Termination w.r.t. Q of the following Term Rewriting System could not be shown:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.


QTRS
  ↳ RRRPoloQTRSProof

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

active(length(nil)) → mark(0)
active(take(0, IL)) → mark(nil)
Used ordering:
Polynomial interpretation [25]:

POL(0) = 0   
POL(U11(x1, x2)) = 2·x1 + x2   
POL(U12(x1, x2)) = 2·x1 + x2   
POL(U21(x1, x2, x3, x4)) = 2 + 2·x1 + 2·x2 + 2·x3 + 2·x4   
POL(U22(x1, x2, x3, x4)) = 2 + x1 + 2·x2 + 2·x3 + x4   
POL(U23(x1, x2, x3, x4)) = 2 + 2·x1 + 2·x2 + 2·x3 + x4   
POL(active(x1)) = x1   
POL(cons(x1, x2)) = x1 + x2   
POL(length(x1)) = x1   
POL(mark(x1)) = x1   
POL(nil) = 1   
POL(s(x1)) = x1   
POL(take(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(tt) = 0   
POL(zeros) = 0   




↳ QTRS
  ↳ RRRPoloQTRSProof
QTRS
      ↳ DependencyPairsProof

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.

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

U211(active(X1), X2, X3, X4) → U211(X1, X2, X3, X4)
ACTIVE(U23(tt, IL, M, N)) → TAKE(M, IL)
MARK(U22(X1, X2, X3, X4)) → MARK(X1)
U231(X1, X2, X3, mark(X4)) → U231(X1, X2, X3, X4)
ACTIVE(U11(tt, L)) → U121(tt, L)
ACTIVE(U21(tt, IL, M, N)) → MARK(U22(tt, IL, M, N))
MARK(tt) → ACTIVE(tt)
ACTIVE(U23(tt, IL, M, N)) → CONS(N, take(M, IL))
MARK(U11(X1, X2)) → MARK(X1)
TAKE(X1, active(X2)) → TAKE(X1, X2)
MARK(cons(X1, X2)) → MARK(X1)
CONS(X1, mark(X2)) → CONS(X1, X2)
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
U121(X1, active(X2)) → U121(X1, X2)
ACTIVE(U12(tt, L)) → S(length(L))
U111(X1, mark(X2)) → U111(X1, X2)
MARK(length(X)) → MARK(X)
MARK(U22(X1, X2, X3, X4)) → ACTIVE(U22(mark(X1), X2, X3, X4))
ACTIVE(zeros) → CONS(0, zeros)
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → TAKE(mark(X1), mark(X2))
U121(active(X1), X2) → U121(X1, X2)
U231(X1, active(X2), X3, X4) → U231(X1, X2, X3, X4)
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
U111(X1, active(X2)) → U111(X1, X2)
S(active(X)) → S(X)
U121(X1, mark(X2)) → U121(X1, X2)
MARK(U12(X1, X2)) → MARK(X1)
MARK(cons(X1, X2)) → CONS(mark(X1), X2)
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
U221(X1, X2, X3, mark(X4)) → U221(X1, X2, X3, X4)
U231(X1, mark(X2), X3, X4) → U231(X1, X2, X3, X4)
TAKE(active(X1), X2) → TAKE(X1, X2)
U221(X1, X2, active(X3), X4) → U221(X1, X2, X3, X4)
MARK(length(X)) → ACTIVE(length(mark(X)))
U231(X1, X2, mark(X3), X4) → U231(X1, X2, X3, X4)
U221(mark(X1), X2, X3, X4) → U221(X1, X2, X3, X4)
U211(X1, active(X2), X3, X4) → U211(X1, X2, X3, X4)
MARK(zeros) → ACTIVE(zeros)
ACTIVE(length(cons(N, L))) → U111(tt, L)
U211(X1, mark(X2), X3, X4) → U211(X1, X2, X3, X4)
MARK(length(X)) → LENGTH(mark(X))
MARK(U21(X1, X2, X3, X4)) → MARK(X1)
TAKE(mark(X1), X2) → TAKE(X1, X2)
U221(active(X1), X2, X3, X4) → U221(X1, X2, X3, X4)
CONS(active(X1), X2) → CONS(X1, X2)
U221(X1, X2, mark(X3), X4) → U221(X1, X2, X3, X4)
ACTIVE(take(s(M), cons(N, IL))) → U211(tt, IL, M, N)
U111(mark(X1), X2) → U111(X1, X2)
U231(X1, X2, active(X3), X4) → U231(X1, X2, X3, X4)
MARK(U23(X1, X2, X3, X4)) → ACTIVE(U23(mark(X1), X2, X3, X4))
ACTIVE(U12(tt, L)) → LENGTH(L)
MARK(U23(X1, X2, X3, X4)) → U231(mark(X1), X2, X3, X4)
CONS(mark(X1), X2) → CONS(X1, X2)
MARK(take(X1, X2)) → MARK(X2)
MARK(s(X)) → MARK(X)
U231(X1, X2, X3, active(X4)) → U231(X1, X2, X3, X4)
U221(X1, X2, X3, active(X4)) → U221(X1, X2, X3, X4)
U221(X1, active(X2), X3, X4) → U221(X1, X2, X3, X4)
ACTIVE(U22(tt, IL, M, N)) → MARK(U23(tt, IL, M, N))
U231(mark(X1), X2, X3, X4) → U231(X1, X2, X3, X4)
MARK(U11(X1, X2)) → U111(mark(X1), X2)
ACTIVE(U21(tt, IL, M, N)) → U221(tt, IL, M, N)
CONS(X1, active(X2)) → CONS(X1, X2)
MARK(U12(X1, X2)) → U121(mark(X1), X2)
U211(X1, X2, mark(X3), X4) → U211(X1, X2, X3, X4)
U211(mark(X1), X2, X3, X4) → U211(X1, X2, X3, X4)
MARK(s(X)) → ACTIVE(s(mark(X)))
U211(X1, X2, X3, mark(X4)) → U211(X1, X2, X3, X4)
LENGTH(mark(X)) → LENGTH(X)
ACTIVE(U22(tt, IL, M, N)) → U231(tt, IL, M, N)
MARK(U23(X1, X2, X3, X4)) → MARK(X1)
LENGTH(active(X)) → LENGTH(X)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
U111(active(X1), X2) → U111(X1, X2)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
S(mark(X)) → S(X)
MARK(s(X)) → S(mark(X))
U211(X1, X2, X3, active(X4)) → U211(X1, X2, X3, X4)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
U211(X1, X2, active(X3), X4) → U211(X1, X2, X3, X4)
ACTIVE(U23(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
MARK(U21(X1, X2, X3, X4)) → U211(mark(X1), X2, X3, X4)
ACTIVE(take(s(M), cons(N, IL))) → MARK(U21(tt, IL, M, N))
MARK(U21(X1, X2, X3, X4)) → ACTIVE(U21(mark(X1), X2, X3, X4))
U221(X1, mark(X2), X3, X4) → U221(X1, X2, X3, X4)
TAKE(X1, mark(X2)) → TAKE(X1, X2)
U231(active(X1), X2, X3, X4) → U231(X1, X2, X3, X4)
MARK(0) → ACTIVE(0)
ACTIVE(zeros) → MARK(cons(0, zeros))
U121(mark(X1), X2) → U121(X1, X2)
MARK(nil) → ACTIVE(nil)
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U22(X1, X2, X3, X4)) → U221(mark(X1), X2, X3, X4)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
QDP
          ↳ DependencyGraphProof

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

U211(active(X1), X2, X3, X4) → U211(X1, X2, X3, X4)
ACTIVE(U23(tt, IL, M, N)) → TAKE(M, IL)
MARK(U22(X1, X2, X3, X4)) → MARK(X1)
U231(X1, X2, X3, mark(X4)) → U231(X1, X2, X3, X4)
ACTIVE(U11(tt, L)) → U121(tt, L)
ACTIVE(U21(tt, IL, M, N)) → MARK(U22(tt, IL, M, N))
MARK(tt) → ACTIVE(tt)
ACTIVE(U23(tt, IL, M, N)) → CONS(N, take(M, IL))
MARK(U11(X1, X2)) → MARK(X1)
TAKE(X1, active(X2)) → TAKE(X1, X2)
MARK(cons(X1, X2)) → MARK(X1)
CONS(X1, mark(X2)) → CONS(X1, X2)
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
U121(X1, active(X2)) → U121(X1, X2)
ACTIVE(U12(tt, L)) → S(length(L))
U111(X1, mark(X2)) → U111(X1, X2)
MARK(length(X)) → MARK(X)
MARK(U22(X1, X2, X3, X4)) → ACTIVE(U22(mark(X1), X2, X3, X4))
ACTIVE(zeros) → CONS(0, zeros)
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → TAKE(mark(X1), mark(X2))
U121(active(X1), X2) → U121(X1, X2)
U231(X1, active(X2), X3, X4) → U231(X1, X2, X3, X4)
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
U111(X1, active(X2)) → U111(X1, X2)
S(active(X)) → S(X)
U121(X1, mark(X2)) → U121(X1, X2)
MARK(U12(X1, X2)) → MARK(X1)
MARK(cons(X1, X2)) → CONS(mark(X1), X2)
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
U221(X1, X2, X3, mark(X4)) → U221(X1, X2, X3, X4)
U231(X1, mark(X2), X3, X4) → U231(X1, X2, X3, X4)
TAKE(active(X1), X2) → TAKE(X1, X2)
U221(X1, X2, active(X3), X4) → U221(X1, X2, X3, X4)
MARK(length(X)) → ACTIVE(length(mark(X)))
U231(X1, X2, mark(X3), X4) → U231(X1, X2, X3, X4)
U221(mark(X1), X2, X3, X4) → U221(X1, X2, X3, X4)
U211(X1, active(X2), X3, X4) → U211(X1, X2, X3, X4)
MARK(zeros) → ACTIVE(zeros)
ACTIVE(length(cons(N, L))) → U111(tt, L)
U211(X1, mark(X2), X3, X4) → U211(X1, X2, X3, X4)
MARK(length(X)) → LENGTH(mark(X))
MARK(U21(X1, X2, X3, X4)) → MARK(X1)
TAKE(mark(X1), X2) → TAKE(X1, X2)
U221(active(X1), X2, X3, X4) → U221(X1, X2, X3, X4)
CONS(active(X1), X2) → CONS(X1, X2)
U221(X1, X2, mark(X3), X4) → U221(X1, X2, X3, X4)
ACTIVE(take(s(M), cons(N, IL))) → U211(tt, IL, M, N)
U111(mark(X1), X2) → U111(X1, X2)
U231(X1, X2, active(X3), X4) → U231(X1, X2, X3, X4)
MARK(U23(X1, X2, X3, X4)) → ACTIVE(U23(mark(X1), X2, X3, X4))
ACTIVE(U12(tt, L)) → LENGTH(L)
MARK(U23(X1, X2, X3, X4)) → U231(mark(X1), X2, X3, X4)
CONS(mark(X1), X2) → CONS(X1, X2)
MARK(take(X1, X2)) → MARK(X2)
MARK(s(X)) → MARK(X)
U231(X1, X2, X3, active(X4)) → U231(X1, X2, X3, X4)
U221(X1, X2, X3, active(X4)) → U221(X1, X2, X3, X4)
U221(X1, active(X2), X3, X4) → U221(X1, X2, X3, X4)
ACTIVE(U22(tt, IL, M, N)) → MARK(U23(tt, IL, M, N))
U231(mark(X1), X2, X3, X4) → U231(X1, X2, X3, X4)
MARK(U11(X1, X2)) → U111(mark(X1), X2)
ACTIVE(U21(tt, IL, M, N)) → U221(tt, IL, M, N)
CONS(X1, active(X2)) → CONS(X1, X2)
MARK(U12(X1, X2)) → U121(mark(X1), X2)
U211(X1, X2, mark(X3), X4) → U211(X1, X2, X3, X4)
U211(mark(X1), X2, X3, X4) → U211(X1, X2, X3, X4)
MARK(s(X)) → ACTIVE(s(mark(X)))
U211(X1, X2, X3, mark(X4)) → U211(X1, X2, X3, X4)
LENGTH(mark(X)) → LENGTH(X)
ACTIVE(U22(tt, IL, M, N)) → U231(tt, IL, M, N)
MARK(U23(X1, X2, X3, X4)) → MARK(X1)
LENGTH(active(X)) → LENGTH(X)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
U111(active(X1), X2) → U111(X1, X2)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
S(mark(X)) → S(X)
MARK(s(X)) → S(mark(X))
U211(X1, X2, X3, active(X4)) → U211(X1, X2, X3, X4)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
U211(X1, X2, active(X3), X4) → U211(X1, X2, X3, X4)
ACTIVE(U23(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
MARK(U21(X1, X2, X3, X4)) → U211(mark(X1), X2, X3, X4)
ACTIVE(take(s(M), cons(N, IL))) → MARK(U21(tt, IL, M, N))
MARK(U21(X1, X2, X3, X4)) → ACTIVE(U21(mark(X1), X2, X3, X4))
U221(X1, mark(X2), X3, X4) → U221(X1, X2, X3, X4)
TAKE(X1, mark(X2)) → TAKE(X1, X2)
U231(active(X1), X2, X3, X4) → U231(X1, X2, X3, X4)
MARK(0) → ACTIVE(0)
ACTIVE(zeros) → MARK(cons(0, zeros))
U121(mark(X1), X2) → U121(X1, X2)
MARK(nil) → ACTIVE(nil)
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U22(X1, X2, X3, X4)) → U221(mark(X1), X2, X3, X4)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 10 SCCs with 22 less nodes.

↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
QDP
                ↳ UsableRulesProof
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP

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

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
                ↳ UsableRulesProof
QDP
                    ↳ QDPSizeChangeProof
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP

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

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

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

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



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
QDP
                ↳ UsableRulesProof
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP

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

U231(X1, X2, X3, mark(X4)) → U231(X1, X2, X3, X4)
U231(X1, X2, X3, active(X4)) → U231(X1, X2, X3, X4)
U231(active(X1), X2, X3, X4) → U231(X1, X2, X3, X4)
U231(X1, mark(X2), X3, X4) → U231(X1, X2, X3, X4)
U231(X1, X2, mark(X3), X4) → U231(X1, X2, X3, X4)
U231(mark(X1), X2, X3, X4) → U231(X1, X2, X3, X4)
U231(X1, X2, active(X3), X4) → U231(X1, X2, X3, X4)
U231(X1, active(X2), X3, X4) → U231(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
QDP
                    ↳ QDPSizeChangeProof
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP

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

U231(X1, X2, X3, mark(X4)) → U231(X1, X2, X3, X4)
U231(active(X1), X2, X3, X4) → U231(X1, X2, X3, X4)
U231(X1, X2, X3, active(X4)) → U231(X1, X2, X3, X4)
U231(X1, mark(X2), X3, X4) → U231(X1, X2, X3, X4)
U231(mark(X1), X2, X3, X4) → U231(X1, X2, X3, X4)
U231(X1, X2, mark(X3), X4) → U231(X1, X2, X3, X4)
U231(X1, X2, active(X3), X4) → U231(X1, X2, X3, X4)
U231(X1, active(X2), X3, X4) → U231(X1, X2, X3, X4)

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

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



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
QDP
                ↳ UsableRulesProof
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP

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

U221(X1, X2, X3, mark(X4)) → U221(X1, X2, X3, X4)
U221(X1, X2, X3, active(X4)) → U221(X1, X2, X3, X4)
U221(X1, active(X2), X3, X4) → U221(X1, X2, X3, X4)
U221(X1, X2, active(X3), X4) → U221(X1, X2, X3, X4)
U221(active(X1), X2, X3, X4) → U221(X1, X2, X3, X4)
U221(X1, X2, mark(X3), X4) → U221(X1, X2, X3, X4)
U221(X1, mark(X2), X3, X4) → U221(X1, X2, X3, X4)
U221(mark(X1), X2, X3, X4) → U221(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
QDP
                    ↳ QDPSizeChangeProof
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP

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

U221(X1, X2, X3, mark(X4)) → U221(X1, X2, X3, X4)
U221(X1, active(X2), X3, X4) → U221(X1, X2, X3, X4)
U221(X1, X2, X3, active(X4)) → U221(X1, X2, X3, X4)
U221(active(X1), X2, X3, X4) → U221(X1, X2, X3, X4)
U221(X1, X2, active(X3), X4) → U221(X1, X2, X3, X4)
U221(X1, X2, mark(X3), X4) → U221(X1, X2, X3, X4)
U221(X1, mark(X2), X3, X4) → U221(X1, X2, X3, X4)
U221(mark(X1), X2, X3, X4) → U221(X1, X2, X3, X4)

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

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



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
QDP
                ↳ UsableRulesProof
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP

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

U211(active(X1), X2, X3, X4) → U211(X1, X2, X3, X4)
U211(X1, active(X2), X3, X4) → U211(X1, X2, X3, X4)
U211(X1, X2, X3, active(X4)) → U211(X1, X2, X3, X4)
U211(X1, X2, mark(X3), X4) → U211(X1, X2, X3, X4)
U211(mark(X1), X2, X3, X4) → U211(X1, X2, X3, X4)
U211(X1, X2, active(X3), X4) → U211(X1, X2, X3, X4)
U211(X1, mark(X2), X3, X4) → U211(X1, X2, X3, X4)
U211(X1, X2, X3, mark(X4)) → U211(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
QDP
                    ↳ QDPSizeChangeProof
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP

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

U211(active(X1), X2, X3, X4) → U211(X1, X2, X3, X4)
U211(X1, active(X2), X3, X4) → U211(X1, X2, X3, X4)
U211(mark(X1), X2, X3, X4) → U211(X1, X2, X3, X4)
U211(X1, X2, mark(X3), X4) → U211(X1, X2, X3, X4)
U211(X1, X2, X3, active(X4)) → U211(X1, X2, X3, X4)
U211(X1, mark(X2), X3, X4) → U211(X1, X2, X3, X4)
U211(X1, X2, active(X3), X4) → U211(X1, X2, X3, X4)
U211(X1, X2, X3, mark(X4)) → U211(X1, X2, X3, X4)

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

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



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
QDP
                ↳ UsableRulesProof
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP

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

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
QDP
                    ↳ QDPSizeChangeProof
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP

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

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

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

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



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
QDP
                ↳ UsableRulesProof
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP

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

S(mark(X)) → S(X)
S(active(X)) → S(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
QDP
                    ↳ QDPSizeChangeProof
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP

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

S(active(X)) → S(X)
S(mark(X)) → S(X)

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

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



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
QDP
                ↳ UsableRulesProof
              ↳ QDP
              ↳ QDP
              ↳ QDP

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

U121(X1, mark(X2)) → U121(X1, X2)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
QDP
                    ↳ QDPSizeChangeProof
              ↳ QDP
              ↳ QDP
              ↳ QDP

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

U121(X1, mark(X2)) → U121(X1, X2)
U121(mark(X1), X2) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(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:



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
QDP
                ↳ UsableRulesProof
              ↳ QDP
              ↳ QDP

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

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
QDP
                    ↳ QDPSizeChangeProof
              ↳ QDP
              ↳ QDP

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

U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U111(mark(X1), X2) → U111(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:



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
QDP
                ↳ UsableRulesProof
              ↳ QDP

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

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
QDP
                    ↳ QDPSizeChangeProof
              ↳ QDP

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

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

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

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



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
QDP
                ↳ RuleRemovalProof

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

MARK(U22(X1, X2, X3, X4)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
ACTIVE(U21(tt, IL, M, N)) → MARK(U22(tt, IL, M, N))
MARK(U11(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
ACTIVE(U22(tt, IL, M, N)) → MARK(U23(tt, IL, M, N))
MARK(cons(X1, X2)) → MARK(X1)
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(length(X)) → MARK(X)
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(U22(X1, X2, X3, X4)) → ACTIVE(U22(mark(X1), X2, X3, X4))
MARK(take(X1, X2)) → MARK(X1)
MARK(U23(X1, X2, X3, X4)) → MARK(X1)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U12(X1, X2)) → MARK(X1)
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(U23(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
MARK(U21(X1, X2, X3, X4)) → ACTIVE(U21(mark(X1), X2, X3, X4))
ACTIVE(take(s(M), cons(N, IL))) → MARK(U21(tt, IL, M, N))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(zeros) → ACTIVE(zeros)
ACTIVE(zeros) → MARK(cons(0, zeros))
MARK(U21(X1, X2, X3, X4)) → MARK(X1)
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U23(X1, X2, X3, X4)) → ACTIVE(U23(mark(X1), X2, X3, X4))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

MARK(U22(X1, X2, X3, X4)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(take(X1, X2)) → MARK(X1)
MARK(U23(X1, X2, X3, X4)) → MARK(X1)
MARK(U21(X1, X2, X3, X4)) → MARK(X1)


Used ordering: POLO with Polynomial interpretation [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = 2·x1   
POL(MARK(x1)) = 2·x1   
POL(U11(x1, x2)) = 2·x1 + x2   
POL(U12(x1, x2)) = x1 + x2   
POL(U21(x1, x2, x3, x4)) = 2 + 2·x1 + 2·x2 + 2·x3 + 2·x4   
POL(U22(x1, x2, x3, x4)) = 2 + 2·x1 + 2·x2 + 2·x3 + 2·x4   
POL(U23(x1, x2, x3, x4)) = 2 + 2·x1 + 2·x2 + 2·x3 + 2·x4   
POL(active(x1)) = x1   
POL(cons(x1, x2)) = 2·x1 + x2   
POL(length(x1)) = x1   
POL(mark(x1)) = x1   
POL(nil) = 0   
POL(s(x1)) = x1   
POL(take(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(tt) = 0   
POL(zeros) = 0   



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
QDP
                    ↳ RuleRemovalProof

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

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(U21(tt, IL, M, N)) → MARK(U22(tt, IL, M, N))
MARK(U12(X1, X2)) → MARK(X1)
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(U11(X1, X2)) → MARK(X1)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(s(X)) → MARK(X)
ACTIVE(U23(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(U22(tt, IL, M, N)) → MARK(U23(tt, IL, M, N))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(take(s(M), cons(N, IL))) → MARK(U21(tt, IL, M, N))
MARK(U21(X1, X2, X3, X4)) → ACTIVE(U21(mark(X1), X2, X3, X4))
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(length(X)) → MARK(X)
MARK(zeros) → ACTIVE(zeros)
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(zeros) → MARK(cons(0, zeros))
MARK(U22(X1, X2, X3, X4)) → ACTIVE(U22(mark(X1), X2, X3, X4))
MARK(U23(X1, X2, X3, X4)) → ACTIVE(U23(mark(X1), X2, X3, X4))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

MARK(U12(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → MARK(X1)
MARK(length(X)) → MARK(X)


Used ordering: POLO with Polynomial interpretation [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = 2·x1   
POL(MARK(x1)) = 2·x1   
POL(U11(x1, x2)) = 2 + 2·x1 + x2   
POL(U12(x1, x2)) = 2 + 2·x1 + x2   
POL(U21(x1, x2, x3, x4)) = x1 + x2 + 2·x3 + 2·x4   
POL(U22(x1, x2, x3, x4)) = x1 + x2 + 2·x3 + 2·x4   
POL(U23(x1, x2, x3, x4)) = 2·x1 + x2 + 2·x3 + 2·x4   
POL(active(x1)) = x1   
POL(cons(x1, x2)) = 2·x1 + x2   
POL(length(x1)) = 2 + x1   
POL(mark(x1)) = x1   
POL(nil) = 0   
POL(s(x1)) = x1   
POL(take(x1, x2)) = 2·x1 + x2   
POL(tt) = 0   
POL(zeros) = 0   



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
QDP
                        ↳ QDPOrderProof

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

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(U21(tt, IL, M, N)) → MARK(U22(tt, IL, M, N))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(s(X)) → MARK(X)
ACTIVE(U23(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(U22(tt, IL, M, N)) → MARK(U23(tt, IL, M, N))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(take(s(M), cons(N, IL))) → MARK(U21(tt, IL, M, N))
MARK(U21(X1, X2, X3, X4)) → ACTIVE(U21(mark(X1), X2, X3, X4))
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(zeros) → ACTIVE(zeros)
MARK(U22(X1, X2, X3, X4)) → ACTIVE(U22(mark(X1), X2, X3, X4))
ACTIVE(zeros) → MARK(cons(0, zeros))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U23(X1, X2, X3, X4)) → ACTIVE(U23(mark(X1), X2, X3, X4))
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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


The following pairs can be oriented strictly and are deleted.


ACTIVE(zeros) → MARK(cons(0, zeros))
The remaining pairs can at least be oriented weakly.

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(U21(tt, IL, M, N)) → MARK(U22(tt, IL, M, N))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(s(X)) → MARK(X)
ACTIVE(U23(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(U22(tt, IL, M, N)) → MARK(U23(tt, IL, M, N))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(take(s(M), cons(N, IL))) → MARK(U21(tt, IL, M, N))
MARK(U21(X1, X2, X3, X4)) → ACTIVE(U21(mark(X1), X2, X3, X4))
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(zeros) → ACTIVE(zeros)
MARK(U22(X1, X2, X3, X4)) → ACTIVE(U22(mark(X1), X2, X3, X4))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U23(X1, X2, X3, X4)) → ACTIVE(U23(mark(X1), X2, X3, X4))
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = x1   
POL(MARK(x1)) = x1   
POL(U11(x1, x2)) = 0   
POL(U12(x1, x2)) = 0   
POL(U21(x1, x2, x3, x4)) = x4   
POL(U22(x1, x2, x3, x4)) = x4   
POL(U23(x1, x2, x3, x4)) = x4   
POL(active(x1)) = x1   
POL(cons(x1, x2)) = x1   
POL(length(x1)) = 0   
POL(mark(x1)) = x1   
POL(nil) = 0   
POL(s(x1)) = x1   
POL(take(x1, x2)) = x2   
POL(tt) = 0   
POL(zeros) = 1   

The following usable rules [17] were oriented:

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)
mark(nil) → active(nil)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
mark(tt) → active(tt)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
length(active(X)) → length(X)
length(mark(X)) → length(X)
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U12(tt, L)) → mark(s(length(L)))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
active(length(cons(N, L))) → mark(U11(tt, L))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(zeros) → mark(cons(0, zeros))
mark(length(X)) → active(length(mark(X)))
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(s(X)) → active(s(mark(X)))
active(U11(tt, L)) → mark(U12(tt, L))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
mark(0) → active(0)
U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(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)



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
QDP
                            ↳ DependencyGraphProof

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

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(U21(tt, IL, M, N)) → MARK(U22(tt, IL, M, N))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(s(X)) → MARK(X)
ACTIVE(U23(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(U22(tt, IL, M, N)) → MARK(U23(tt, IL, M, N))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(take(s(M), cons(N, IL))) → MARK(U21(tt, IL, M, N))
MARK(U21(X1, X2, X3, X4)) → ACTIVE(U21(mark(X1), X2, X3, X4))
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(zeros) → ACTIVE(zeros)
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(U22(X1, X2, X3, X4)) → ACTIVE(U22(mark(X1), X2, X3, X4))
MARK(U23(X1, X2, X3, X4)) → ACTIVE(U23(mark(X1), X2, X3, X4))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ DependencyGraphProof
QDP
                                ↳ QDPOrderProof

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

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(U21(tt, IL, M, N)) → MARK(U22(tt, IL, M, N))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(s(X)) → MARK(X)
ACTIVE(U23(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(U22(tt, IL, M, N)) → MARK(U23(tt, IL, M, N))
ACTIVE(take(s(M), cons(N, IL))) → MARK(U21(tt, IL, M, N))
MARK(U21(X1, X2, X3, X4)) → ACTIVE(U21(mark(X1), X2, X3, X4))
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(U22(X1, X2, X3, X4)) → ACTIVE(U22(mark(X1), X2, X3, X4))
MARK(U23(X1, X2, X3, X4)) → ACTIVE(U23(mark(X1), X2, X3, X4))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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


The following pairs can be oriented strictly and are deleted.


MARK(cons(X1, X2)) → MARK(X1)
The remaining pairs can at least be oriented weakly.

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(U21(tt, IL, M, N)) → MARK(U22(tt, IL, M, N))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(s(X)) → MARK(X)
ACTIVE(U23(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(U22(tt, IL, M, N)) → MARK(U23(tt, IL, M, N))
ACTIVE(take(s(M), cons(N, IL))) → MARK(U21(tt, IL, M, N))
MARK(U21(X1, X2, X3, X4)) → ACTIVE(U21(mark(X1), X2, X3, X4))
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(U22(X1, X2, X3, X4)) → ACTIVE(U22(mark(X1), X2, X3, X4))
MARK(U23(X1, X2, X3, X4)) → ACTIVE(U23(mark(X1), X2, X3, X4))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = x1   
POL(MARK(x1)) = x1   
POL(U11(x1, x2)) = 0   
POL(U12(x1, x2)) = 0   
POL(U21(x1, x2, x3, x4)) = 1 + x4   
POL(U22(x1, x2, x3, x4)) = 1 + x4   
POL(U23(x1, x2, x3, x4)) = 1 + x4   
POL(active(x1)) = x1   
POL(cons(x1, x2)) = 1 + x1   
POL(length(x1)) = 0   
POL(mark(x1)) = x1   
POL(nil) = 0   
POL(s(x1)) = x1   
POL(take(x1, x2)) = x2   
POL(tt) = 0   
POL(zeros) = 1   

The following usable rules [17] were oriented:

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)
mark(nil) → active(nil)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
mark(tt) → active(tt)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
length(active(X)) → length(X)
length(mark(X)) → length(X)
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U12(tt, L)) → mark(s(length(L)))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
active(length(cons(N, L))) → mark(U11(tt, L))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(zeros) → mark(cons(0, zeros))
mark(length(X)) → active(length(mark(X)))
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(s(X)) → active(s(mark(X)))
active(U11(tt, L)) → mark(U12(tt, L))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
mark(0) → active(0)
U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(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)



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ QDPOrderProof
QDP
                                    ↳ QDPOrderProof

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

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(U21(tt, IL, M, N)) → MARK(U22(tt, IL, M, N))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(s(X)) → MARK(X)
ACTIVE(U23(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(U22(tt, IL, M, N)) → MARK(U23(tt, IL, M, N))
ACTIVE(take(s(M), cons(N, IL))) → MARK(U21(tt, IL, M, N))
MARK(U21(X1, X2, X3, X4)) → ACTIVE(U21(mark(X1), X2, X3, X4))
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(U22(X1, X2, X3, X4)) → ACTIVE(U22(mark(X1), X2, X3, X4))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U23(X1, X2, X3, X4)) → ACTIVE(U23(mark(X1), X2, X3, X4))
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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


The following pairs can be oriented strictly and are deleted.


MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
The remaining pairs can at least be oriented weakly.

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(U21(tt, IL, M, N)) → MARK(U22(tt, IL, M, N))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(s(X)) → MARK(X)
ACTIVE(U23(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(U22(tt, IL, M, N)) → MARK(U23(tt, IL, M, N))
ACTIVE(take(s(M), cons(N, IL))) → MARK(U21(tt, IL, M, N))
MARK(U21(X1, X2, X3, X4)) → ACTIVE(U21(mark(X1), X2, X3, X4))
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(U22(X1, X2, X3, X4)) → ACTIVE(U22(mark(X1), X2, X3, X4))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U23(X1, X2, X3, X4)) → ACTIVE(U23(mark(X1), X2, X3, X4))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = 0   
POL(MARK(x1)) = x1   
POL(U11(x1, x2)) = 0   
POL(U12(x1, x2)) = 0   
POL(U21(x1, x2, x3, x4)) = 0   
POL(U22(x1, x2, x3, x4)) = 0   
POL(U23(x1, x2, x3, x4)) = 0   
POL(active(x1)) = x1   
POL(cons(x1, x2)) = 0   
POL(length(x1)) = 0   
POL(mark(x1)) = x1   
POL(nil) = 0   
POL(s(x1)) = x1   
POL(take(x1, x2)) = 1   
POL(tt) = 0   
POL(zeros) = 0   

The following usable rules [17] were oriented:

cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
length(active(X)) → length(X)
length(mark(X)) → length(X)
U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
QDP
                                        ↳ QDPOrderProof

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

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(U21(tt, IL, M, N)) → MARK(U22(tt, IL, M, N))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(s(X)) → MARK(X)
ACTIVE(U23(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(U22(tt, IL, M, N)) → MARK(U23(tt, IL, M, N))
ACTIVE(take(s(M), cons(N, IL))) → MARK(U21(tt, IL, M, N))
MARK(U21(X1, X2, X3, X4)) → ACTIVE(U21(mark(X1), X2, X3, X4))
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(U22(X1, X2, X3, X4)) → ACTIVE(U22(mark(X1), X2, X3, X4))
MARK(U23(X1, X2, X3, X4)) → ACTIVE(U23(mark(X1), X2, X3, X4))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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


The following pairs can be oriented strictly and are deleted.


ACTIVE(take(s(M), cons(N, IL))) → MARK(U21(tt, IL, M, N))
The remaining pairs can at least be oriented weakly.

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(U21(tt, IL, M, N)) → MARK(U22(tt, IL, M, N))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(s(X)) → MARK(X)
ACTIVE(U23(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(U22(tt, IL, M, N)) → MARK(U23(tt, IL, M, N))
MARK(U21(X1, X2, X3, X4)) → ACTIVE(U21(mark(X1), X2, X3, X4))
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(U22(X1, X2, X3, X4)) → ACTIVE(U22(mark(X1), X2, X3, X4))
MARK(U23(X1, X2, X3, X4)) → ACTIVE(U23(mark(X1), X2, X3, X4))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = x1   
POL(MARK(x1)) = 0   
POL(U11(x1, x2)) = 0   
POL(U12(x1, x2)) = 0   
POL(U21(x1, x2, x3, x4)) = 0   
POL(U22(x1, x2, x3, x4)) = 0   
POL(U23(x1, x2, x3, x4)) = 0   
POL(active(x1)) = 0   
POL(cons(x1, x2)) = 0   
POL(length(x1)) = 0   
POL(mark(x1)) = 0   
POL(nil) = 0   
POL(s(x1)) = 0   
POL(take(x1, x2)) = 1   
POL(tt) = 0   
POL(zeros) = 0   

The following usable rules [17] were oriented:

cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
length(active(X)) → length(X)
length(mark(X)) → length(X)
U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
QDP
                                            ↳ QDPOrderProof

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

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(U21(tt, IL, M, N)) → MARK(U22(tt, IL, M, N))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(s(X)) → MARK(X)
ACTIVE(U23(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(U22(tt, IL, M, N)) → MARK(U23(tt, IL, M, N))
MARK(U21(X1, X2, X3, X4)) → ACTIVE(U21(mark(X1), X2, X3, X4))
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(U22(X1, X2, X3, X4)) → ACTIVE(U22(mark(X1), X2, X3, X4))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U23(X1, X2, X3, X4)) → ACTIVE(U23(mark(X1), X2, X3, X4))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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


The following pairs can be oriented strictly and are deleted.


MARK(U21(X1, X2, X3, X4)) → ACTIVE(U21(mark(X1), X2, X3, X4))
The remaining pairs can at least be oriented weakly.

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(U21(tt, IL, M, N)) → MARK(U22(tt, IL, M, N))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(s(X)) → MARK(X)
ACTIVE(U23(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(U22(tt, IL, M, N)) → MARK(U23(tt, IL, M, N))
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(U22(X1, X2, X3, X4)) → ACTIVE(U22(mark(X1), X2, X3, X4))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U23(X1, X2, X3, X4)) → ACTIVE(U23(mark(X1), X2, X3, X4))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = 0   
POL(MARK(x1)) = x1   
POL(U11(x1, x2)) = 0   
POL(U12(x1, x2)) = 0   
POL(U21(x1, x2, x3, x4)) = 1 + x4   
POL(U22(x1, x2, x3, x4)) = 0   
POL(U23(x1, x2, x3, x4)) = 0   
POL(active(x1)) = x1   
POL(cons(x1, x2)) = 0   
POL(length(x1)) = 0   
POL(mark(x1)) = x1   
POL(nil) = 0   
POL(s(x1)) = x1   
POL(take(x1, x2)) = 0   
POL(tt) = 0   
POL(zeros) = 0   

The following usable rules [17] were oriented:

cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
length(active(X)) → length(X)
length(mark(X)) → length(X)
U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
QDP
                                                ↳ QDPOrderProof

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

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(U21(tt, IL, M, N)) → MARK(U22(tt, IL, M, N))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(s(X)) → MARK(X)
ACTIVE(U23(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(U22(tt, IL, M, N)) → MARK(U23(tt, IL, M, N))
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(U22(X1, X2, X3, X4)) → ACTIVE(U22(mark(X1), X2, X3, X4))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U23(X1, X2, X3, X4)) → ACTIVE(U23(mark(X1), X2, X3, X4))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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


The following pairs can be oriented strictly and are deleted.


ACTIVE(U21(tt, IL, M, N)) → MARK(U22(tt, IL, M, N))
The remaining pairs can at least be oriented weakly.

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(s(X)) → MARK(X)
ACTIVE(U23(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(U22(tt, IL, M, N)) → MARK(U23(tt, IL, M, N))
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(U22(X1, X2, X3, X4)) → ACTIVE(U22(mark(X1), X2, X3, X4))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U23(X1, X2, X3, X4)) → ACTIVE(U23(mark(X1), X2, X3, X4))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = x1   
POL(MARK(x1)) = 0   
POL(U11(x1, x2)) = 0   
POL(U12(x1, x2)) = 0   
POL(U21(x1, x2, x3, x4)) = 1   
POL(U22(x1, x2, x3, x4)) = 0   
POL(U23(x1, x2, x3, x4)) = 0   
POL(active(x1)) = 0   
POL(cons(x1, x2)) = 0   
POL(length(x1)) = 0   
POL(mark(x1)) = 0   
POL(nil) = 0   
POL(s(x1)) = 0   
POL(take(x1, x2)) = 0   
POL(tt) = 0   
POL(zeros) = 0   

The following usable rules [17] were oriented:

cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
length(active(X)) → length(X)
length(mark(X)) → length(X)
U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
QDP
                                                    ↳ QDPOrderProof

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

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(s(X)) → MARK(X)
ACTIVE(U23(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(U22(tt, IL, M, N)) → MARK(U23(tt, IL, M, N))
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(U22(X1, X2, X3, X4)) → ACTIVE(U22(mark(X1), X2, X3, X4))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U23(X1, X2, X3, X4)) → ACTIVE(U23(mark(X1), X2, X3, X4))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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


The following pairs can be oriented strictly and are deleted.


MARK(U22(X1, X2, X3, X4)) → ACTIVE(U22(mark(X1), X2, X3, X4))
The remaining pairs can at least be oriented weakly.

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(s(X)) → MARK(X)
ACTIVE(U23(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(U22(tt, IL, M, N)) → MARK(U23(tt, IL, M, N))
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U23(X1, X2, X3, X4)) → ACTIVE(U23(mark(X1), X2, X3, X4))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = 0   
POL(MARK(x1)) = x1   
POL(U11(x1, x2)) = 0   
POL(U12(x1, x2)) = 0   
POL(U21(x1, x2, x3, x4)) = 0   
POL(U22(x1, x2, x3, x4)) = 1 + x4   
POL(U23(x1, x2, x3, x4)) = 0   
POL(active(x1)) = x1   
POL(cons(x1, x2)) = 0   
POL(length(x1)) = 0   
POL(mark(x1)) = x1   
POL(nil) = 0   
POL(s(x1)) = x1   
POL(take(x1, x2)) = 0   
POL(tt) = 0   
POL(zeros) = 0   

The following usable rules [17] were oriented:

cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
length(active(X)) → length(X)
length(mark(X)) → length(X)
U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
QDP
                                                        ↳ QDPOrderProof

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

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(U23(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(U22(tt, IL, M, N)) → MARK(U23(tt, IL, M, N))
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U23(X1, X2, X3, X4)) → ACTIVE(U23(mark(X1), X2, X3, X4))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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


The following pairs can be oriented strictly and are deleted.


ACTIVE(U22(tt, IL, M, N)) → MARK(U23(tt, IL, M, N))
The remaining pairs can at least be oriented weakly.

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(U23(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U23(X1, X2, X3, X4)) → ACTIVE(U23(mark(X1), X2, X3, X4))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
Used ordering: Polynomial interpretation [25,35]:

POL(U23(x1, x2, x3, x4)) = 0   
POL(U22(x1, x2, x3, x4)) = 3 + x_1 + (3)x_2 + (2)x_3 + x_4   
POL(mark(x1)) = 0   
POL(U12(x1, x2)) = 0   
POL(U11(x1, x2)) = 0   
POL(U21(x1, x2, x3, x4)) = (3)x_2 + (2)x_3 + (4)x_4   
POL(take(x1, x2)) = (4)x_1 + (2)x_2   
POL(0) = 4   
POL(ACTIVE(x1)) = (4)x_1   
POL(cons(x1, x2)) = 0   
POL(active(x1)) = 3   
POL(MARK(x1)) = 0   
POL(tt) = 1   
POL(zeros) = 3   
POL(s(x1)) = 0   
POL(length(x1)) = 0   
POL(nil) = 2   
The value of delta used in the strict ordering is 16.
The following usable rules [17] were oriented:

cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
length(active(X)) → length(X)
length(mark(X)) → length(X)
U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
QDP
                                                            ↳ QDPOrderProof

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

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(s(X)) → MARK(X)
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(U23(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U23(X1, X2, X3, X4)) → ACTIVE(U23(mark(X1), X2, X3, X4))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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


The following pairs can be oriented strictly and are deleted.


MARK(U23(X1, X2, X3, X4)) → ACTIVE(U23(mark(X1), X2, X3, X4))
The remaining pairs can at least be oriented weakly.

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(s(X)) → MARK(X)
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(U23(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = 0   
POL(MARK(x1)) = x1   
POL(U11(x1, x2)) = 0   
POL(U12(x1, x2)) = 0   
POL(U21(x1, x2, x3, x4)) = 0   
POL(U22(x1, x2, x3, x4)) = 0   
POL(U23(x1, x2, x3, x4)) = 1 + x4   
POL(active(x1)) = x1   
POL(cons(x1, x2)) = 0   
POL(length(x1)) = 0   
POL(mark(x1)) = x1   
POL(nil) = 0   
POL(s(x1)) = x1   
POL(take(x1, x2)) = 0   
POL(tt) = 0   
POL(zeros) = 0   

The following usable rules [17] were oriented:

cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ QDPOrderProof
QDP
                                                                ↳ QDPOrderProof

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

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(U23(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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


The following pairs can be oriented strictly and are deleted.


ACTIVE(U23(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
The remaining pairs can at least be oriented weakly.

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = x1   
POL(MARK(x1)) = 0   
POL(U11(x1, x2)) = 0   
POL(U12(x1, x2)) = 0   
POL(U21(x1, x2, x3, x4)) = 0   
POL(U22(x1, x2, x3, x4)) = 0   
POL(U23(x1, x2, x3, x4)) = 1 + x2   
POL(active(x1)) = 0   
POL(cons(x1, x2)) = 0   
POL(length(x1)) = 0   
POL(mark(x1)) = 0   
POL(nil) = 0   
POL(s(x1)) = 0   
POL(take(x1, x2)) = 0   
POL(tt) = 0   
POL(zeros) = 0   

The following usable rules [17] were oriented:

cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ QDPOrderProof
                                                              ↳ QDP
                                                                ↳ QDPOrderProof
QDP
                                                                    ↳ QDPOrderProof

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

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(s(X)) → MARK(X)
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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


The following pairs can be oriented strictly and are deleted.


MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(s(X)) → ACTIVE(s(mark(X)))
The remaining pairs can at least be oriented weakly.

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(s(X)) → MARK(X)
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = x1   
POL(MARK(x1)) = 1   
POL(U11(x1, x2)) = 1   
POL(U12(x1, x2)) = 1   
POL(U21(x1, x2, x3, x4)) = 0   
POL(U22(x1, x2, x3, x4)) = 0   
POL(U23(x1, x2, x3, x4)) = 0   
POL(active(x1)) = 0   
POL(cons(x1, x2)) = 0   
POL(length(x1)) = 1   
POL(mark(x1)) = 0   
POL(nil) = 0   
POL(s(x1)) = 0   
POL(take(x1, x2)) = 0   
POL(tt) = 0   
POL(zeros) = 0   

The following usable rules [17] were oriented:

cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ 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
                                                                        ↳ Narrowing

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

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(s(X)) → MARK(X)
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2)) at position [0] we obtained the following new rules:

MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))
MARK(U11(zeros, y1)) → ACTIVE(U11(active(zeros), y1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(U11(U23(x0, x1, x2, x3), y1)) → ACTIVE(U11(active(U23(mark(x0), x1, x2, x3)), y1))
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(U11(U21(x0, x1, x2, x3), y1)) → ACTIVE(U11(active(U21(mark(x0), x1, x2, x3)), y1))
MARK(U11(U12(x0, x1), y1)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(U11(U22(x0, x1, x2, x3), y1)) → ACTIVE(U11(active(U22(mark(x0), x1, x2, x3)), y1))
MARK(U11(take(x0, x1), y1)) → ACTIVE(U11(active(take(mark(x0), mark(x1))), y1))



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ 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
                                                                        ↳ Narrowing
QDP
                                                                            ↳ Narrowing

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

MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(s(X)) → MARK(X)
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))
MARK(U11(zeros, y1)) → ACTIVE(U11(active(zeros), y1))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(U23(x0, x1, x2, x3), y1)) → ACTIVE(U11(active(U23(mark(x0), x1, x2, x3)), y1))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(U11(U21(x0, x1, x2, x3), y1)) → ACTIVE(U11(active(U21(mark(x0), x1, x2, x3)), y1))
MARK(U11(U12(x0, x1), y1)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U11(U22(x0, x1, x2, x3), y1)) → ACTIVE(U11(active(U22(mark(x0), x1, x2, x3)), y1))
MARK(U11(take(x0, x1), y1)) → ACTIVE(U11(active(take(mark(x0), mark(x1))), y1))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2)) at position [0] we obtained the following new rules:

MARK(U12(U21(x0, x1, x2, x3), y1)) → ACTIVE(U12(active(U21(mark(x0), x1, x2, x3)), y1))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(x0))), y1))
MARK(U12(U23(x0, x1, x2, x3), y1)) → ACTIVE(U12(active(U23(mark(x0), x1, x2, x3)), y1))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U12(U11(x0, x1), y1)) → ACTIVE(U12(active(U11(mark(x0), x1)), y1))
MARK(U12(cons(x0, x1), y1)) → ACTIVE(U12(active(cons(mark(x0), x1)), y1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(U12(zeros, y1)) → ACTIVE(U12(active(zeros), y1))
MARK(U12(nil, y1)) → ACTIVE(U12(active(nil), y1))
MARK(U12(0, y1)) → ACTIVE(U12(active(0), y1))
MARK(U12(U22(x0, x1, x2, x3), y1)) → ACTIVE(U12(active(U22(mark(x0), x1, x2, x3)), y1))
MARK(U12(take(x0, x1), y1)) → ACTIVE(U12(active(take(mark(x0), mark(x1))), y1))
MARK(U12(length(x0), y1)) → ACTIVE(U12(active(length(mark(x0))), y1))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ 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
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ Narrowing
QDP
                                                                                ↳ Narrowing

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

MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(x0))), y1))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(s(X)) → MARK(X)
MARK(U12(U23(x0, x1, x2, x3), y1)) → ACTIVE(U12(active(U23(mark(x0), x1, x2, x3)), y1))
MARK(U12(cons(x0, x1), y1)) → ACTIVE(U12(active(cons(mark(x0), x1)), y1))
MARK(U12(U11(x0, x1), y1)) → ACTIVE(U12(active(U11(mark(x0), x1)), y1))
MARK(U12(zeros, y1)) → ACTIVE(U12(active(zeros), y1))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(U11(U21(x0, x1, x2, x3), y1)) → ACTIVE(U11(active(U21(mark(x0), x1, x2, x3)), y1))
MARK(U12(0, y1)) → ACTIVE(U12(active(0), y1))
MARK(U12(take(x0, x1), y1)) → ACTIVE(U12(active(take(mark(x0), mark(x1))), y1))
MARK(U11(U12(x0, x1), y1)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1))
MARK(U12(length(x0), y1)) → ACTIVE(U12(active(length(mark(x0))), y1))
MARK(U11(U22(x0, x1, x2, x3), y1)) → ACTIVE(U11(active(U22(mark(x0), x1, x2, x3)), y1))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
MARK(U12(U21(x0, x1, x2, x3), y1)) → ACTIVE(U12(active(U21(mark(x0), x1, x2, x3)), y1))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U11(zeros, y1)) → ACTIVE(U11(active(zeros), y1))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(U12(nil, y1)) → ACTIVE(U12(active(nil), y1))
MARK(U11(U23(x0, x1, x2, x3), y1)) → ACTIVE(U11(active(U23(mark(x0), x1, x2, x3)), y1))
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(U12(U22(x0, x1, x2, x3), y1)) → ACTIVE(U12(active(U22(mark(x0), x1, x2, x3)), y1))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))
MARK(U11(take(x0, x1), y1)) → ACTIVE(U11(active(take(mark(x0), mark(x1))), y1))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(length(tt)) → ACTIVE(length(active(tt)))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(length(0)) → ACTIVE(length(active(0)))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(length(U22(x0, x1, x2, x3))) → ACTIVE(length(active(U22(mark(x0), x1, x2, x3))))
MARK(length(U23(x0, x1, x2, x3))) → ACTIVE(length(active(U23(mark(x0), x1, x2, x3))))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(nil)) → ACTIVE(length(active(nil)))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(length(U21(x0, x1, x2, x3))) → ACTIVE(length(active(U21(mark(x0), x1, x2, x3))))
MARK(length(x0)) → ACTIVE(length(x0))



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ 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
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ Narrowing
                                                                              ↳ QDP
                                                                                ↳ Narrowing
QDP
                                                                                    ↳ Narrowing

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

MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(length(tt)) → ACTIVE(length(active(tt)))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(x0))), y1))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(U12(U23(x0, x1, x2, x3), y1)) → ACTIVE(U12(active(U23(mark(x0), x1, x2, x3)), y1))
MARK(U12(U11(x0, x1), y1)) → ACTIVE(U12(active(U11(mark(x0), x1)), y1))
MARK(U11(U21(x0, x1, x2, x3), y1)) → ACTIVE(U11(active(U21(mark(x0), x1, x2, x3)), y1))
MARK(length(nil)) → ACTIVE(length(active(nil)))
MARK(U11(U12(x0, x1), y1)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
MARK(U12(U21(x0, x1, x2, x3), y1)) → ACTIVE(U12(active(U21(mark(x0), x1, x2, x3)), y1))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(length(0)) → ACTIVE(length(active(0)))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U22(x0, x1, x2, x3))) → ACTIVE(length(active(U22(mark(x0), x1, x2, x3))))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(length(U21(x0, x1, x2, x3))) → ACTIVE(length(active(U21(mark(x0), x1, x2, x3))))
MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
MARK(U12(cons(x0, x1), y1)) → ACTIVE(U12(active(cons(mark(x0), x1)), y1))
MARK(U12(zeros, y1)) → ACTIVE(U12(active(zeros), y1))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(U12(0, y1)) → ACTIVE(U12(active(0), y1))
MARK(U12(take(x0, x1), y1)) → ACTIVE(U12(active(take(mark(x0), mark(x1))), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U12(length(x0), y1)) → ACTIVE(U12(active(length(mark(x0))), y1))
MARK(U11(U22(x0, x1, x2, x3), y1)) → ACTIVE(U11(active(U22(mark(x0), x1, x2, x3)), y1))
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))
MARK(length(U23(x0, x1, x2, x3))) → ACTIVE(length(active(U23(mark(x0), x1, x2, x3))))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U11(zeros, y1)) → ACTIVE(U11(active(zeros), y1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U12(nil, y1)) → ACTIVE(U12(active(nil), y1))
MARK(U11(U23(x0, x1, x2, x3), y1)) → ACTIVE(U11(active(U23(mark(x0), x1, x2, x3)), y1))
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(U12(U22(x0, x1, x2, x3), y1)) → ACTIVE(U12(active(U22(mark(x0), x1, x2, x3)), y1))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(length(x0)) → ACTIVE(length(x0))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))
MARK(U11(take(x0, x1), y1)) → ACTIVE(U11(active(take(mark(x0), mark(x1))), y1))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

MARK(length(tt)) → ACTIVE(length(tt))



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ 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
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ Narrowing
                                                                              ↳ QDP
                                                                                ↳ Narrowing
                                                                                  ↳ QDP
                                                                                    ↳ Narrowing
QDP
                                                                                        ↳ DependencyGraphProof

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

MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(x0))), y1))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(U12(U23(x0, x1, x2, x3), y1)) → ACTIVE(U12(active(U23(mark(x0), x1, x2, x3)), y1))
MARK(U12(U11(x0, x1), y1)) → ACTIVE(U12(active(U11(mark(x0), x1)), y1))
MARK(U11(U21(x0, x1, x2, x3), y1)) → ACTIVE(U11(active(U21(mark(x0), x1, x2, x3)), y1))
MARK(length(nil)) → ACTIVE(length(active(nil)))
MARK(U11(U12(x0, x1), y1)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
MARK(U12(U21(x0, x1, x2, x3), y1)) → ACTIVE(U12(active(U21(mark(x0), x1, x2, x3)), y1))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(length(0)) → ACTIVE(length(active(0)))
MARK(length(tt)) → ACTIVE(length(tt))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U22(x0, x1, x2, x3))) → ACTIVE(length(active(U22(mark(x0), x1, x2, x3))))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(length(U21(x0, x1, x2, x3))) → ACTIVE(length(active(U21(mark(x0), x1, x2, x3))))
MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
MARK(U12(cons(x0, x1), y1)) → ACTIVE(U12(active(cons(mark(x0), x1)), y1))
MARK(U12(zeros, y1)) → ACTIVE(U12(active(zeros), y1))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(U12(0, y1)) → ACTIVE(U12(active(0), y1))
MARK(U12(take(x0, x1), y1)) → ACTIVE(U12(active(take(mark(x0), mark(x1))), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U12(length(x0), y1)) → ACTIVE(U12(active(length(mark(x0))), y1))
MARK(U11(U22(x0, x1, x2, x3), y1)) → ACTIVE(U11(active(U22(mark(x0), x1, x2, x3)), y1))
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(length(U23(x0, x1, x2, x3))) → ACTIVE(length(active(U23(mark(x0), x1, x2, x3))))
MARK(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U11(zeros, y1)) → ACTIVE(U11(active(zeros), y1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(U23(x0, x1, x2, x3), y1)) → ACTIVE(U11(active(U23(mark(x0), x1, x2, x3)), y1))
MARK(U12(nil, y1)) → ACTIVE(U12(active(nil), y1))
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(U12(U22(x0, x1, x2, x3), y1)) → ACTIVE(U12(active(U22(mark(x0), x1, x2, x3)), y1))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))
MARK(length(x0)) → ACTIVE(length(x0))
MARK(U11(take(x0, x1), y1)) → ACTIVE(U11(active(take(mark(x0), mark(x1))), y1))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ 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
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ Narrowing
                                                                              ↳ QDP
                                                                                ↳ Narrowing
                                                                                  ↳ QDP
                                                                                    ↳ Narrowing
                                                                                      ↳ QDP
                                                                                        ↳ DependencyGraphProof
QDP
                                                                                            ↳ Narrowing

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

MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(x0))), y1))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(U12(U23(x0, x1, x2, x3), y1)) → ACTIVE(U12(active(U23(mark(x0), x1, x2, x3)), y1))
MARK(U12(U11(x0, x1), y1)) → ACTIVE(U12(active(U11(mark(x0), x1)), y1))
MARK(U11(U21(x0, x1, x2, x3), y1)) → ACTIVE(U11(active(U21(mark(x0), x1, x2, x3)), y1))
MARK(length(nil)) → ACTIVE(length(active(nil)))
MARK(U11(U12(x0, x1), y1)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
MARK(U12(U21(x0, x1, x2, x3), y1)) → ACTIVE(U12(active(U21(mark(x0), x1, x2, x3)), y1))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(length(0)) → ACTIVE(length(active(0)))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U22(x0, x1, x2, x3))) → ACTIVE(length(active(U22(mark(x0), x1, x2, x3))))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(length(U21(x0, x1, x2, x3))) → ACTIVE(length(active(U21(mark(x0), x1, x2, x3))))
MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
MARK(U12(cons(x0, x1), y1)) → ACTIVE(U12(active(cons(mark(x0), x1)), y1))
MARK(U12(zeros, y1)) → ACTIVE(U12(active(zeros), y1))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(U12(0, y1)) → ACTIVE(U12(active(0), y1))
MARK(U12(take(x0, x1), y1)) → ACTIVE(U12(active(take(mark(x0), mark(x1))), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U12(length(x0), y1)) → ACTIVE(U12(active(length(mark(x0))), y1))
MARK(U11(U22(x0, x1, x2, x3), y1)) → ACTIVE(U11(active(U22(mark(x0), x1, x2, x3)), y1))
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))
MARK(length(U23(x0, x1, x2, x3))) → ACTIVE(length(active(U23(mark(x0), x1, x2, x3))))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U11(zeros, y1)) → ACTIVE(U11(active(zeros), y1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(U23(x0, x1, x2, x3), y1)) → ACTIVE(U11(active(U23(mark(x0), x1, x2, x3)), y1))
MARK(U12(nil, y1)) → ACTIVE(U12(active(nil), y1))
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(U12(U22(x0, x1, x2, x3), y1)) → ACTIVE(U12(active(U22(mark(x0), x1, x2, x3)), y1))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))
MARK(length(x0)) → ACTIVE(length(x0))
MARK(U11(take(x0, x1), y1)) → ACTIVE(U11(active(take(mark(x0), mark(x1))), y1))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

MARK(length(0)) → ACTIVE(length(0))



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ 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
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ Narrowing
                                                                              ↳ QDP
                                                                                ↳ Narrowing
                                                                                  ↳ QDP
                                                                                    ↳ Narrowing
                                                                                      ↳ QDP
                                                                                        ↳ DependencyGraphProof
                                                                                          ↳ QDP
                                                                                            ↳ Narrowing
QDP
                                                                                                ↳ DependencyGraphProof

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

MARK(length(0)) → ACTIVE(length(0))
MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(x0))), y1))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(U12(U23(x0, x1, x2, x3), y1)) → ACTIVE(U12(active(U23(mark(x0), x1, x2, x3)), y1))
MARK(U12(U11(x0, x1), y1)) → ACTIVE(U12(active(U11(mark(x0), x1)), y1))
MARK(U11(U21(x0, x1, x2, x3), y1)) → ACTIVE(U11(active(U21(mark(x0), x1, x2, x3)), y1))
MARK(length(nil)) → ACTIVE(length(active(nil)))
MARK(U11(U12(x0, x1), y1)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
MARK(U12(U21(x0, x1, x2, x3), y1)) → ACTIVE(U12(active(U21(mark(x0), x1, x2, x3)), y1))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U22(x0, x1, x2, x3))) → ACTIVE(length(active(U22(mark(x0), x1, x2, x3))))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(length(U21(x0, x1, x2, x3))) → ACTIVE(length(active(U21(mark(x0), x1, x2, x3))))
MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
MARK(U12(cons(x0, x1), y1)) → ACTIVE(U12(active(cons(mark(x0), x1)), y1))
MARK(U12(zeros, y1)) → ACTIVE(U12(active(zeros), y1))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(U12(0, y1)) → ACTIVE(U12(active(0), y1))
MARK(U12(take(x0, x1), y1)) → ACTIVE(U12(active(take(mark(x0), mark(x1))), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U12(length(x0), y1)) → ACTIVE(U12(active(length(mark(x0))), y1))
MARK(U11(U22(x0, x1, x2, x3), y1)) → ACTIVE(U11(active(U22(mark(x0), x1, x2, x3)), y1))
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(length(U23(x0, x1, x2, x3))) → ACTIVE(length(active(U23(mark(x0), x1, x2, x3))))
MARK(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U11(zeros, y1)) → ACTIVE(U11(active(zeros), y1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U12(nil, y1)) → ACTIVE(U12(active(nil), y1))
MARK(U11(U23(x0, x1, x2, x3), y1)) → ACTIVE(U11(active(U23(mark(x0), x1, x2, x3)), y1))
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(U12(U22(x0, x1, x2, x3), y1)) → ACTIVE(U12(active(U22(mark(x0), x1, x2, x3)), y1))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))
MARK(U11(take(x0, x1), y1)) → ACTIVE(U11(active(take(mark(x0), mark(x1))), y1))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ 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
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ Narrowing
                                                                              ↳ QDP
                                                                                ↳ Narrowing
                                                                                  ↳ QDP
                                                                                    ↳ Narrowing
                                                                                      ↳ QDP
                                                                                        ↳ DependencyGraphProof
                                                                                          ↳ QDP
                                                                                            ↳ Narrowing
                                                                                              ↳ QDP
                                                                                                ↳ DependencyGraphProof
QDP
                                                                                                    ↳ Narrowing

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

MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(x0))), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(s(X)) → MARK(X)
MARK(U12(U23(x0, x1, x2, x3), y1)) → ACTIVE(U12(active(U23(mark(x0), x1, x2, x3)), y1))
MARK(U12(cons(x0, x1), y1)) → ACTIVE(U12(active(cons(mark(x0), x1)), y1))
MARK(U12(U11(x0, x1), y1)) → ACTIVE(U12(active(U11(mark(x0), x1)), y1))
MARK(U12(zeros, y1)) → ACTIVE(U12(active(zeros), y1))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(U11(U21(x0, x1, x2, x3), y1)) → ACTIVE(U11(active(U21(mark(x0), x1, x2, x3)), y1))
MARK(U12(0, y1)) → ACTIVE(U12(active(0), y1))
MARK(length(nil)) → ACTIVE(length(active(nil)))
MARK(U12(take(x0, x1), y1)) → ACTIVE(U12(active(take(mark(x0), mark(x1))), y1))
MARK(U11(U12(x0, x1), y1)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U12(length(x0), y1)) → ACTIVE(U12(active(length(mark(x0))), y1))
MARK(U11(U22(x0, x1, x2, x3), y1)) → ACTIVE(U11(active(U22(mark(x0), x1, x2, x3)), y1))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
MARK(U12(U21(x0, x1, x2, x3), y1)) → ACTIVE(U12(active(U21(mark(x0), x1, x2, x3)), y1))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(length(U22(x0, x1, x2, x3))) → ACTIVE(length(active(U22(mark(x0), x1, x2, x3))))
MARK(length(U23(x0, x1, x2, x3))) → ACTIVE(length(active(U23(mark(x0), x1, x2, x3))))
MARK(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U11(zeros, y1)) → ACTIVE(U11(active(zeros), y1))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(U12(nil, y1)) → ACTIVE(U12(active(nil), y1))
MARK(U11(U23(x0, x1, x2, x3), y1)) → ACTIVE(U11(active(U23(mark(x0), x1, x2, x3)), y1))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(U12(U22(x0, x1, x2, x3), y1)) → ACTIVE(U12(active(U22(mark(x0), x1, x2, x3)), y1))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(length(U21(x0, x1, x2, x3))) → ACTIVE(length(active(U21(mark(x0), x1, x2, x3))))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))
MARK(U11(take(x0, x1), y1)) → ACTIVE(U11(active(take(mark(x0), mark(x1))), y1))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

MARK(length(nil)) → ACTIVE(length(nil))



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ 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
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ Narrowing
                                                                              ↳ QDP
                                                                                ↳ Narrowing
                                                                                  ↳ QDP
                                                                                    ↳ Narrowing
                                                                                      ↳ QDP
                                                                                        ↳ DependencyGraphProof
                                                                                          ↳ QDP
                                                                                            ↳ Narrowing
                                                                                              ↳ QDP
                                                                                                ↳ DependencyGraphProof
                                                                                                  ↳ QDP
                                                                                                    ↳ Narrowing
QDP
                                                                                                        ↳ DependencyGraphProof

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

MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(x0))), y1))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(U12(U23(x0, x1, x2, x3), y1)) → ACTIVE(U12(active(U23(mark(x0), x1, x2, x3)), y1))
MARK(U12(U11(x0, x1), y1)) → ACTIVE(U12(active(U11(mark(x0), x1)), y1))
MARK(length(nil)) → ACTIVE(length(nil))
MARK(U11(U21(x0, x1, x2, x3), y1)) → ACTIVE(U11(active(U21(mark(x0), x1, x2, x3)), y1))
MARK(U11(U12(x0, x1), y1)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
MARK(U12(U21(x0, x1, x2, x3), y1)) → ACTIVE(U12(active(U21(mark(x0), x1, x2, x3)), y1))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U22(x0, x1, x2, x3))) → ACTIVE(length(active(U22(mark(x0), x1, x2, x3))))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(length(U21(x0, x1, x2, x3))) → ACTIVE(length(active(U21(mark(x0), x1, x2, x3))))
MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
MARK(U12(cons(x0, x1), y1)) → ACTIVE(U12(active(cons(mark(x0), x1)), y1))
MARK(U12(zeros, y1)) → ACTIVE(U12(active(zeros), y1))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(U12(0, y1)) → ACTIVE(U12(active(0), y1))
MARK(U12(take(x0, x1), y1)) → ACTIVE(U12(active(take(mark(x0), mark(x1))), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U12(length(x0), y1)) → ACTIVE(U12(active(length(mark(x0))), y1))
MARK(U11(U22(x0, x1, x2, x3), y1)) → ACTIVE(U11(active(U22(mark(x0), x1, x2, x3)), y1))
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))
MARK(length(U23(x0, x1, x2, x3))) → ACTIVE(length(active(U23(mark(x0), x1, x2, x3))))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U11(zeros, y1)) → ACTIVE(U11(active(zeros), y1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(U23(x0, x1, x2, x3), y1)) → ACTIVE(U11(active(U23(mark(x0), x1, x2, x3)), y1))
MARK(U12(nil, y1)) → ACTIVE(U12(active(nil), y1))
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(U12(U22(x0, x1, x2, x3), y1)) → ACTIVE(U12(active(U22(mark(x0), x1, x2, x3)), y1))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))
MARK(length(x0)) → ACTIVE(length(x0))
MARK(U11(take(x0, x1), y1)) → ACTIVE(U11(active(take(mark(x0), mark(x1))), y1))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ 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
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ Narrowing
                                                                              ↳ QDP
                                                                                ↳ Narrowing
                                                                                  ↳ QDP
                                                                                    ↳ Narrowing
                                                                                      ↳ QDP
                                                                                        ↳ DependencyGraphProof
                                                                                          ↳ QDP
                                                                                            ↳ Narrowing
                                                                                              ↳ QDP
                                                                                                ↳ DependencyGraphProof
                                                                                                  ↳ QDP
                                                                                                    ↳ Narrowing
                                                                                                      ↳ QDP
                                                                                                        ↳ DependencyGraphProof
QDP
                                                                                                            ↳ QDPOrderProof

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

MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(x0))), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(s(X)) → MARK(X)
MARK(U12(U23(x0, x1, x2, x3), y1)) → ACTIVE(U12(active(U23(mark(x0), x1, x2, x3)), y1))
MARK(U12(cons(x0, x1), y1)) → ACTIVE(U12(active(cons(mark(x0), x1)), y1))
MARK(U12(U11(x0, x1), y1)) → ACTIVE(U12(active(U11(mark(x0), x1)), y1))
MARK(U12(zeros, y1)) → ACTIVE(U12(active(zeros), y1))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(U11(U21(x0, x1, x2, x3), y1)) → ACTIVE(U11(active(U21(mark(x0), x1, x2, x3)), y1))
MARK(U12(0, y1)) → ACTIVE(U12(active(0), y1))
MARK(U12(take(x0, x1), y1)) → ACTIVE(U12(active(take(mark(x0), mark(x1))), y1))
MARK(U11(U12(x0, x1), y1)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U12(length(x0), y1)) → ACTIVE(U12(active(length(mark(x0))), y1))
MARK(U11(U22(x0, x1, x2, x3), y1)) → ACTIVE(U11(active(U22(mark(x0), x1, x2, x3)), y1))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
MARK(U12(U21(x0, x1, x2, x3), y1)) → ACTIVE(U12(active(U21(mark(x0), x1, x2, x3)), y1))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(length(U22(x0, x1, x2, x3))) → ACTIVE(length(active(U22(mark(x0), x1, x2, x3))))
MARK(length(U23(x0, x1, x2, x3))) → ACTIVE(length(active(U23(mark(x0), x1, x2, x3))))
MARK(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U11(zeros, y1)) → ACTIVE(U11(active(zeros), y1))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(U12(nil, y1)) → ACTIVE(U12(active(nil), y1))
MARK(U11(U23(x0, x1, x2, x3), y1)) → ACTIVE(U11(active(U23(mark(x0), x1, x2, x3)), y1))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(U12(U22(x0, x1, x2, x3), y1)) → ACTIVE(U12(active(U22(mark(x0), x1, x2, x3)), y1))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(length(U21(x0, x1, x2, x3))) → ACTIVE(length(active(U21(mark(x0), x1, x2, x3))))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))
MARK(U11(take(x0, x1), y1)) → ACTIVE(U11(active(take(mark(x0), mark(x1))), y1))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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


The following pairs can be oriented strictly and are deleted.


MARK(U12(U23(x0, x1, x2, x3), y1)) → ACTIVE(U12(active(U23(mark(x0), x1, x2, x3)), y1))
MARK(U12(cons(x0, x1), y1)) → ACTIVE(U12(active(cons(mark(x0), x1)), y1))
MARK(U12(zeros, y1)) → ACTIVE(U12(active(zeros), y1))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(U11(U21(x0, x1, x2, x3), y1)) → ACTIVE(U11(active(U21(mark(x0), x1, x2, x3)), y1))
MARK(U12(0, y1)) → ACTIVE(U12(active(0), y1))
MARK(U12(take(x0, x1), y1)) → ACTIVE(U12(active(take(mark(x0), mark(x1))), y1))
MARK(U11(U22(x0, x1, x2, x3), y1)) → ACTIVE(U11(active(U22(mark(x0), x1, x2, x3)), y1))
MARK(U12(U21(x0, x1, x2, x3), y1)) → ACTIVE(U12(active(U21(mark(x0), x1, x2, x3)), y1))
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
MARK(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))
MARK(U11(zeros, y1)) → ACTIVE(U11(active(zeros), y1))
MARK(U12(nil, y1)) → ACTIVE(U12(active(nil), y1))
MARK(U11(U23(x0, x1, x2, x3), y1)) → ACTIVE(U11(active(U23(mark(x0), x1, x2, x3)), y1))
MARK(U12(U22(x0, x1, x2, x3), y1)) → ACTIVE(U12(active(U22(mark(x0), x1, x2, x3)), y1))
MARK(U11(take(x0, x1), y1)) → ACTIVE(U11(active(take(mark(x0), mark(x1))), y1))
The remaining pairs can at least be oriented weakly.

MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(x0))), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(s(X)) → MARK(X)
MARK(U12(U11(x0, x1), y1)) → ACTIVE(U12(active(U11(mark(x0), x1)), y1))
MARK(U11(U12(x0, x1), y1)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U12(length(x0), y1)) → ACTIVE(U12(active(length(mark(x0))), y1))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(length(U22(x0, x1, x2, x3))) → ACTIVE(length(active(U22(mark(x0), x1, x2, x3))))
MARK(length(U23(x0, x1, x2, x3))) → ACTIVE(length(active(U23(mark(x0), x1, x2, x3))))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(length(U21(x0, x1, x2, x3))) → ACTIVE(length(active(U21(mark(x0), x1, x2, x3))))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))
Used ordering: Polynomial interpretation [25]:

POL(0) = 1   
POL(ACTIVE(x1)) = 0   
POL(MARK(x1)) = x1   
POL(U11(x1, x2)) = x1   
POL(U12(x1, x2)) = x1   
POL(U21(x1, x2, x3, x4)) = 1   
POL(U22(x1, x2, x3, x4)) = 1 + x3   
POL(U23(x1, x2, x3, x4)) = 1 + x1   
POL(active(x1)) = x1   
POL(cons(x1, x2)) = 1 + x2   
POL(length(x1)) = 0   
POL(mark(x1)) = x1   
POL(nil) = 1   
POL(s(x1)) = x1   
POL(take(x1, x2)) = 1   
POL(tt) = 0   
POL(zeros) = 1   

The following usable rules [17] were oriented:

U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ 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
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ Narrowing
                                                                              ↳ QDP
                                                                                ↳ Narrowing
                                                                                  ↳ QDP
                                                                                    ↳ Narrowing
                                                                                      ↳ QDP
                                                                                        ↳ DependencyGraphProof
                                                                                          ↳ QDP
                                                                                            ↳ Narrowing
                                                                                              ↳ QDP
                                                                                                ↳ DependencyGraphProof
                                                                                                  ↳ QDP
                                                                                                    ↳ Narrowing
                                                                                                      ↳ QDP
                                                                                                        ↳ DependencyGraphProof
                                                                                                          ↳ QDP
                                                                                                            ↳ QDPOrderProof
QDP
                                                                                                                ↳ QDPOrderProof

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

MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(x0))), y1))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
MARK(U12(U11(x0, x1), y1)) → ACTIVE(U12(active(U11(mark(x0), x1)), y1))
MARK(U11(U12(x0, x1), y1)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U12(length(x0), y1)) → ACTIVE(U12(active(length(mark(x0))), y1))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(length(U22(x0, x1, x2, x3))) → ACTIVE(length(active(U22(mark(x0), x1, x2, x3))))
MARK(length(U23(x0, x1, x2, x3))) → ACTIVE(length(active(U23(mark(x0), x1, x2, x3))))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(length(U21(x0, x1, x2, x3))) → ACTIVE(length(active(U21(mark(x0), x1, x2, x3))))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))
MARK(length(x0)) → ACTIVE(length(x0))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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


The following pairs can be oriented strictly and are deleted.


MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(U12(U11(x0, x1), y1)) → ACTIVE(U12(active(U11(mark(x0), x1)), y1))
MARK(U11(U12(x0, x1), y1)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1))
MARK(U12(length(x0), y1)) → ACTIVE(U12(active(length(mark(x0))), y1))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))
The remaining pairs can at least be oriented weakly.

MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(x0))), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(length(U22(x0, x1, x2, x3))) → ACTIVE(length(active(U22(mark(x0), x1, x2, x3))))
MARK(length(U23(x0, x1, x2, x3))) → ACTIVE(length(active(U23(mark(x0), x1, x2, x3))))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(length(U21(x0, x1, x2, x3))) → ACTIVE(length(active(U21(mark(x0), x1, x2, x3))))
MARK(length(x0)) → ACTIVE(length(x0))
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = 1   
POL(MARK(x1)) = x1   
POL(U11(x1, x2)) = 1 + x1   
POL(U12(x1, x2)) = 1 + x1   
POL(U21(x1, x2, x3, x4)) = 0   
POL(U22(x1, x2, x3, x4)) = 0   
POL(U23(x1, x2, x3, x4)) = 0   
POL(active(x1)) = x1   
POL(cons(x1, x2)) = 0   
POL(length(x1)) = 1   
POL(mark(x1)) = x1   
POL(nil) = 0   
POL(s(x1)) = x1   
POL(take(x1, x2)) = x1   
POL(tt) = 0   
POL(zeros) = 0   

The following usable rules [17] were oriented:

U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ 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
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ Narrowing
                                                                              ↳ QDP
                                                                                ↳ Narrowing
                                                                                  ↳ QDP
                                                                                    ↳ Narrowing
                                                                                      ↳ QDP
                                                                                        ↳ DependencyGraphProof
                                                                                          ↳ QDP
                                                                                            ↳ Narrowing
                                                                                              ↳ QDP
                                                                                                ↳ DependencyGraphProof
                                                                                                  ↳ QDP
                                                                                                    ↳ Narrowing
                                                                                                      ↳ QDP
                                                                                                        ↳ DependencyGraphProof
                                                                                                          ↳ QDP
                                                                                                            ↳ QDPOrderProof
                                                                                                              ↳ QDP
                                                                                                                ↳ QDPOrderProof
QDP
                                                                                                                    ↳ QDPOrderProof

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

MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(x0))), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(length(U22(x0, x1, x2, x3))) → ACTIVE(length(active(U22(mark(x0), x1, x2, x3))))
MARK(length(U23(x0, x1, x2, x3))) → ACTIVE(length(active(U23(mark(x0), x1, x2, x3))))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(length(U21(x0, x1, x2, x3))) → ACTIVE(length(active(U21(mark(x0), x1, x2, x3))))
MARK(length(x0)) → ACTIVE(length(x0))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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


The following pairs can be oriented strictly and are deleted.


MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
The remaining pairs can at least be oriented weakly.

MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(x0))), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(length(U22(x0, x1, x2, x3))) → ACTIVE(length(active(U22(mark(x0), x1, x2, x3))))
MARK(length(U23(x0, x1, x2, x3))) → ACTIVE(length(active(U23(mark(x0), x1, x2, x3))))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(length(U21(x0, x1, x2, x3))) → ACTIVE(length(active(U21(mark(x0), x1, x2, x3))))
MARK(length(x0)) → ACTIVE(length(x0))
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( U23(x1, ..., x4) ) =
/0\
\0/
+
/10\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3+
/00\
\00/
·x4

M( U22(x1, ..., x4) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3+
/00\
\00/
·x4

M( mark(x1) ) =
/0\
\0/
+
/11\
\00/
·x1

M( U12(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( U11(x1, x2) ) =
/0\
\0/
+
/11\
\00/
·x1+
/00\
\00/
·x2

M( U21(x1, ..., x4) ) =
/0\
\0/
+
/00\
\11/
·x1+
/11\
\10/
·x2+
/10\
\00/
·x3+
/00\
\11/
·x4

M( take(x1, x2) ) =
/0\
\0/
+
/10\
\00/
·x1+
/00\
\00/
·x2

M( 0 ) =
/0\
\1/

M( active(x1) ) =
/0\
\0/
+
/11\
\00/
·x1

M( cons(x1, x2) ) =
/0\
\0/
+
/11\
\00/
·x1+
/00\
\11/
·x2

M( tt ) =
/0\
\0/

M( zeros ) =
/0\
\0/

M( s(x1) ) =
/0\
\1/
+
/10\
\00/
·x1

M( length(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( nil ) =
/0\
\0/

Tuple symbols:
M( MARK(x1) ) = 0+
[1,0]
·x1

M( ACTIVE(x1) ) = 0+
[0,0]
·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:

U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
length(active(X)) → length(X)
length(mark(X)) → length(X)
U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ 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
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ Narrowing
                                                                              ↳ QDP
                                                                                ↳ Narrowing
                                                                                  ↳ QDP
                                                                                    ↳ Narrowing
                                                                                      ↳ QDP
                                                                                        ↳ DependencyGraphProof
                                                                                          ↳ QDP
                                                                                            ↳ Narrowing
                                                                                              ↳ QDP
                                                                                                ↳ DependencyGraphProof
                                                                                                  ↳ QDP
                                                                                                    ↳ Narrowing
                                                                                                      ↳ QDP
                                                                                                        ↳ DependencyGraphProof
                                                                                                          ↳ QDP
                                                                                                            ↳ QDPOrderProof
                                                                                                              ↳ QDP
                                                                                                                ↳ QDPOrderProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ QDPOrderProof
QDP
                                                                                                                        ↳ QDPOrderProof

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

MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(x0))), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(s(X)) → MARK(X)
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(length(U22(x0, x1, x2, x3))) → ACTIVE(length(active(U22(mark(x0), x1, x2, x3))))
MARK(length(U23(x0, x1, x2, x3))) → ACTIVE(length(active(U23(mark(x0), x1, x2, x3))))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(length(U21(x0, x1, x2, x3))) → ACTIVE(length(active(U21(mark(x0), x1, x2, x3))))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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


The following pairs can be oriented strictly and are deleted.


MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(x0))), y1))
The remaining pairs can at least be oriented weakly.

MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(s(X)) → MARK(X)
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(length(U22(x0, x1, x2, x3))) → ACTIVE(length(active(U22(mark(x0), x1, x2, x3))))
MARK(length(U23(x0, x1, x2, x3))) → ACTIVE(length(active(U23(mark(x0), x1, x2, x3))))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(length(U21(x0, x1, x2, x3))) → ACTIVE(length(active(U21(mark(x0), x1, x2, x3))))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( U23(x1, ..., x4) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3+
/00\
\00/
·x4

M( U22(x1, ..., x4) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3+
/00\
\00/
·x4

M( mark(x1) ) =
/0\
\0/
+
/10\
\01/
·x1

M( U12(x1, x2) ) =
/0\
\0/
+
/00\
\10/
·x1+
/00\
\00/
·x2

M( U11(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( U21(x1, ..., x4) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3+
/00\
\00/
·x4

M( take(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/10\
\00/
·x2

M( 0 ) =
/0\
\0/

M( active(x1) ) =
/0\
\0/
+
/10\
\01/
·x1

M( cons(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( tt ) =
/0\
\0/

M( zeros ) =
/0\
\0/

M( s(x1) ) =
/1\
\0/
+
/00\
\01/
·x1

M( length(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( nil ) =
/0\
\0/

Tuple symbols:
M( MARK(x1) ) = 0+
[0,1]
·x1

M( ACTIVE(x1) ) = 0+
[0,0]
·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:

U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ 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
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ Narrowing
                                                                              ↳ QDP
                                                                                ↳ Narrowing
                                                                                  ↳ QDP
                                                                                    ↳ Narrowing
                                                                                      ↳ QDP
                                                                                        ↳ DependencyGraphProof
                                                                                          ↳ QDP
                                                                                            ↳ Narrowing
                                                                                              ↳ QDP
                                                                                                ↳ DependencyGraphProof
                                                                                                  ↳ QDP
                                                                                                    ↳ Narrowing
                                                                                                      ↳ QDP
                                                                                                        ↳ DependencyGraphProof
                                                                                                          ↳ QDP
                                                                                                            ↳ QDPOrderProof
                                                                                                              ↳ QDP
                                                                                                                ↳ QDPOrderProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ QDPOrderProof
                                                                                                                      ↳ QDP
                                                                                                                        ↳ QDPOrderProof
QDP
                                                                                                                            ↳ QDPOrderProof

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

MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(s(X)) → MARK(X)
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(length(U22(x0, x1, x2, x3))) → ACTIVE(length(active(U22(mark(x0), x1, x2, x3))))
MARK(length(U23(x0, x1, x2, x3))) → ACTIVE(length(active(U23(mark(x0), x1, x2, x3))))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(length(U21(x0, x1, x2, x3))) → ACTIVE(length(active(U21(mark(x0), x1, x2, x3))))
MARK(length(x0)) → ACTIVE(length(x0))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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


The following pairs can be oriented strictly and are deleted.


MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
The remaining pairs can at least be oriented weakly.

MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(s(X)) → MARK(X)
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(length(U22(x0, x1, x2, x3))) → ACTIVE(length(active(U22(mark(x0), x1, x2, x3))))
MARK(length(U23(x0, x1, x2, x3))) → ACTIVE(length(active(U23(mark(x0), x1, x2, x3))))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(length(U21(x0, x1, x2, x3))) → ACTIVE(length(active(U21(mark(x0), x1, x2, x3))))
MARK(length(x0)) → ACTIVE(length(x0))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( U23(x1, ..., x4) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3+
/00\
\00/
·x4

M( U22(x1, ..., x4) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3+
/00\
\00/
·x4

M( mark(x1) ) =
/1\
\0/
+
/10\
\01/
·x1

M( U12(x1, x2) ) =
/0\
\0/
+
/00\
\01/
·x1+
/00\
\01/
·x2

M( U11(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\11/
·x2

M( U21(x1, ..., x4) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3+
/00\
\00/
·x4

M( take(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( 0 ) =
/0\
\0/

M( active(x1) ) =
/1\
\0/
+
/00\
\11/
·x1

M( cons(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\11/
·x2

M( tt ) =
/0\
\0/

M( zeros ) =
/0\
\0/

M( s(x1) ) =
/0\
\0/
+
/00\
\01/
·x1

M( length(x1) ) =
/0\
\0/
+
/00\
\01/
·x1

M( nil ) =
/0\
\0/

Tuple symbols:
M( MARK(x1) ) = 0+
[0,1]
·x1

M( ACTIVE(x1) ) = 0+
[0,1]
·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:

U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U12(tt, L)) → mark(s(length(L)))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
active(length(cons(N, L))) → mark(U11(tt, L))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(zeros) → mark(cons(0, zeros))
mark(length(X)) → active(length(mark(X)))
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(s(X)) → active(s(mark(X)))
active(U11(tt, L)) → mark(U12(tt, L))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
mark(0) → active(0)
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)
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)
mark(nil) → active(nil)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
mark(tt) → active(tt)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
length(active(X)) → length(X)
length(mark(X)) → length(X)
U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ 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
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ Narrowing
                                                                              ↳ QDP
                                                                                ↳ Narrowing
                                                                                  ↳ QDP
                                                                                    ↳ Narrowing
                                                                                      ↳ QDP
                                                                                        ↳ DependencyGraphProof
                                                                                          ↳ QDP
                                                                                            ↳ Narrowing
                                                                                              ↳ QDP
                                                                                                ↳ DependencyGraphProof
                                                                                                  ↳ QDP
                                                                                                    ↳ Narrowing
                                                                                                      ↳ QDP
                                                                                                        ↳ DependencyGraphProof
                                                                                                          ↳ QDP
                                                                                                            ↳ QDPOrderProof
                                                                                                              ↳ QDP
                                                                                                                ↳ QDPOrderProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ QDPOrderProof
                                                                                                                      ↳ QDP
                                                                                                                        ↳ QDPOrderProof
                                                                                                                          ↳ QDP
                                                                                                                            ↳ QDPOrderProof
QDP
                                                                                                                                ↳ QDPOrderProof

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

MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(s(X)) → MARK(X)
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(length(U22(x0, x1, x2, x3))) → ACTIVE(length(active(U22(mark(x0), x1, x2, x3))))
MARK(length(U23(x0, x1, x2, x3))) → ACTIVE(length(active(U23(mark(x0), x1, x2, x3))))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(length(U21(x0, x1, x2, x3))) → ACTIVE(length(active(U21(mark(x0), x1, x2, x3))))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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


The following pairs can be oriented strictly and are deleted.


MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
The remaining pairs can at least be oriented weakly.

MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(s(X)) → MARK(X)
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(length(U22(x0, x1, x2, x3))) → ACTIVE(length(active(U22(mark(x0), x1, x2, x3))))
MARK(length(U23(x0, x1, x2, x3))) → ACTIVE(length(active(U23(mark(x0), x1, x2, x3))))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(length(U21(x0, x1, x2, x3))) → ACTIVE(length(active(U21(mark(x0), x1, x2, x3))))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( U23(x1, ..., x4) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3+
/00\
\00/
·x4

M( U22(x1, ..., x4) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3+
/00\
\00/
·x4

M( mark(x1) ) =
/0\
\1/
+
/10\
\01/
·x1

M( U12(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/11\
\00/
·x2

M( U11(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/11\
\00/
·x2

M( U21(x1, ..., x4) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3+
/00\
\00/
·x4

M( take(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( 0 ) =
/0\
\0/

M( cons(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/11\
\00/
·x2

M( active(x1) ) =
/0\
\1/
+
/10\
\01/
·x1

M( tt ) =
/0\
\0/

M( zeros ) =
/0\
\0/

M( s(x1) ) =
/0\
\0/
+
/10\
\00/
·x1

M( length(x1) ) =
/0\
\0/
+
/10\
\00/
·x1

M( nil ) =
/0\
\0/

Tuple symbols:
M( MARK(x1) ) = 0+
[1,0]
·x1

M( ACTIVE(x1) ) = 0+
[1,0]
·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:

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)
mark(nil) → active(nil)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
mark(tt) → active(tt)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
length(active(X)) → length(X)
length(mark(X)) → length(X)
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U12(tt, L)) → mark(s(length(L)))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
active(length(cons(N, L))) → mark(U11(tt, L))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(zeros) → mark(cons(0, zeros))
mark(length(X)) → active(length(mark(X)))
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(s(X)) → active(s(mark(X)))
active(U11(tt, L)) → mark(U12(tt, L))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
mark(0) → active(0)
U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(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)



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ 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
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ Narrowing
                                                                              ↳ QDP
                                                                                ↳ Narrowing
                                                                                  ↳ QDP
                                                                                    ↳ Narrowing
                                                                                      ↳ QDP
                                                                                        ↳ DependencyGraphProof
                                                                                          ↳ QDP
                                                                                            ↳ Narrowing
                                                                                              ↳ QDP
                                                                                                ↳ DependencyGraphProof
                                                                                                  ↳ QDP
                                                                                                    ↳ Narrowing
                                                                                                      ↳ QDP
                                                                                                        ↳ DependencyGraphProof
                                                                                                          ↳ QDP
                                                                                                            ↳ QDPOrderProof
                                                                                                              ↳ QDP
                                                                                                                ↳ QDPOrderProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ QDPOrderProof
                                                                                                                      ↳ QDP
                                                                                                                        ↳ QDPOrderProof
                                                                                                                          ↳ QDP
                                                                                                                            ↳ QDPOrderProof
                                                                                                                              ↳ QDP
                                                                                                                                ↳ QDPOrderProof
QDP
                                                                                                                                    ↳ QDPOrderProof

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

MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(s(X)) → MARK(X)
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(length(U22(x0, x1, x2, x3))) → ACTIVE(length(active(U22(mark(x0), x1, x2, x3))))
MARK(length(U23(x0, x1, x2, x3))) → ACTIVE(length(active(U23(mark(x0), x1, x2, x3))))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(length(U21(x0, x1, x2, x3))) → ACTIVE(length(active(U21(mark(x0), x1, x2, x3))))
MARK(length(x0)) → ACTIVE(length(x0))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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


The following pairs can be oriented strictly and are deleted.


MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
The remaining pairs can at least be oriented weakly.

MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(s(X)) → MARK(X)
MARK(length(U22(x0, x1, x2, x3))) → ACTIVE(length(active(U22(mark(x0), x1, x2, x3))))
MARK(length(U23(x0, x1, x2, x3))) → ACTIVE(length(active(U23(mark(x0), x1, x2, x3))))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(length(U21(x0, x1, x2, x3))) → ACTIVE(length(active(U21(mark(x0), x1, x2, x3))))
MARK(length(x0)) → ACTIVE(length(x0))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( U23(x1, ..., x4) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3+
/00\
\00/
·x4

M( U22(x1, ..., x4) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3+
/00\
\00/
·x4

M( mark(x1) ) =
/0\
\0/
+
/10\
\01/
·x1

M( U12(x1, x2) ) =
/0\
\1/
+
/00\
\00/
·x1+
/11\
\11/
·x2

M( U11(x1, x2) ) =
/0\
\1/
+
/00\
\00/
·x1+
/11\
\11/
·x2

M( U21(x1, ..., x4) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3+
/00\
\00/
·x4

M( take(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( 0 ) =
/0\
\0/

M( cons(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/11\
\00/
·x2

M( active(x1) ) =
/0\
\0/
+
/10\
\01/
·x1

M( tt ) =
/0\
\0/

M( zeros ) =
/0\
\0/

M( s(x1) ) =
/0\
\0/
+
/10\
\00/
·x1

M( length(x1) ) =
/0\
\1/
+
/11\
\10/
·x1

M( nil ) =
/0\
\0/

Tuple symbols:
M( MARK(x1) ) = 1+
[1,0]
·x1

M( ACTIVE(x1) ) = 0+
[0,1]
·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:

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)
mark(nil) → active(nil)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
mark(tt) → active(tt)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
length(active(X)) → length(X)
length(mark(X)) → length(X)
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U12(tt, L)) → mark(s(length(L)))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
active(length(cons(N, L))) → mark(U11(tt, L))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(zeros) → mark(cons(0, zeros))
mark(length(X)) → active(length(mark(X)))
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(s(X)) → active(s(mark(X)))
active(U11(tt, L)) → mark(U12(tt, L))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
mark(0) → active(0)
U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(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)



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ 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
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ Narrowing
                                                                              ↳ QDP
                                                                                ↳ Narrowing
                                                                                  ↳ QDP
                                                                                    ↳ Narrowing
                                                                                      ↳ QDP
                                                                                        ↳ DependencyGraphProof
                                                                                          ↳ QDP
                                                                                            ↳ Narrowing
                                                                                              ↳ QDP
                                                                                                ↳ DependencyGraphProof
                                                                                                  ↳ QDP
                                                                                                    ↳ Narrowing
                                                                                                      ↳ QDP
                                                                                                        ↳ DependencyGraphProof
                                                                                                          ↳ QDP
                                                                                                            ↳ QDPOrderProof
                                                                                                              ↳ QDP
                                                                                                                ↳ QDPOrderProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ QDPOrderProof
                                                                                                                      ↳ QDP
                                                                                                                        ↳ QDPOrderProof
                                                                                                                          ↳ QDP
                                                                                                                            ↳ QDPOrderProof
                                                                                                                              ↳ QDP
                                                                                                                                ↳ QDPOrderProof
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ QDPOrderProof
QDP
                                                                                                                                        ↳ QDPOrderProof

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

MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(s(X)) → MARK(X)
MARK(length(U22(x0, x1, x2, x3))) → ACTIVE(length(active(U22(mark(x0), x1, x2, x3))))
MARK(length(U23(x0, x1, x2, x3))) → ACTIVE(length(active(U23(mark(x0), x1, x2, x3))))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(U21(x0, x1, x2, x3))) → ACTIVE(length(active(U21(mark(x0), x1, x2, x3))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(length(x0)) → ACTIVE(length(x0))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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


The following pairs can be oriented strictly and are deleted.


MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
The remaining pairs can at least be oriented weakly.

MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(s(X)) → MARK(X)
MARK(length(U22(x0, x1, x2, x3))) → ACTIVE(length(active(U22(mark(x0), x1, x2, x3))))
MARK(length(U23(x0, x1, x2, x3))) → ACTIVE(length(active(U23(mark(x0), x1, x2, x3))))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(length(U21(x0, x1, x2, x3))) → ACTIVE(length(active(U21(mark(x0), x1, x2, x3))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(length(x0)) → ACTIVE(length(x0))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( U23(x1, ..., x4) ) =
/1\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3+
/00\
\00/
·x4

M( U22(x1, ..., x4) ) =
/0\
\0/
+
/01\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3+
/00\
\00/
·x4

M( mark(x1) ) =
/0\
\0/
+
/10\
\01/
·x1

M( U12(x1, x2) ) =
/0\
\0/
+
/10\
\10/
·x1+
/11\
\11/
·x2

M( U11(x1, x2) ) =
/0\
\0/
+
/01\
\01/
·x1+
/11\
\11/
·x2

M( U21(x1, ..., x4) ) =
/0\
\0/
+
/01\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3+
/00\
\00/
·x4

M( take(x1, x2) ) =
/1\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( 0 ) =
/0\
\0/

M( cons(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/11\
\00/
·x2

M( active(x1) ) =
/0\
\0/
+
/10\
\01/
·x1

M( tt ) =
/1\
\1/

M( zeros ) =
/1\
\0/

M( s(x1) ) =
/0\
\1/
+
/10\
\00/
·x1

M( length(x1) ) =
/1\
\1/
+
/11\
\10/
·x1

M( nil ) =
/0\
\0/

Tuple symbols:
M( MARK(x1) ) = 0+
[1,0]
·x1

M( ACTIVE(x1) ) = 0+
[0,1]
·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:

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)
mark(nil) → active(nil)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
mark(tt) → active(tt)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
length(active(X)) → length(X)
length(mark(X)) → length(X)
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U12(tt, L)) → mark(s(length(L)))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
active(length(cons(N, L))) → mark(U11(tt, L))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(zeros) → mark(cons(0, zeros))
mark(length(X)) → active(length(mark(X)))
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(s(X)) → active(s(mark(X)))
active(U11(tt, L)) → mark(U12(tt, L))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
mark(0) → active(0)
U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(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)



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ 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
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ Narrowing
                                                                              ↳ QDP
                                                                                ↳ Narrowing
                                                                                  ↳ QDP
                                                                                    ↳ Narrowing
                                                                                      ↳ QDP
                                                                                        ↳ DependencyGraphProof
                                                                                          ↳ QDP
                                                                                            ↳ Narrowing
                                                                                              ↳ QDP
                                                                                                ↳ DependencyGraphProof
                                                                                                  ↳ QDP
                                                                                                    ↳ Narrowing
                                                                                                      ↳ QDP
                                                                                                        ↳ DependencyGraphProof
                                                                                                          ↳ QDP
                                                                                                            ↳ QDPOrderProof
                                                                                                              ↳ QDP
                                                                                                                ↳ QDPOrderProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ QDPOrderProof
                                                                                                                      ↳ QDP
                                                                                                                        ↳ QDPOrderProof
                                                                                                                          ↳ QDP
                                                                                                                            ↳ QDPOrderProof
                                                                                                                              ↳ QDP
                                                                                                                                ↳ QDPOrderProof
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ QDPOrderProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ QDPOrderProof
QDP

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

MARK(length(take(x0, x1))) → ACTIVE(length(active(take(mark(x0), mark(x1)))))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(s(X)) → MARK(X)
MARK(length(U22(x0, x1, x2, x3))) → ACTIVE(length(active(U22(mark(x0), x1, x2, x3))))
MARK(length(U23(x0, x1, x2, x3))) → ACTIVE(length(active(U23(mark(x0), x1, x2, x3))))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(length(U21(x0, x1, x2, x3))) → ACTIVE(length(active(U21(mark(x0), x1, x2, x3))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(length(x0)) → ACTIVE(length(x0))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(U21(tt, IL, M, N)) → mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) → mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(U11(tt, L))
active(take(s(M), cons(N, IL))) → mark(U21(tt, IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X1, X2, X3, X4)) → active(U21(mark(X1), X2, X3, X4))
mark(U22(X1, X2, X3, X4)) → active(U22(mark(X1), X2, X3, X4))
mark(U23(X1, X2, X3, X4)) → active(U23(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, mark(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, mark(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, mark(X4)) → U21(X1, X2, X3, X4)
U21(active(X1), X2, X3, X4) → U21(X1, X2, X3, X4)
U21(X1, active(X2), X3, X4) → U21(X1, X2, X3, X4)
U21(X1, X2, active(X3), X4) → U21(X1, X2, X3, X4)
U21(X1, X2, X3, active(X4)) → U21(X1, X2, X3, X4)
U22(mark(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, mark(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, mark(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, mark(X4)) → U22(X1, X2, X3, X4)
U22(active(X1), X2, X3, X4) → U22(X1, X2, X3, X4)
U22(X1, active(X2), X3, X4) → U22(X1, X2, X3, X4)
U22(X1, X2, active(X3), X4) → U22(X1, X2, X3, X4)
U22(X1, X2, X3, active(X4)) → U22(X1, X2, X3, X4)
U23(mark(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, mark(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, mark(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, mark(X4)) → U23(X1, X2, X3, X4)
U23(active(X1), X2, X3, X4) → U23(X1, X2, X3, X4)
U23(X1, active(X2), X3, X4) → U23(X1, X2, X3, X4)
U23(X1, X2, active(X3), X4) → U23(X1, X2, X3, X4)
U23(X1, X2, X3, active(X4)) → U23(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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