0))) → ACTIVE(and(mark(y0), active(isNatIList(x0))))
ACTIVE(and(tt, T)) → MARK(T)
MARK(and(x0, y1)) → ACTIVE(and(x0, mark(y1)))
MARK(and(isNatList(x0), y1)) → ACTIVE(and(active(isNatList(x0)), mark(y1)))

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(and(length(x0), y1)) → ACTIVE(and(active(length(mark(x0))), mark(y1)))
MARK(and(uLength(x0, x1), y1)) → ACTIVE(and(active(uLength(mark(x0), x1)), mark(y1)))

The remaining pairs can at least be oriented weakly.

MARK(and(y0, cons(x0, x1))) → ACTIVE(and(mark(y0), active(cons(mark(x0), x1))))
MARK(and(tt, y1)) → ACTIVE(and(active(tt), mark(y1)))
MARK(and(y0, tt)) → ACTIVE(and(mark(y0), active(tt)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(and(isNat(x0), y1)) → ACTIVE(and(active(isNat(x0)), mark(y1)))
MARK(and(zeros, y1)) → ACTIVE(and(active(zeros), mark(y1)))
MARK(and(y0, zeros)) → ACTIVE(and(mark(y0), active(zeros)))
MARK(and(y0, uTake1(x0))) → ACTIVE(and(mark(y0), active(uTake1(mark(x0)))))
MARK(and(y0, isNatList(x0))) → ACTIVE(and(mark(y0), active(isNatList(x0))))
MARK(and(y0, and(x0, x1))) → ACTIVE(and(mark(y0), active(and(mark(x0), mark(x1)))))
MARK(and(cons(x0, x1), y1)) → ACTIVE(and(active(cons(mark(x0), x1)), mark(y1)))
MARK(and(y0, x1)) → ACTIVE(and(mark(y0), x1))
MARK(and(y0, isNat(x0))) → ACTIVE(and(mark(y0), active(isNat(x0))))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(and(x0, x1), y1)) → ACTIVE(and(active(and(mark(x0), mark(x1))), mark(y1)))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(y0, take(x0, x1))) → ACTIVE(and(mark(y0), active(take(mark(x0), mark(x1)))))
MARK(and(y0, nil)) → ACTIVE(and(mark(y0), active(nil)))
MARK(and(isNatIList(x0), y1)) → ACTIVE(and(active(isNatIList(x0)), mark(y1)))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)
MARK(and(s(x0), y1)) → ACTIVE(and(active(s(mark(x0))), mark(y1)))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(y0, s(x0))) → ACTIVE(and(mark(y0), active(s(mark(x0)))))
MARK(and(y0, uLength(x0, x1))) → ACTIVE(and(mark(y0), active(uLength(mark(x0), x1))))
MARK(and(y0, 0)) → ACTIVE(and(mark(y0), active(0)))
MARK(and(y0, length(x0))) → ACTIVE(and(mark(y0), active(length(mark(x0)))))
MARK(and(y0, uTake2(x0, x1, x2, x3))) → ACTIVE(and(mark(y0), active(uTake2(mark(x0), x1, x2, x3))))
MARK(and(y0, isNatIList(x0))) → ACTIVE(and(mark(y0), active(isNatIList(x0))))
ACTIVE(and(tt, T)) → MARK(T)
MARK(and(x0, y1)) → ACTIVE(and(x0, mark(y1)))
MARK(and(isNatList(x0), y1)) → ACTIVE(and(active(isNatList(x0)), mark(y1)))

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( mark(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

/	1	0	\
\	1	0	/

x₂

M( take(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	0	/

x₂

M( 0 ) =

/	0	\
\	0	/

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( active(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( tt ) =

/	0	\
\	0	/

M( uTake2(x₁, ..., x₄) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	1	0	/

x₂

/	0	0	\
\	0	0	/

x₃

/	0	0	\
\	0	0	/

x₄

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( isNatIList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( uTake1(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( length(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( uLength(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( nil ) =

/	0	\
\	0	/

Tuple symbols:

M( MARK(x₁) ) =

[

]

x₁

M( ACTIVE(x₁) ) =

[

]

x₁

uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
active(isNat(0)) → mark(tt)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
mark(tt) → active(tt)
active(isNatIList(zeros)) → mark(tt)
active(isNatList(nil)) → mark(tt)
active(uTake1(tt)) → mark(nil)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
mark(0) → active(0)
mark(nil) → active(nil)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

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

MARK(and(y0, cons(x0, x1))) → ACTIVE(and(mark(y0), active(cons(mark(x0), x1))))
MARK(and(y0, tt)) → ACTIVE(and(mark(y0), active(tt)))
MARK(and(tt, y1)) → ACTIVE(and(active(tt), mark(y1)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(and(y0, zeros)) → ACTIVE(and(mark(y0), active(zeros)))
MARK(and(zeros, y1)) → ACTIVE(and(active(zeros), mark(y1)))
MARK(and(isNat(x0), y1)) → ACTIVE(and(active(isNat(x0)), mark(y1)))
MARK(and(y0, uTake1(x0))) → ACTIVE(and(mark(y0), active(uTake1(mark(x0)))))
MARK(and(cons(x0, x1), y1)) → ACTIVE(and(active(cons(mark(x0), x1)), mark(y1)))
MARK(and(y0, and(x0, x1))) → ACTIVE(and(mark(y0), active(and(mark(x0), mark(x1)))))
MARK(and(y0, isNatList(x0))) → ACTIVE(and(mark(y0), active(isNatList(x0))))
MARK(and(y0, x1)) → ACTIVE(and(mark(y0), x1))
MARK(and(y0, isNat(x0))) → ACTIVE(and(mark(y0), active(isNat(x0))))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(and(x0, x1), y1)) → ACTIVE(and(active(and(mark(x0), mark(x1))), mark(y1)))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(y0, take(x0, x1))) → ACTIVE(and(mark(y0), active(take(mark(x0), mark(x1)))))
MARK(and(y0, nil)) → ACTIVE(and(mark(y0), active(nil)))
MARK(and(isNatIList(x0), y1)) → ACTIVE(and(active(isNatIList(x0)), mark(y1)))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)
MARK(and(s(x0), y1)) → ACTIVE(and(active(s(mark(x0))), mark(y1)))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(y0, s(x0))) → ACTIVE(and(mark(y0), active(s(mark(x0)))))
MARK(and(y0, uLength(x0, x1))) → ACTIVE(and(mark(y0), active(uLength(mark(x0), x1))))
MARK(and(y0, 0)) → ACTIVE(and(mark(y0), active(0)))
MARK(and(y0, length(x0))) → ACTIVE(and(mark(y0), active(length(mark(x0)))))
MARK(and(y0, uTake2(x0, x1, x2, x3))) → ACTIVE(and(mark(y0), active(uTake2(mark(x0), x1, x2, x3))))
MARK(and(y0, isNatIList(x0))) → ACTIVE(and(mark(y0), active(isNatIList(x0))))
MARK(and(x0, y1)) → ACTIVE(and(x0, mark(y1)))
ACTIVE(and(tt, T)) → MARK(T)
MARK(and(isNatList(x0), y1)) → ACTIVE(and(active(isNatList(x0)), mark(y1)))

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(and(zeros, y1)) → ACTIVE(and(active(zeros), mark(y1)))

The remaining pairs can at least be oriented weakly.

MARK(and(y0, cons(x0, x1))) → ACTIVE(and(mark(y0), active(cons(mark(x0), x1))))
MARK(and(y0, tt)) → ACTIVE(and(mark(y0), active(tt)))
MARK(and(tt, y1)) → ACTIVE(and(active(tt), mark(y1)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(and(y0, zeros)) → ACTIVE(and(mark(y0), active(zeros)))
MARK(and(isNat(x0), y1)) → ACTIVE(and(active(isNat(x0)), mark(y1)))
MARK(and(y0, uTake1(x0))) → ACTIVE(and(mark(y0), active(uTake1(mark(x0)))))
MARK(and(cons(x0, x1), y1)) → ACTIVE(and(active(cons(mark(x0), x1)), mark(y1)))
MARK(and(y0, and(x0, x1))) → ACTIVE(and(mark(y0), active(and(mark(x0), mark(x1)))))
MARK(and(y0, isNatList(x0))) → ACTIVE(and(mark(y0), active(isNatList(x0))))
MARK(and(y0, x1)) → ACTIVE(and(mark(y0), x1))
MARK(and(y0, isNat(x0))) → ACTIVE(and(mark(y0), active(isNat(x0))))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(and(x0, x1), y1)) → ACTIVE(and(active(and(mark(x0), mark(x1))), mark(y1)))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(y0, take(x0, x1))) → ACTIVE(and(mark(y0), active(take(mark(x0), mark(x1)))))
MARK(and(y0, nil)) → ACTIVE(and(mark(y0), active(nil)))
MARK(and(isNatIList(x0), y1)) → ACTIVE(and(active(isNatIList(x0)), mark(y1)))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)
MARK(and(s(x0), y1)) → ACTIVE(and(active(s(mark(x0))), mark(y1)))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(y0, s(x0))) → ACTIVE(and(mark(y0), active(s(mark(x0)))))
MARK(and(y0, uLength(x0, x1))) → ACTIVE(and(mark(y0), active(uLength(mark(x0), x1))))
MARK(and(y0, 0)) → ACTIVE(and(mark(y0), active(0)))
MARK(and(y0, length(x0))) → ACTIVE(and(mark(y0), active(length(mark(x0)))))
MARK(and(y0, uTake2(x0, x1, x2, x3))) → ACTIVE(and(mark(y0), active(uTake2(mark(x0), x1, x2, x3))))
MARK(and(y0, isNatIList(x0))) → ACTIVE(and(mark(y0), active(isNatIList(x0))))
MARK(and(x0, y1)) → ACTIVE(and(x0, mark(y1)))
ACTIVE(and(tt, T)) → MARK(T)
MARK(and(isNatList(x0), y1)) → ACTIVE(and(active(isNatList(x0)), mark(y1)))

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( mark(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	1	0	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

/	1	0	\
\	1	0	/

x₂

M( take(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( 0 ) =

/	0	\
\	0	/

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( active(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	1	0	/

x₁

M( tt ) =

/	0	\
\	0	/

M( uTake2(x₁, ..., x₄) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

/	0	0	\
\	0	0	/

x₄

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( zeros ) =

/	1	\
\	0	/

M( isNatIList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( uTake1(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( length(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( uLength(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( nil ) =

/	0	\
\	0	/

Tuple symbols:

M( MARK(x₁) ) =

[

]

x₁

M( ACTIVE(x₁) ) =

[

]

x₁

uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
active(isNat(0)) → mark(tt)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
mark(tt) → active(tt)
active(isNatIList(zeros)) → mark(tt)
active(isNatList(nil)) → mark(tt)
active(uTake1(tt)) → mark(nil)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
mark(0) → active(0)
mark(nil) → active(nil)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

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

MARK(and(y0, cons(x0, x1))) → ACTIVE(and(mark(y0), active(cons(mark(x0), x1))))
MARK(and(tt, y1)) → ACTIVE(and(active(tt), mark(y1)))
MARK(and(y0, tt)) → ACTIVE(and(mark(y0), active(tt)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(and(isNat(x0), y1)) → ACTIVE(and(active(isNat(x0)), mark(y1)))
MARK(and(y0, zeros)) → ACTIVE(and(mark(y0), active(zeros)))
MARK(and(y0, uTake1(x0))) → ACTIVE(and(mark(y0), active(uTake1(mark(x0)))))
MARK(and(y0, isNatList(x0))) → ACTIVE(and(mark(y0), active(isNatList(x0))))
MARK(and(y0, and(x0, x1))) → ACTIVE(and(mark(y0), active(and(mark(x0), mark(x1)))))
MARK(and(cons(x0, x1), y1)) → ACTIVE(and(active(cons(mark(x0), x1)), mark(y1)))
MARK(and(y0, x1)) → ACTIVE(and(mark(y0), x1))
MARK(and(y0, isNat(x0))) → ACTIVE(and(mark(y0), active(isNat(x0))))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(and(x0, x1), y1)) → ACTIVE(and(active(and(mark(x0), mark(x1))), mark(y1)))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(y0, take(x0, x1))) → ACTIVE(and(mark(y0), active(take(mark(x0), mark(x1)))))
MARK(and(y0, nil)) → ACTIVE(and(mark(y0), active(nil)))
MARK(and(isNatIList(x0), y1)) → ACTIVE(and(active(isNatIList(x0)), mark(y1)))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)
MARK(and(s(x0), y1)) → ACTIVE(and(active(s(mark(x0))), mark(y1)))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(y0, s(x0))) → ACTIVE(and(mark(y0), active(s(mark(x0)))))
MARK(and(y0, uLength(x0, x1))) → ACTIVE(and(mark(y0), active(uLength(mark(x0), x1))))
MARK(and(y0, 0)) → ACTIVE(and(mark(y0), active(0)))
MARK(and(y0, length(x0))) → ACTIVE(and(mark(y0), active(length(mark(x0)))))
MARK(and(y0, uTake2(x0, x1, x2, x3))) → ACTIVE(and(mark(y0), active(uTake2(mark(x0), x1, x2, x3))))
MARK(and(y0, isNatIList(x0))) → ACTIVE(and(mark(y0), active(isNatIList(x0))))
ACTIVE(and(tt, T)) → MARK(T)
MARK(and(x0, y1)) → ACTIVE(and(x0, mark(y1)))
MARK(and(isNatList(x0), y1)) → ACTIVE(and(active(isNatList(x0)), mark(y1)))

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(and(s(x0), y1)) → ACTIVE(and(active(s(mark(x0))), mark(y1)))

The remaining pairs can at least be oriented weakly.

MARK(and(y0, cons(x0, x1))) → ACTIVE(and(mark(y0), active(cons(mark(x0), x1))))
MARK(and(tt, y1)) → ACTIVE(and(active(tt), mark(y1)))
MARK(and(y0, tt)) → ACTIVE(and(mark(y0), active(tt)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(and(isNat(x0), y1)) → ACTIVE(and(active(isNat(x0)), mark(y1)))
MARK(and(y0, zeros)) → ACTIVE(and(mark(y0), active(zeros)))
MARK(and(y0, uTake1(x0))) → ACTIVE(and(mark(y0), active(uTake1(mark(x0)))))
MARK(and(y0, isNatList(x0))) → ACTIVE(and(mark(y0), active(isNatList(x0))))
MARK(and(y0, and(x0, x1))) → ACTIVE(and(mark(y0), active(and(mark(x0), mark(x1)))))
MARK(and(cons(x0, x1), y1)) → ACTIVE(and(active(cons(mark(x0), x1)), mark(y1)))
MARK(and(y0, x1)) → ACTIVE(and(mark(y0), x1))
MARK(and(y0, isNat(x0))) → ACTIVE(and(mark(y0), active(isNat(x0))))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(and(x0, x1), y1)) → ACTIVE(and(active(and(mark(x0), mark(x1))), mark(y1)))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(y0, take(x0, x1))) → ACTIVE(and(mark(y0), active(take(mark(x0), mark(x1)))))
MARK(and(y0, nil)) → ACTIVE(and(mark(y0), active(nil)))
MARK(and(isNatIList(x0), y1)) → ACTIVE(and(active(isNatIList(x0)), mark(y1)))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(y0, s(x0))) → ACTIVE(and(mark(y0), active(s(mark(x0)))))
MARK(and(y0, uLength(x0, x1))) → ACTIVE(and(mark(y0), active(uLength(mark(x0), x1))))
MARK(and(y0, 0)) → ACTIVE(and(mark(y0), active(0)))
MARK(and(y0, length(x0))) → ACTIVE(and(mark(y0), active(length(mark(x0)))))
MARK(and(y0, uTake2(x0, x1, x2, x3))) → ACTIVE(and(mark(y0), active(uTake2(mark(x0), x1, x2, x3))))
MARK(and(y0, isNatIList(x0))) → ACTIVE(and(mark(y0), active(isNatIList(x0))))
ACTIVE(and(tt, T)) → MARK(T)
MARK(and(x0, y1)) → ACTIVE(and(x0, mark(y1)))
MARK(and(isNatList(x0), y1)) → ACTIVE(and(active(isNatList(x0)), mark(y1)))

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( mark(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	0	1	\
\	0	1	/

x₂

M( take(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( 0 ) =

/	0	\
\	0	/

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( active(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( tt ) =

/	0	\
\	0	/

M( uTake2(x₁, ..., x₄) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

/	0	0	\
\	0	0	/

x₄

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( isNatIList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( uTake1(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	0	1	/

x₁

M( s(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( length(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( uLength(x₁, x₂) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( nil ) =

/	0	\
\	0	/

Tuple symbols:

M( MARK(x₁) ) =

[

]

x₁

M( ACTIVE(x₁) ) =

[

]

x₁

uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
active(isNat(0)) → mark(tt)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
mark(tt) → active(tt)
active(isNatIList(zeros)) → mark(tt)
active(isNatList(nil)) → mark(tt)
active(uTake1(tt)) → mark(nil)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
mark(0) → active(0)
mark(nil) → active(nil)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

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

MARK(and(y0, cons(x0, x1))) → ACTIVE(and(mark(y0), active(cons(mark(x0), x1))))
MARK(and(y0, tt)) → ACTIVE(and(mark(y0), active(tt)))
MARK(and(tt, y1)) → ACTIVE(and(active(tt), mark(y1)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(and(y0, zeros)) → ACTIVE(and(mark(y0), active(zeros)))
MARK(and(isNat(x0), y1)) → ACTIVE(and(active(isNat(x0)), mark(y1)))
MARK(and(y0, uTake1(x0))) → ACTIVE(and(mark(y0), active(uTake1(mark(x0)))))
MARK(and(cons(x0, x1), y1)) → ACTIVE(and(active(cons(mark(x0), x1)), mark(y1)))
MARK(and(y0, and(x0, x1))) → ACTIVE(and(mark(y0), active(and(mark(x0), mark(x1)))))
MARK(and(y0, isNatList(x0))) → ACTIVE(and(mark(y0), active(isNatList(x0))))
MARK(and(y0, x1)) → ACTIVE(and(mark(y0), x1))
MARK(and(y0, isNat(x0))) → ACTIVE(and(mark(y0), active(isNat(x0))))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(and(x0, x1), y1)) → ACTIVE(and(active(and(mark(x0), mark(x1))), mark(y1)))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(y0, take(x0, x1))) → ACTIVE(and(mark(y0), active(take(mark(x0), mark(x1)))))
MARK(and(y0, nil)) → ACTIVE(and(mark(y0), active(nil)))
MARK(and(isNatIList(x0), y1)) → ACTIVE(and(active(isNatIList(x0)), mark(y1)))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(y0, s(x0))) → ACTIVE(and(mark(y0), active(s(mark(x0)))))
MARK(and(y0, uLength(x0, x1))) → ACTIVE(and(mark(y0), active(uLength(mark(x0), x1))))
MARK(and(y0, 0)) → ACTIVE(and(mark(y0), active(0)))
MARK(and(y0, length(x0))) → ACTIVE(and(mark(y0), active(length(mark(x0)))))
MARK(and(y0, uTake2(x0, x1, x2, x3))) → ACTIVE(and(mark(y0), active(uTake2(mark(x0), x1, x2, x3))))
MARK(and(y0, isNatIList(x0))) → ACTIVE(and(mark(y0), active(isNatIList(x0))))
MARK(and(x0, y1)) → ACTIVE(and(x0, mark(y1)))
ACTIVE(and(tt, T)) → MARK(T)
MARK(and(isNatList(x0), y1)) → ACTIVE(and(active(isNatList(x0)), mark(y1)))

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(and(cons(x0, x1), y1)) → ACTIVE(and(active(cons(mark(x0), x1)), mark(y1)))

The remaining pairs can at least be oriented weakly.

MARK(and(y0, cons(x0, x1))) → ACTIVE(and(mark(y0), active(cons(mark(x0), x1))))
MARK(and(y0, tt)) → ACTIVE(and(mark(y0), active(tt)))
MARK(and(tt, y1)) → ACTIVE(and(active(tt), mark(y1)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(and(y0, zeros)) → ACTIVE(and(mark(y0), active(zeros)))
MARK(and(isNat(x0), y1)) → ACTIVE(and(active(isNat(x0)), mark(y1)))
MARK(and(y0, uTake1(x0))) → ACTIVE(and(mark(y0), active(uTake1(mark(x0)))))
MARK(and(y0, and(x0, x1))) → ACTIVE(and(mark(y0), active(and(mark(x0), mark(x1)))))
MARK(and(y0, isNatList(x0))) → ACTIVE(and(mark(y0), active(isNatList(x0))))
MARK(and(y0, x1)) → ACTIVE(and(mark(y0), x1))
MARK(and(y0, isNat(x0))) → ACTIVE(and(mark(y0), active(isNat(x0))))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(and(x0, x1), y1)) → ACTIVE(and(active(and(mark(x0), mark(x1))), mark(y1)))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(y0, take(x0, x1))) → ACTIVE(and(mark(y0), active(take(mark(x0), mark(x1)))))
MARK(and(y0, nil)) → ACTIVE(and(mark(y0), active(nil)))
MARK(and(isNatIList(x0), y1)) → ACTIVE(and(active(isNatIList(x0)), mark(y1)))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(y0, s(x0))) → ACTIVE(and(mark(y0), active(s(mark(x0)))))
MARK(and(y0, uLength(x0, x1))) → ACTIVE(and(mark(y0), active(uLength(mark(x0), x1))))
MARK(and(y0, 0)) → ACTIVE(and(mark(y0), active(0)))
MARK(and(y0, length(x0))) → ACTIVE(and(mark(y0), active(length(mark(x0)))))
MARK(and(y0, uTake2(x0, x1, x2, x3))) → ACTIVE(and(mark(y0), active(uTake2(mark(x0), x1, x2, x3))))
MARK(and(y0, isNatIList(x0))) → ACTIVE(and(mark(y0), active(isNatIList(x0))))
MARK(and(x0, y1)) → ACTIVE(and(x0, mark(y1)))
ACTIVE(and(tt, T)) → MARK(T)
MARK(and(isNatList(x0), y1)) → ACTIVE(and(active(isNatList(x0)), mark(y1)))

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( mark(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	1	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	1	1	/

x₁

/	1	1	\
\	0	0	/

x₂

M( take(x₁, x₂) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	1	1	\
\	0	0	/

x₂

M( 0 ) =

/	0	\
\	0	/

M( cons(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	1	\
\	0	0	/

x₂

M( active(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	1	/

x₁

M( tt ) =

/	0	\
\	0	/

M( uTake2(x₁, ..., x₄) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

/	0	0	\
\	0	0	/

x₄

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( zeros ) =

/	1	\
\	0	/

M( isNatIList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( uTake1(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( length(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	1	/

x₁

M( uLength(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( nil ) =

/	0	\
\	0	/

Tuple symbols:

M( MARK(x₁) ) =

[

]

x₁

M( ACTIVE(x₁) ) =

[

]

x₁

uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
active(isNat(0)) → mark(tt)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
mark(tt) → active(tt)
active(isNatIList(zeros)) → mark(tt)
active(isNatList(nil)) → mark(tt)
active(uTake1(tt)) → mark(nil)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
mark(0) → active(0)
mark(nil) → active(nil)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QDPOrderProof

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

MARK(and(y0, cons(x0, x1))) → ACTIVE(and(mark(y0), active(cons(mark(x0), x1))))
MARK(and(tt, y1)) → ACTIVE(and(active(tt), mark(y1)))
MARK(and(y0, tt)) → ACTIVE(and(mark(y0), active(tt)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(and(isNat(x0), y1)) → ACTIVE(and(active(isNat(x0)), mark(y1)))
MARK(and(y0, zeros)) → ACTIVE(and(mark(y0), active(zeros)))
MARK(and(y0, uTake1(x0))) → ACTIVE(and(mark(y0), active(uTake1(mark(x0)))))
MARK(and(y0, isNatList(x0))) → ACTIVE(and(mark(y0), active(isNatList(x0))))
MARK(and(y0, and(x0, x1))) → ACTIVE(and(mark(y0), active(and(mark(x0), mark(x1)))))
MARK(and(y0, x1)) → ACTIVE(and(mark(y0), x1))
MARK(and(y0, isNat(x0))) → ACTIVE(and(mark(y0), active(isNat(x0))))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(and(x0, x1), y1)) → ACTIVE(and(active(and(mark(x0), mark(x1))), mark(y1)))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(y0, take(x0, x1))) → ACTIVE(and(mark(y0), active(take(mark(x0), mark(x1)))))
MARK(and(y0, nil)) → ACTIVE(and(mark(y0), active(nil)))
MARK(and(isNatIList(x0), y1)) → ACTIVE(and(active(isNatIList(x0)), mark(y1)))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(y0, s(x0))) → ACTIVE(and(mark(y0), active(s(mark(x0)))))
MARK(and(y0, uLength(x0, x1))) → ACTIVE(and(mark(y0), active(uLength(mark(x0), x1))))
MARK(and(y0, 0)) → ACTIVE(and(mark(y0), active(0)))
MARK(and(y0, length(x0))) → ACTIVE(and(mark(y0), active(length(mark(x0)))))
MARK(and(y0, uTake2(x0, x1, x2, x3))) → ACTIVE(and(mark(y0), active(uTake2(mark(x0), x1, x2, x3))))
MARK(and(y0, isNatIList(x0))) → ACTIVE(and(mark(y0), active(isNatIList(x0))))
ACTIVE(and(tt, T)) → MARK(T)
MARK(and(x0, y1)) → ACTIVE(and(x0, mark(y1)))
MARK(and(isNatList(x0), y1)) → ACTIVE(and(active(isNatList(x0)), mark(y1)))

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(and(isNatIList(x0), y1)) → ACTIVE(and(active(isNatIList(x0)), mark(y1)))

The remaining pairs can at least be oriented weakly.

MARK(and(y0, cons(x0, x1))) → ACTIVE(and(mark(y0), active(cons(mark(x0), x1))))
MARK(and(tt, y1)) → ACTIVE(and(active(tt), mark(y1)))
MARK(and(y0, tt)) → ACTIVE(and(mark(y0), active(tt)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(and(isNat(x0), y1)) → ACTIVE(and(active(isNat(x0)), mark(y1)))
MARK(and(y0, zeros)) → ACTIVE(and(mark(y0), active(zeros)))
MARK(and(y0, uTake1(x0))) → ACTIVE(and(mark(y0), active(uTake1(mark(x0)))))
MARK(and(y0, isNatList(x0))) → ACTIVE(and(mark(y0), active(isNatList(x0))))
MARK(and(y0, and(x0, x1))) → ACTIVE(and(mark(y0), active(and(mark(x0), mark(x1)))))
MARK(and(y0, x1)) → ACTIVE(and(mark(y0), x1))
MARK(and(y0, isNat(x0))) → ACTIVE(and(mark(y0), active(isNat(x0))))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(and(x0, x1), y1)) → ACTIVE(and(active(and(mark(x0), mark(x1))), mark(y1)))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(y0, take(x0, x1))) → ACTIVE(and(mark(y0), active(take(mark(x0), mark(x1)))))
MARK(and(y0, nil)) → ACTIVE(and(mark(y0), active(nil)))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(y0, s(x0))) → ACTIVE(and(mark(y0), active(s(mark(x0)))))
MARK(and(y0, uLength(x0, x1))) → ACTIVE(and(mark(y0), active(uLength(mark(x0), x1))))
MARK(and(y0, 0)) → ACTIVE(and(mark(y0), active(0)))
MARK(and(y0, length(x0))) → ACTIVE(and(mark(y0), active(length(mark(x0)))))
MARK(and(y0, uTake2(x0, x1, x2, x3))) → ACTIVE(and(mark(y0), active(uTake2(mark(x0), x1, x2, x3))))
MARK(and(y0, isNatIList(x0))) → ACTIVE(and(mark(y0), active(isNatIList(x0))))
ACTIVE(and(tt, T)) → MARK(T)
MARK(and(x0, y1)) → ACTIVE(and(x0, mark(y1)))
MARK(and(isNatList(x0), y1)) → ACTIVE(and(active(isNatList(x0)), mark(y1)))

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( mark(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

/	1	0	\
\	1	0	/

x₂

M( take(x₁, x₂) ) =

/	1	\
\	0	/

/	1	0	\
\	0	0	/

x₁

/	1	0	\
\	0	0	/

x₂

M( 0 ) =

/	0	\
\	0	/

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

/	1	0	\
\	0	0	/

x₂

M( active(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( tt ) =

/	0	\
\	0	/

M( uTake2(x₁, ..., x₄) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	0	/

x₂

/	1	0	\
\	0	0	/

x₃

/	1	0	\
\	0	0	/

x₄

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	1	0	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( isNatIList(x₁) ) =

/	1	\
\	1	/

/	1	0	\
\	1	0	/

x₁

M( uTake1(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( length(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( uLength(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	0	/

x₂

M( nil ) =

/	0	\
\	0	/

Tuple symbols:

M( MARK(x₁) ) =

[

]

x₁

M( ACTIVE(x₁) ) =

[

]

x₁

uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
active(isNat(0)) → mark(tt)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
mark(tt) → active(tt)
active(isNatIList(zeros)) → mark(tt)
active(isNatList(nil)) → mark(tt)
active(uTake1(tt)) → mark(nil)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
mark(0) → active(0)
mark(nil) → active(nil)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QDPOrderProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ QDPOrderProof

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

MARK(and(y0, cons(x0, x1))) → ACTIVE(and(mark(y0), active(cons(mark(x0), x1))))
MARK(and(y0, tt)) → ACTIVE(and(mark(y0), active(tt)))
MARK(and(tt, y1)) → ACTIVE(and(active(tt), mark(y1)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(and(y0, zeros)) → ACTIVE(and(mark(y0), active(zeros)))
MARK(and(isNat(x0), y1)) → ACTIVE(and(active(isNat(x0)), mark(y1)))
MARK(and(y0, uTake1(x0))) → ACTIVE(and(mark(y0), active(uTake1(mark(x0)))))
MARK(and(y0, and(x0, x1))) → ACTIVE(and(mark(y0), active(and(mark(x0), mark(x1)))))
MARK(and(y0, isNatList(x0))) → ACTIVE(and(mark(y0), active(isNatList(x0))))
MARK(and(y0, x1)) → ACTIVE(and(mark(y0), x1))
MARK(and(y0, isNat(x0))) → ACTIVE(and(mark(y0), active(isNat(x0))))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(and(x0, x1), y1)) → ACTIVE(and(active(and(mark(x0), mark(x1))), mark(y1)))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(y0, take(x0, x1))) → ACTIVE(and(mark(y0), active(take(mark(x0), mark(x1)))))
MARK(and(y0, nil)) → ACTIVE(and(mark(y0), active(nil)))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(y0, s(x0))) → ACTIVE(and(mark(y0), active(s(mark(x0)))))
MARK(and(y0, uLength(x0, x1))) → ACTIVE(and(mark(y0), active(uLength(mark(x0), x1))))
MARK(and(y0, 0)) → ACTIVE(and(mark(y0), active(0)))
MARK(and(y0, length(x0))) → ACTIVE(and(mark(y0), active(length(mark(x0)))))
MARK(and(y0, uTake2(x0, x1, x2, x3))) → ACTIVE(and(mark(y0), active(uTake2(mark(x0), x1, x2, x3))))
MARK(and(y0, isNatIList(x0))) → ACTIVE(and(mark(y0), active(isNatIList(x0))))
MARK(and(x0, y1)) → ACTIVE(and(x0, mark(y1)))
ACTIVE(and(tt, T)) → MARK(T)
MARK(and(isNatList(x0), y1)) → ACTIVE(and(active(isNatList(x0)), mark(y1)))

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(and(isNatList(x0), y1)) → ACTIVE(and(active(isNatList(x0)), mark(y1)))

The remaining pairs can at least be oriented weakly.

MARK(and(y0, cons(x0, x1))) → ACTIVE(and(mark(y0), active(cons(mark(x0), x1))))
MARK(and(y0, tt)) → ACTIVE(and(mark(y0), active(tt)))
MARK(and(tt, y1)) → ACTIVE(and(active(tt), mark(y1)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(and(y0, zeros)) → ACTIVE(and(mark(y0), active(zeros)))
MARK(and(isNat(x0), y1)) → ACTIVE(and(active(isNat(x0)), mark(y1)))
MARK(and(y0, uTake1(x0))) → ACTIVE(and(mark(y0), active(uTake1(mark(x0)))))
MARK(and(y0, and(x0, x1))) → ACTIVE(and(mark(y0), active(and(mark(x0), mark(x1)))))
MARK(and(y0, isNatList(x0))) → ACTIVE(and(mark(y0), active(isNatList(x0))))
MARK(and(y0, x1)) → ACTIVE(and(mark(y0), x1))
MARK(and(y0, isNat(x0))) → ACTIVE(and(mark(y0), active(isNat(x0))))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(and(x0, x1), y1)) → ACTIVE(and(active(and(mark(x0), mark(x1))), mark(y1)))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(y0, take(x0, x1))) → ACTIVE(and(mark(y0), active(take(mark(x0), mark(x1)))))
MARK(and(y0, nil)) → ACTIVE(and(mark(y0), active(nil)))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(y0, s(x0))) → ACTIVE(and(mark(y0), active(s(mark(x0)))))
MARK(and(y0, uLength(x0, x1))) → ACTIVE(and(mark(y0), active(uLength(mark(x0), x1))))
MARK(and(y0, 0)) → ACTIVE(and(mark(y0), active(0)))
MARK(and(y0, length(x0))) → ACTIVE(and(mark(y0), active(length(mark(x0)))))
MARK(and(y0, uTake2(x0, x1, x2, x3))) → ACTIVE(and(mark(y0), active(uTake2(mark(x0), x1, x2, x3))))
MARK(and(y0, isNatIList(x0))) → ACTIVE(and(mark(y0), active(isNatIList(x0))))
MARK(and(x0, y1)) → ACTIVE(and(x0, mark(y1)))
ACTIVE(and(tt, T)) → MARK(T)

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( mark(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	0	1	\
\	0	1	/

x₂

M( take(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	0	0	\
\	0	1	/

x₂

M( 0 ) =

/	0	\
\	0	/

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	0	1	\
\	0	1	/

x₂

M( active(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( tt ) =

/	0	\
\	0	/

M( uTake2(x₁, ..., x₄) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	1	/

x₂

/	0	0	\
\	0	1	/

x₃

/	0	0	\
\	0	1	/

x₄

M( isNatList(x₁) ) =

/	1	\
\	1	/

/	0	1	\
\	0	1	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( isNatIList(x₁) ) =

/	1	\
\	1	/

/	0	1	\
\	0	1	/

x₁

M( uTake1(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( length(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	1	/

x₁

M( uLength(x₁, x₂) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	1	\
\	0	1	/

x₂

M( nil ) =

/	0	\
\	0	/

Tuple symbols:

M( MARK(x₁) ) =

[

]

x₁

M( ACTIVE(x₁) ) =

[

]

x₁

uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
active(isNat(0)) → mark(tt)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
mark(tt) → active(tt)
active(isNatIList(zeros)) → mark(tt)
active(isNatList(nil)) → mark(tt)
active(uTake1(tt)) → mark(nil)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
mark(0) → active(0)
mark(nil) → active(nil)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QDPOrderProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ QDPOrderProof

                                                                                                                          ↳ QDP

                                                                                                                            ↳ QDPOrderProof

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

MARK(and(y0, cons(x0, x1))) → ACTIVE(and(mark(y0), active(cons(mark(x0), x1))))
MARK(and(tt, y1)) → ACTIVE(and(active(tt), mark(y1)))
MARK(and(y0, tt)) → ACTIVE(and(mark(y0), active(tt)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(and(isNat(x0), y1)) → ACTIVE(and(active(isNat(x0)), mark(y1)))
MARK(and(y0, zeros)) → ACTIVE(and(mark(y0), active(zeros)))
MARK(and(y0, uTake1(x0))) → ACTIVE(and(mark(y0), active(uTake1(mark(x0)))))
MARK(and(y0, isNatList(x0))) → ACTIVE(and(mark(y0), active(isNatList(x0))))
MARK(and(y0, and(x0, x1))) → ACTIVE(and(mark(y0), active(and(mark(x0), mark(x1)))))
MARK(and(y0, x1)) → ACTIVE(and(mark(y0), x1))
MARK(and(y0, isNat(x0))) → ACTIVE(and(mark(y0), active(isNat(x0))))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(and(x0, x1), y1)) → ACTIVE(and(active(and(mark(x0), mark(x1))), mark(y1)))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(y0, take(x0, x1))) → ACTIVE(and(mark(y0), active(take(mark(x0), mark(x1)))))
MARK(and(y0, nil)) → ACTIVE(and(mark(y0), active(nil)))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(y0, s(x0))) → ACTIVE(and(mark(y0), active(s(mark(x0)))))
MARK(and(y0, uLength(x0, x1))) → ACTIVE(and(mark(y0), active(uLength(mark(x0), x1))))
MARK(and(y0, 0)) → ACTIVE(and(mark(y0), active(0)))
MARK(and(y0, length(x0))) → ACTIVE(and(mark(y0), active(length(mark(x0)))))
MARK(and(y0, uTake2(x0, x1, x2, x3))) → ACTIVE(and(mark(y0), active(uTake2(mark(x0), x1, x2, x3))))
MARK(and(y0, isNatIList(x0))) → ACTIVE(and(mark(y0), active(isNatIList(x0))))
ACTIVE(and(tt, T)) → MARK(T)
MARK(and(x0, y1)) → ACTIVE(and(x0, mark(y1)))

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(and(y0, zeros)) → ACTIVE(and(mark(y0), active(zeros)))
MARK(and(y0, 0)) → ACTIVE(and(mark(y0), active(0)))

The remaining pairs can at least be oriented weakly.

MARK(and(y0, cons(x0, x1))) → ACTIVE(and(mark(y0), active(cons(mark(x0), x1))))
MARK(and(tt, y1)) → ACTIVE(and(active(tt), mark(y1)))
MARK(and(y0, tt)) → ACTIVE(and(mark(y0), active(tt)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(and(isNat(x0), y1)) → ACTIVE(and(active(isNat(x0)), mark(y1)))
MARK(and(y0, uTake1(x0))) → ACTIVE(and(mark(y0), active(uTake1(mark(x0)))))
MARK(and(y0, isNatList(x0))) → ACTIVE(and(mark(y0), active(isNatList(x0))))
MARK(and(y0, and(x0, x1))) → ACTIVE(and(mark(y0), active(and(mark(x0), mark(x1)))))
MARK(and(y0, x1)) → ACTIVE(and(mark(y0), x1))
MARK(and(y0, isNat(x0))) → ACTIVE(and(mark(y0), active(isNat(x0))))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(and(x0, x1), y1)) → ACTIVE(and(active(and(mark(x0), mark(x1))), mark(y1)))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(y0, take(x0, x1))) → ACTIVE(and(mark(y0), active(take(mark(x0), mark(x1)))))
MARK(and(y0, nil)) → ACTIVE(and(mark(y0), active(nil)))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(y0, s(x0))) → ACTIVE(and(mark(y0), active(s(mark(x0)))))
MARK(and(y0, uLength(x0, x1))) → ACTIVE(and(mark(y0), active(uLength(mark(x0), x1))))
MARK(and(y0, length(x0))) → ACTIVE(and(mark(y0), active(length(mark(x0)))))
MARK(and(y0, uTake2(x0, x1, x2, x3))) → ACTIVE(and(mark(y0), active(uTake2(mark(x0), x1, x2, x3))))
MARK(and(y0, isNatIList(x0))) → ACTIVE(and(mark(y0), active(isNatIList(x0))))
ACTIVE(and(tt, T)) → MARK(T)
MARK(and(x0, y1)) → ACTIVE(and(x0, mark(y1)))

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( mark(x₁) ) =

/	0	\
\	0	/

/	1	1	\
\	1	0	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	1	1	\
\	1	0	/

x₁

/	1	1	\
\	1	0	/

x₂

M( take(x₁, x₂) ) =

/	0	\
\	0	/

/	1	1	\
\	1	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( 0 ) =

/	1	\
\	1	/

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( active(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( tt ) =

/	0	\
\	0	/

M( uTake2(x₁, ..., x₄) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

/	0	0	\
\	0	0	/

x₄

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( zeros ) =

/	1	\
\	1	/

M( isNatIList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( uTake1(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	1	1	\
\	1	0	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( length(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( uLength(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( nil ) =

/	0	\
\	0	/

Tuple symbols:

M( MARK(x₁) ) =

[

]

x₁

M( ACTIVE(x₁) ) =

[

]

x₁

uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
active(isNat(0)) → mark(tt)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
mark(tt) → active(tt)
active(isNatIList(zeros)) → mark(tt)
active(isNatList(nil)) → mark(tt)
active(uTake1(tt)) → mark(nil)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
mark(0) → active(0)
mark(nil) → active(nil)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QDPOrderProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ QDPOrderProof

                                                                                                                          ↳ QDP

                                                                                                                            ↳ QDPOrderProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ QDPOrderProof

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

MARK(and(y0, cons(x0, x1))) → ACTIVE(and(mark(y0), active(cons(mark(x0), x1))))
MARK(and(y0, tt)) → ACTIVE(and(mark(y0), active(tt)))
MARK(and(tt, y1)) → ACTIVE(and(active(tt), mark(y1)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(and(isNat(x0), y1)) → ACTIVE(and(active(isNat(x0)), mark(y1)))
MARK(and(y0, uTake1(x0))) → ACTIVE(and(mark(y0), active(uTake1(mark(x0)))))
MARK(and(y0, and(x0, x1))) → ACTIVE(and(mark(y0), active(and(mark(x0), mark(x1)))))
MARK(and(y0, isNatList(x0))) → ACTIVE(and(mark(y0), active(isNatList(x0))))
MARK(and(y0, x1)) → ACTIVE(and(mark(y0), x1))
MARK(and(y0, isNat(x0))) → ACTIVE(and(mark(y0), active(isNat(x0))))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(and(x0, x1), y1)) → ACTIVE(and(active(and(mark(x0), mark(x1))), mark(y1)))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(y0, take(x0, x1))) → ACTIVE(and(mark(y0), active(take(mark(x0), mark(x1)))))
MARK(and(y0, nil)) → ACTIVE(and(mark(y0), active(nil)))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(y0, s(x0))) → ACTIVE(and(mark(y0), active(s(mark(x0)))))
MARK(and(y0, uLength(x0, x1))) → ACTIVE(and(mark(y0), active(uLength(mark(x0), x1))))
MARK(and(y0, length(x0))) → ACTIVE(and(mark(y0), active(length(mark(x0)))))
MARK(and(y0, uTake2(x0, x1, x2, x3))) → ACTIVE(and(mark(y0), active(uTake2(mark(x0), x1, x2, x3))))
MARK(and(y0, isNatIList(x0))) → ACTIVE(and(mark(y0), active(isNatIList(x0))))
ACTIVE(and(tt, T)) → MARK(T)
MARK(and(x0, y1)) → ACTIVE(and(x0, mark(y1)))

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(and(y0, length(x0))) → ACTIVE(and(mark(y0), active(length(mark(x0)))))

The remaining pairs can at least be oriented weakly.

MARK(and(y0, cons(x0, x1))) → ACTIVE(and(mark(y0), active(cons(mark(x0), x1))))
MARK(and(y0, tt)) → ACTIVE(and(mark(y0), active(tt)))
MARK(and(tt, y1)) → ACTIVE(and(active(tt), mark(y1)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(and(isNat(x0), y1)) → ACTIVE(and(active(isNat(x0)), mark(y1)))
MARK(and(y0, uTake1(x0))) → ACTIVE(and(mark(y0), active(uTake1(mark(x0)))))
MARK(and(y0, and(x0, x1))) → ACTIVE(and(mark(y0), active(and(mark(x0), mark(x1)))))
MARK(and(y0, isNatList(x0))) → ACTIVE(and(mark(y0), active(isNatList(x0))))
MARK(and(y0, x1)) → ACTIVE(and(mark(y0), x1))
MARK(and(y0, isNat(x0))) → ACTIVE(and(mark(y0), active(isNat(x0))))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(and(x0, x1), y1)) → ACTIVE(and(active(and(mark(x0), mark(x1))), mark(y1)))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(y0, take(x0, x1))) → ACTIVE(and(mark(y0), active(take(mark(x0), mark(x1)))))
MARK(and(y0, nil)) → ACTIVE(and(mark(y0), active(nil)))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(y0, s(x0))) → ACTIVE(and(mark(y0), active(s(mark(x0)))))
MARK(and(y0, uLength(x0, x1))) → ACTIVE(and(mark(y0), active(uLength(mark(x0), x1))))
MARK(and(y0, uTake2(x0, x1, x2, x3))) → ACTIVE(and(mark(y0), active(uTake2(mark(x0), x1, x2, x3))))
MARK(and(y0, isNatIList(x0))) → ACTIVE(and(mark(y0), active(isNatIList(x0))))
ACTIVE(and(tt, T)) → MARK(T)
MARK(and(x0, y1)) → ACTIVE(and(x0, mark(y1)))

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( mark(x₁) ) =

/	0	\
\	0	/

/	1	1	\
\	1	0	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	1	1	\
\	1	0	/

x₁

/	1	1	\
\	1	0	/

x₂

M( take(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( 0 ) =

/	0	\
\	0	/

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( active(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( tt ) =

/	0	\
\	0	/

M( uTake2(x₁, ..., x₄) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

/	0	0	\
\	0	0	/

x₄

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( isNatIList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( uTake1(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( length(x₁) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( uLength(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( nil ) =

/	0	\
\	0	/

Tuple symbols:

M( MARK(x₁) ) =

[

]

x₁

M( ACTIVE(x₁) ) =

[

]

x₁

uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
active(isNat(0)) → mark(tt)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
mark(tt) → active(tt)
active(isNatIList(zeros)) → mark(tt)
active(isNatList(nil)) → mark(tt)
active(uTake1(tt)) → mark(nil)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
mark(0) → active(0)
mark(nil) → active(nil)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QDPOrderProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ QDPOrderProof

                                                                                                                          ↳ QDP

                                                                                                                            ↳ QDPOrderProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ QDPOrderProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QDPOrderProof

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

MARK(and(y0, cons(x0, x1))) → ACTIVE(and(mark(y0), active(cons(mark(x0), x1))))
MARK(and(y0, tt)) → ACTIVE(and(mark(y0), active(tt)))
MARK(and(tt, y1)) → ACTIVE(and(active(tt), mark(y1)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(and(isNat(x0), y1)) → ACTIVE(and(active(isNat(x0)), mark(y1)))
MARK(and(y0, s(x0))) → ACTIVE(and(mark(y0), active(s(mark(x0)))))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(y0, uLength(x0, x1))) → ACTIVE(and(mark(y0), active(uLength(mark(x0), x1))))
MARK(and(y0, uTake1(x0))) → ACTIVE(and(mark(y0), active(uTake1(mark(x0)))))
MARK(and(y0, and(x0, x1))) → ACTIVE(and(mark(y0), active(and(mark(x0), mark(x1)))))
MARK(and(y0, isNatList(x0))) → ACTIVE(and(mark(y0), active(isNatList(x0))))
MARK(and(y0, x1)) → ACTIVE(and(mark(y0), x1))
MARK(and(y0, isNat(x0))) → ACTIVE(and(mark(y0), active(isNat(x0))))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(y0, uTake2(x0, x1, x2, x3))) → ACTIVE(and(mark(y0), active(uTake2(mark(x0), x1, x2, x3))))
MARK(and(and(x0, x1), y1)) → ACTIVE(and(active(and(mark(x0), mark(x1))), mark(y1)))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(y0, take(x0, x1))) → ACTIVE(and(mark(y0), active(take(mark(x0), mark(x1)))))
MARK(and(y0, nil)) → ACTIVE(and(mark(y0), active(nil)))
MARK(and(y0, isNatIList(x0))) → ACTIVE(and(mark(y0), active(isNatIList(x0))))
MARK(and(x0, y1)) → ACTIVE(and(x0, mark(y1)))
ACTIVE(and(tt, T)) → MARK(T)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(and(y0, uLength(x0, x1))) → ACTIVE(and(mark(y0), active(uLength(mark(x0), x1))))
MARK(and(y0, uTake2(x0, x1, x2, x3))) → ACTIVE(and(mark(y0), active(uTake2(mark(x0), x1, x2, x3))))
MARK(and(y0, take(x0, x1))) → ACTIVE(and(mark(y0), active(take(mark(x0), mark(x1)))))

The remaining pairs can at least be oriented weakly.

MARK(and(y0, cons(x0, x1))) → ACTIVE(and(mark(y0), active(cons(mark(x0), x1))))
MARK(and(y0, tt)) → ACTIVE(and(mark(y0), active(tt)))
MARK(and(tt, y1)) → ACTIVE(and(active(tt), mark(y1)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(and(isNat(x0), y1)) → ACTIVE(and(active(isNat(x0)), mark(y1)))
MARK(and(y0, s(x0))) → ACTIVE(and(mark(y0), active(s(mark(x0)))))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(y0, uTake1(x0))) → ACTIVE(and(mark(y0), active(uTake1(mark(x0)))))
MARK(and(y0, and(x0, x1))) → ACTIVE(and(mark(y0), active(and(mark(x0), mark(x1)))))
MARK(and(y0, isNatList(x0))) → ACTIVE(and(mark(y0), active(isNatList(x0))))
MARK(and(y0, x1)) → ACTIVE(and(mark(y0), x1))
MARK(and(y0, isNat(x0))) → ACTIVE(and(mark(y0), active(isNat(x0))))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(and(x0, x1), y1)) → ACTIVE(and(active(and(mark(x0), mark(x1))), mark(y1)))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(y0, nil)) → ACTIVE(and(mark(y0), active(nil)))
MARK(and(y0, isNatIList(x0))) → ACTIVE(and(mark(y0), active(isNatIList(x0))))
MARK(and(x0, y1)) → ACTIVE(and(x0, mark(y1)))
ACTIVE(and(tt, T)) → MARK(T)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( mark(x₁) ) =

/	0	\
\	0	/

/	1	1	\
\	1	0	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	1	1	\
\	1	0	/

x₁

/	1	1	\
\	1	0	/

x₂

M( take(x₁, x₂) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	1	1	\
\	1	0	/

x₂

M( 0 ) =

/	0	\
\	0	/

M( cons(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( active(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( tt ) =

/	0	\
\	0	/

M( uTake2(x₁, ..., x₄) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

/	0	0	\
\	0	0	/

x₄

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( zeros ) =

/	1	\
\	1	/

M( isNatIList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( uTake1(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( length(x₁) ) =

/	1	\
\	0	/

/	1	1	\
\	1	0	/

x₁

M( uLength(x₁, x₂) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( nil ) =

/	0	\
\	0	/

Tuple symbols:

M( MARK(x₁) ) =

[

]

x₁

M( ACTIVE(x₁) ) =

[

]

x₁

uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
active(isNat(0)) → mark(tt)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
mark(tt) → active(tt)
active(isNatIList(zeros)) → mark(tt)
active(isNatList(nil)) → mark(tt)
active(uTake1(tt)) → mark(nil)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
mark(0) → active(0)
mark(nil) → active(nil)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QDPOrderProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ QDPOrderProof

                                                                                                                          ↳ QDP

                                                                                                                            ↳ QDPOrderProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ QDPOrderProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

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

MARK(and(y0, cons(x0, x1))) → ACTIVE(and(mark(y0), active(cons(mark(x0), x1))))
MARK(and(y0, tt)) → ACTIVE(and(mark(y0), active(tt)))
MARK(and(tt, y1)) → ACTIVE(and(active(tt), mark(y1)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(and(isNat(x0), y1)) → ACTIVE(and(active(isNat(x0)), mark(y1)))
MARK(and(y0, s(x0))) → ACTIVE(and(mark(y0), active(s(mark(x0)))))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(y0, uTake1(x0))) → ACTIVE(and(mark(y0), active(uTake1(mark(x0)))))
MARK(and(y0, and(x0, x1))) → ACTIVE(and(mark(y0), active(and(mark(x0), mark(x1)))))
MARK(and(y0, isNatList(x0))) → ACTIVE(and(mark(y0), active(isNatList(x0))))
MARK(and(y0, x1)) → ACTIVE(and(mark(y0), x1))
MARK(and(y0, isNat(x0))) → ACTIVE(and(mark(y0), active(isNat(x0))))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(and(x0, x1), y1)) → ACTIVE(and(active(and(mark(x0), mark(x1))), mark(y1)))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(y0, nil)) → ACTIVE(and(mark(y0), active(nil)))
MARK(and(y0, isNatIList(x0))) → ACTIVE(and(mark(y0), active(isNatIList(x0))))
ACTIVE(and(tt, T)) → MARK(T)
MARK(and(x0, y1)) → ACTIVE(and(x0, mark(y1)))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(and(y0, uTake1(x0))) → ACTIVE(and(mark(y0), active(uTake1(mark(x0)))))

The remaining pairs can at least be oriented weakly.

MARK(and(y0, cons(x0, x1))) → ACTIVE(and(mark(y0), active(cons(mark(x0), x1))))
MARK(and(y0, tt)) → ACTIVE(and(mark(y0), active(tt)))
MARK(and(tt, y1)) → ACTIVE(and(active(tt), mark(y1)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(and(isNat(x0), y1)) → ACTIVE(and(active(isNat(x0)), mark(y1)))
MARK(and(y0, s(x0))) → ACTIVE(and(mark(y0), active(s(mark(x0)))))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(y0, and(x0, x1))) → ACTIVE(and(mark(y0), active(and(mark(x0), mark(x1)))))
MARK(and(y0, isNatList(x0))) → ACTIVE(and(mark(y0), active(isNatList(x0))))
MARK(and(y0, x1)) → ACTIVE(and(mark(y0), x1))
MARK(and(y0, isNat(x0))) → ACTIVE(and(mark(y0), active(isNat(x0))))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(and(x0, x1), y1)) → ACTIVE(and(active(and(mark(x0), mark(x1))), mark(y1)))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(y0, nil)) → ACTIVE(and(mark(y0), active(nil)))
MARK(and(y0, isNatIList(x0))) → ACTIVE(and(mark(y0), active(isNatIList(x0))))
ACTIVE(and(tt, T)) → MARK(T)
MARK(and(x0, y1)) → ACTIVE(and(x0, mark(y1)))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( mark(x₁) ) =

/	0	\
\	0	/

/	1	1	\
\	1	0	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	1	1	\
\	1	0	/

x₁

/	1	1	\
\	1	0	/

x₂

M( take(x₁, x₂) ) =

/	1	\
\	0	/

/	1	1	\
\	1	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( 0 ) =

/	1	\
\	0	/

M( active(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( tt ) =

/	0	\
\	0	/

M( uTake2(x₁, ..., x₄) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

/	0	0	\
\	0	0	/

x₄

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( isNatIList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( uTake1(x₁) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( length(x₁) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( nil ) =

/	0	\
\	0	/

M( uLength(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

Tuple symbols:

M( MARK(x₁) ) =

[

]

x₁

M( ACTIVE(x₁) ) =

[

]

x₁

uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
active(isNat(0)) → mark(tt)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
mark(0) → active(0)
mark(nil) → active(nil)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
mark(tt) → active(tt)
active(isNatIList(zeros)) → mark(tt)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
active(isNatList(nil)) → mark(tt)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
active(uTake1(tt)) → mark(nil)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QDPOrderProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ QDPOrderProof

                                                                                                                          ↳ QDP

                                                                                                                            ↳ QDPOrderProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ QDPOrderProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

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

MARK(and(y0, cons(x0, x1))) → ACTIVE(and(mark(y0), active(cons(mark(x0), x1))))
MARK(and(y0, tt)) → ACTIVE(and(mark(y0), active(tt)))
MARK(and(tt, y1)) → ACTIVE(and(active(tt), mark(y1)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(and(isNat(x0), y1)) → ACTIVE(and(active(isNat(x0)), mark(y1)))
MARK(and(y0, s(x0))) → ACTIVE(and(mark(y0), active(s(mark(x0)))))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(y0, and(x0, x1))) → ACTIVE(and(mark(y0), active(and(mark(x0), mark(x1)))))
MARK(and(y0, isNatList(x0))) → ACTIVE(and(mark(y0), active(isNatList(x0))))
MARK(and(y0, x1)) → ACTIVE(and(mark(y0), x1))
MARK(and(y0, isNat(x0))) → ACTIVE(and(mark(y0), active(isNat(x0))))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(and(x0, x1), y1)) → ACTIVE(and(active(and(mark(x0), mark(x1))), mark(y1)))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(y0, nil)) → ACTIVE(and(mark(y0), active(nil)))
MARK(and(y0, isNatIList(x0))) → ACTIVE(and(mark(y0), active(isNatIList(x0))))
MARK(and(x0, y1)) → ACTIVE(and(x0, mark(y1)))
ACTIVE(and(tt, T)) → MARK(T)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(and(y0, nil)) → ACTIVE(and(mark(y0), active(nil)))

The remaining pairs can at least be oriented weakly.

MARK(and(y0, cons(x0, x1))) → ACTIVE(and(mark(y0), active(cons(mark(x0), x1))))
MARK(and(y0, tt)) → ACTIVE(and(mark(y0), active(tt)))
MARK(and(tt, y1)) → ACTIVE(and(active(tt), mark(y1)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(and(isNat(x0), y1)) → ACTIVE(and(active(isNat(x0)), mark(y1)))
MARK(and(y0, s(x0))) → ACTIVE(and(mark(y0), active(s(mark(x0)))))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(y0, and(x0, x1))) → ACTIVE(and(mark(y0), active(and(mark(x0), mark(x1)))))
MARK(and(y0, isNatList(x0))) → ACTIVE(and(mark(y0), active(isNatList(x0))))
MARK(and(y0, x1)) → ACTIVE(and(mark(y0), x1))
MARK(and(y0, isNat(x0))) → ACTIVE(and(mark(y0), active(isNat(x0))))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(and(x0, x1), y1)) → ACTIVE(and(active(and(mark(x0), mark(x1))), mark(y1)))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(y0, isNatIList(x0))) → ACTIVE(and(mark(y0), active(isNatIList(x0))))
MARK(and(x0, y1)) → ACTIVE(and(x0, mark(y1)))
ACTIVE(and(tt, T)) → MARK(T)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( mark(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	1	\
\	1	0	/

x₁

/	1	0	\
\	1	1	/

x₂

M( take(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( 0 ) =

/	0	\
\	0	/

M( active(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( tt ) =

/	0	\
\	0	/

M( uTake2(x₁, ..., x₄) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

/	0	0	\
\	0	0	/

x₄

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( isNatIList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( uTake1(x₁) ) =

/	1	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( length(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( nil ) =

/	1	\
\	0	/

M( uLength(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

Tuple symbols:

M( MARK(x₁) ) =

[

]

x₁

M( ACTIVE(x₁) ) =

[

]

x₁

uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
active(isNat(0)) → mark(tt)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
mark(0) → active(0)
mark(nil) → active(nil)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
mark(tt) → active(tt)
active(isNatIList(zeros)) → mark(tt)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
active(isNatList(nil)) → mark(tt)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
active(uTake1(tt)) → mark(nil)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ 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:

MARK(and(y0, cons(x0, x1))) → ACTIVE(and(mark(y0), active(cons(mark(x0), x1))))
MARK(and(y0, tt)) → ACTIVE(and(mark(y0), active(tt)))
MARK(and(tt, y1)) → ACTIVE(and(active(tt), mark(y1)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(and(isNat(x0), y1)) → ACTIVE(and(active(isNat(x0)), mark(y1)))
MARK(and(y0, s(x0))) → ACTIVE(and(mark(y0), active(s(mark(x0)))))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(and(y0, and(x0, x1))) → ACTIVE(and(mark(y0), active(and(mark(x0), mark(x1)))))
MARK(and(y0, isNatList(x0))) → ACTIVE(and(mark(y0), active(isNatList(x0))))
MARK(and(y0, x1)) → ACTIVE(and(mark(y0), x1))
MARK(and(y0, isNat(x0))) → ACTIVE(and(mark(y0), active(isNat(x0))))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(and(and(x0, x1), y1)) → ACTIVE(and(active(and(mark(x0), mark(x1))), mark(y1)))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(y0, isNatIList(x0))) → ACTIVE(and(mark(y0), active(isNatIList(x0))))
ACTIVE(and(tt, T)) → MARK(T)
MARK(and(x0, y1)) → ACTIVE(and(x0, mark(y1)))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

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