Termination w.r.t. Q proof of /tmp/tmpMzCZNk/OvConsOS_nokinds

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(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.

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

ACTIVE(take(X1, X2)) → TAKE(X1, active(X2))
ACTIVE(cons(X1, X2)) → CONS(active(X1), X2)
ACTIVE(length(cons(N, L))) → AND(isNatList(L), isNat(N))
PROPER(U11(X1, X2)) → U11¹(proper(X1), proper(X2))
PROPER(isNatList(X)) → PROPER(X)
PROPER(length(X)) → LENGTH(proper(X))
ACTIVE(take(X1, X2)) → ACTIVE(X1)
ACTIVE(take(s(M), cons(N, IL))) → ISNAT(N)
ACTIVE(U31(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → TAKE(active(X1), X2)
PROPER(U11(X1, X2)) → PROPER(X2)
PROPER(length(X)) → PROPER(X)
PROPER(isNatList(X)) → ISNATLIST(proper(X))
ACTIVE(U11(X1, X2)) → ACTIVE(X1)
PROPER(U31(X1, X2, X3, X4)) → PROPER(X4)
ACTIVE(take(s(M), cons(N, IL))) → AND(isNat(M), isNat(N))
ACTIVE(s(X)) → ACTIVE(X)
PROPER(U31(X1, X2, X3, X4)) → PROPER(X1)
PROPER(U31(X1, X2, X3, X4)) → PROPER(X2)
S(ok(X)) → S(X)
CONS(mark(X1), X2) → CONS(X1, X2)
TOP(mark(X)) → PROPER(X)
ACTIVE(U31(tt, IL, M, N)) → TAKE(M, IL)
ACTIVE(and(X1, X2)) → AND(active(X1), X2)
TOP(ok(X)) → ACTIVE(X)
ACTIVE(length(X)) → ACTIVE(X)
PROPER(and(X1, X2)) → AND(proper(X1), proper(X2))
ACTIVE(take(s(M), cons(N, IL))) → U31¹(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(U21(X)) → U21¹(proper(X))
LENGTH(mark(X)) → LENGTH(X)
ACTIVE(isNatList(take(V1, V2))) → ISNATILIST(V2)
S(mark(X)) → S(X)
PROPER(cons(X1, X2)) → CONS(proper(X1), proper(X2))
ACTIVE(length(X)) → LENGTH(active(X))
PROPER(and(X1, X2)) → PROPER(X1)
ISNATLIST(ok(X)) → ISNATLIST(X)
ACTIVE(isNatList(cons(V1, V2))) → ISNATLIST(V2)
ACTIVE(isNat(s(V1))) → ISNAT(V1)
AND(mark(X1), X2) → AND(X1, X2)
PROPER(U21(X)) → PROPER(X)
ACTIVE(take(0, IL)) → U21¹(isNatIList(IL))
ACTIVE(isNatList(take(V1, V2))) → AND(isNat(V1), isNatIList(V2))
U21¹(ok(X)) → U21¹(X)
ACTIVE(isNatIList(V)) → ISNATLIST(V)
ACTIVE(zeros) → CONS(0, zeros)
ACTIVE(isNatList(take(V1, V2))) → ISNAT(V1)
PROPER(cons(X1, X2)) → PROPER(X2)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(isNatIList(cons(V1, V2))) → ISNAT(V1)
ISNATILIST(ok(X)) → ISNATILIST(X)
ACTIVE(isNatIList(cons(V1, V2))) → ISNATILIST(V2)
U21¹(mark(X)) → U21¹(X)
PROPER(s(X)) → S(proper(X))
ACTIVE(and(X1, X2)) → ACTIVE(X1)
ACTIVE(isNatList(cons(V1, V2))) → ISNAT(V1)
U11¹(ok(X1), ok(X2)) → U11¹(X1, X2)
ACTIVE(isNatIList(cons(V1, V2))) → AND(isNat(V1), isNatIList(V2))
PROPER(and(X1, X2)) → PROPER(X2)
PROPER(isNatIList(X)) → ISNATILIST(proper(X))
ACTIVE(take(0, IL)) → ISNATILIST(IL)
ACTIVE(U11(tt, L)) → S(length(L))
TAKE(mark(X1), X2) → TAKE(X1, X2)
U11¹(mark(X1), X2) → U11¹(X1, X2)
ACTIVE(U31(X1, X2, X3, X4)) → U31¹(active(X1), X2, X3, X4)
PROPER(isNat(X)) → PROPER(X)
ACTIVE(take(s(M), cons(N, IL))) → AND(isNatIList(IL), and(isNat(M), isNat(N)))
LENGTH(ok(X)) → LENGTH(X)
ACTIVE(take(X1, X2)) → ACTIVE(X2)
PROPER(take(X1, X2)) → TAKE(proper(X1), proper(X2))
PROPER(U31(X1, X2, X3, X4)) → PROPER(X3)
ACTIVE(isNat(length(V1))) → ISNATLIST(V1)
AND(ok(X1), ok(X2)) → AND(X1, X2)
PROPER(isNatIList(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
ACTIVE(take(s(M), cons(N, IL))) → ISNATILIST(IL)
ACTIVE(U11(X1, X2)) → U11¹(active(X1), X2)
PROPER(take(X1, X2)) → PROPER(X2)
TOP(ok(X)) → TOP(active(X))
PROPER(isNat(X)) → ISNAT(proper(X))
ACTIVE(U21(X)) → ACTIVE(X)
U31¹(ok(X1), ok(X2), ok(X3), ok(X4)) → U31¹(X1, X2, X3, X4)
TAKE(ok(X1), ok(X2)) → TAKE(X1, X2)
CONS(ok(X1), ok(X2)) → CONS(X1, X2)
ISNAT(ok(X)) → ISNAT(X)
ACTIVE(take(s(M), cons(N, IL))) → ISNAT(M)
ACTIVE(U31(tt, IL, M, N)) → CONS(N, take(M, IL))
ACTIVE(length(cons(N, L))) → ISNATLIST(L)
ACTIVE(isNatList(cons(V1, V2))) → AND(isNat(V1), isNatList(V2))
ACTIVE(U11(tt, L)) → LENGTH(L)
PROPER(U31(X1, X2, X3, X4)) → U31¹(proper(X1), proper(X2), proper(X3), proper(X4))
TAKE(X1, mark(X2)) → TAKE(X1, X2)
ACTIVE(U21(X)) → U21¹(active(X))
ACTIVE(length(cons(N, L))) → ISNAT(N)
PROPER(take(X1, X2)) → PROPER(X1)
ACTIVE(length(cons(N, L))) → U11¹(and(isNatList(L), isNat(N)), L)
TOP(mark(X)) → TOP(proper(X))
PROPER(U11(X1, X2)) → PROPER(X1)
ACTIVE(s(X)) → S(active(X))
U31¹(mark(X1), X2, X3, X4) → U31¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

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

ACTIVE(take(X1, X2)) → TAKE(X1, active(X2))
ACTIVE(cons(X1, X2)) → CONS(active(X1), X2)
ACTIVE(length(cons(N, L))) → AND(isNatList(L), isNat(N))
PROPER(U11(X1, X2)) → U11¹(proper(X1), proper(X2))
PROPER(isNatList(X)) → PROPER(X)
PROPER(length(X)) → LENGTH(proper(X))
ACTIVE(take(X1, X2)) → ACTIVE(X1)
ACTIVE(take(s(M), cons(N, IL))) → ISNAT(N)
ACTIVE(U31(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → TAKE(active(X1), X2)
PROPER(U11(X1, X2)) → PROPER(X2)
PROPER(length(X)) → PROPER(X)
PROPER(isNatList(X)) → ISNATLIST(proper(X))
ACTIVE(U11(X1, X2)) → ACTIVE(X1)
PROPER(U31(X1, X2, X3, X4)) → PROPER(X4)
ACTIVE(take(s(M), cons(N, IL))) → AND(isNat(M), isNat(N))
ACTIVE(s(X)) → ACTIVE(X)
PROPER(U31(X1, X2, X3, X4)) → PROPER(X1)
PROPER(U31(X1, X2, X3, X4)) → PROPER(X2)
S(ok(X)) → S(X)
CONS(mark(X1), X2) → CONS(X1, X2)
TOP(mark(X)) → PROPER(X)
ACTIVE(U31(tt, IL, M, N)) → TAKE(M, IL)
ACTIVE(and(X1, X2)) → AND(active(X1), X2)
TOP(ok(X)) → ACTIVE(X)
ACTIVE(length(X)) → ACTIVE(X)
PROPER(and(X1, X2)) → AND(proper(X1), proper(X2))
ACTIVE(take(s(M), cons(N, IL))) → U31¹(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(U21(X)) → U21¹(proper(X))
LENGTH(mark(X)) → LENGTH(X)
ACTIVE(isNatList(take(V1, V2))) → ISNATILIST(V2)
S(mark(X)) → S(X)
PROPER(cons(X1, X2)) → CONS(proper(X1), proper(X2))
ACTIVE(length(X)) → LENGTH(active(X))
PROPER(and(X1, X2)) → PROPER(X1)
ISNATLIST(ok(X)) → ISNATLIST(X)
ACTIVE(isNatList(cons(V1, V2))) → ISNATLIST(V2)
ACTIVE(isNat(s(V1))) → ISNAT(V1)
AND(mark(X1), X2) → AND(X1, X2)
PROPER(U21(X)) → PROPER(X)
ACTIVE(take(0, IL)) → U21¹(isNatIList(IL))
ACTIVE(isNatList(take(V1, V2))) → AND(isNat(V1), isNatIList(V2))
U21¹(ok(X)) → U21¹(X)
ACTIVE(isNatIList(V)) → ISNATLIST(V)
ACTIVE(zeros) → CONS(0, zeros)
ACTIVE(isNatList(take(V1, V2))) → ISNAT(V1)
PROPER(cons(X1, X2)) → PROPER(X2)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(isNatIList(cons(V1, V2))) → ISNAT(V1)
ISNATILIST(ok(X)) → ISNATILIST(X)
ACTIVE(isNatIList(cons(V1, V2))) → ISNATILIST(V2)
U21¹(mark(X)) → U21¹(X)
PROPER(s(X)) → S(proper(X))
ACTIVE(and(X1, X2)) → ACTIVE(X1)
ACTIVE(isNatList(cons(V1, V2))) → ISNAT(V1)
U11¹(ok(X1), ok(X2)) → U11¹(X1, X2)
ACTIVE(isNatIList(cons(V1, V2))) → AND(isNat(V1), isNatIList(V2))
PROPER(and(X1, X2)) → PROPER(X2)
PROPER(isNatIList(X)) → ISNATILIST(proper(X))
ACTIVE(take(0, IL)) → ISNATILIST(IL)
ACTIVE(U11(tt, L)) → S(length(L))
TAKE(mark(X1), X2) → TAKE(X1, X2)
U11¹(mark(X1), X2) → U11¹(X1, X2)
ACTIVE(U31(X1, X2, X3, X4)) → U31¹(active(X1), X2, X3, X4)
PROPER(isNat(X)) → PROPER(X)
ACTIVE(take(s(M), cons(N, IL))) → AND(isNatIList(IL), and(isNat(M), isNat(N)))
LENGTH(ok(X)) → LENGTH(X)
ACTIVE(take(X1, X2)) → ACTIVE(X2)
PROPER(take(X1, X2)) → TAKE(proper(X1), proper(X2))
PROPER(U31(X1, X2, X3, X4)) → PROPER(X3)
ACTIVE(isNat(length(V1))) → ISNATLIST(V1)
AND(ok(X1), ok(X2)) → AND(X1, X2)
PROPER(isNatIList(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
ACTIVE(take(s(M), cons(N, IL))) → ISNATILIST(IL)
ACTIVE(U11(X1, X2)) → U11¹(active(X1), X2)
PROPER(take(X1, X2)) → PROPER(X2)
TOP(ok(X)) → TOP(active(X))
PROPER(isNat(X)) → ISNAT(proper(X))
ACTIVE(U21(X)) → ACTIVE(X)
U31¹(ok(X1), ok(X2), ok(X3), ok(X4)) → U31¹(X1, X2, X3, X4)
TAKE(ok(X1), ok(X2)) → TAKE(X1, X2)
CONS(ok(X1), ok(X2)) → CONS(X1, X2)
ISNAT(ok(X)) → ISNAT(X)
ACTIVE(take(s(M), cons(N, IL))) → ISNAT(M)
ACTIVE(U31(tt, IL, M, N)) → CONS(N, take(M, IL))
ACTIVE(length(cons(N, L))) → ISNATLIST(L)
ACTIVE(isNatList(cons(V1, V2))) → AND(isNat(V1), isNatList(V2))
ACTIVE(U11(tt, L)) → LENGTH(L)
PROPER(U31(X1, X2, X3, X4)) → U31¹(proper(X1), proper(X2), proper(X3), proper(X4))
TAKE(X1, mark(X2)) → TAKE(X1, X2)
ACTIVE(U21(X)) → U21¹(active(X))
ACTIVE(length(cons(N, L))) → ISNAT(N)
PROPER(take(X1, X2)) → PROPER(X1)
ACTIVE(length(cons(N, L))) → U11¹(and(isNatList(L), isNat(N)), L)
TOP(mark(X)) → TOP(proper(X))
PROPER(U11(X1, X2)) → PROPER(X1)
ACTIVE(s(X)) → S(active(X))
U31¹(mark(X1), X2, X3, X4) → U31¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 14 SCCs with 51 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ISNATILIST(ok(X)) → ISNATILIST(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ISNATILIST(ok(X)) → ISNATILIST(X)

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

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ISNATLIST(ok(X)) → ISNATLIST(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ISNATLIST(ok(X)) → ISNATLIST(X)

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ISNAT(ok(X)) → ISNAT(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ISNAT(ok(X)) → ISNAT(X)

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U31¹(ok(X1), ok(X2), ok(X3), ok(X4)) → U31¹(X1, X2, X3, X4)
U31¹(mark(X1), X2, X3, X4) → U31¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U31¹(ok(X1), ok(X2), ok(X3), ok(X4)) → U31¹(X1, X2, X3, X4)
U31¹(mark(X1), X2, X3, X4) → U31¹(X1, X2, X3, X4)

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

U31¹(ok(X1), ok(X2), ok(X3), ok(X4)) → U31¹(X1, X2, X3, X4)
The graph contains the following edges 1 > 1, 2 > 2, 3 > 3, 4 > 4
U31¹(mark(X1), X2, X3, X4) → U31¹(X1, X2, X3, X4)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U21¹(ok(X)) → U21¹(X)
U21¹(mark(X)) → U21¹(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U21¹(ok(X)) → U21¹(X)
U21¹(mark(X)) → U21¹(X)

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

U21¹(ok(X)) → U21¹(X)
The graph contains the following edges 1 > 1
U21¹(mark(X)) → U21¹(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

S(ok(X)) → S(X)
S(mark(X)) → S(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

S(ok(X)) → S(X)
S(mark(X)) → S(X)

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

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

PROPER(isNat(X)) → PROPER(X)
PROPER(and(X1, X2)) → PROPER(X1)
PROPER(U11(X1, X2)) → PROPER(X2)
PROPER(length(X)) → PROPER(X)
PROPER(U31(X1, X2, X3, X4)) → PROPER(X3)
PROPER(isNatIList(X)) → PROPER(X)
PROPER(and(X1, X2)) → PROPER(X2)
PROPER(isNatList(X)) → PROPER(X)
PROPER(U31(X1, X2, X3, X4)) → PROPER(X4)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(s(X)) → PROPER(X)
PROPER(take(X1, X2)) → PROPER(X1)
PROPER(U21(X)) → PROPER(X)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(take(X1, X2)) → PROPER(X2)
PROPER(U31(X1, X2, X3, X4)) → PROPER(X1)
PROPER(U31(X1, X2, X3, X4)) → PROPER(X2)
PROPER(U11(X1, X2)) → PROPER(X1)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

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

PROPER(isNat(X)) → PROPER(X)
PROPER(and(X1, X2)) → PROPER(X1)
PROPER(U11(X1, X2)) → PROPER(X2)
PROPER(length(X)) → PROPER(X)
PROPER(U31(X1, X2, X3, X4)) → PROPER(X3)
PROPER(isNatIList(X)) → PROPER(X)
PROPER(and(X1, X2)) → PROPER(X2)
PROPER(isNatList(X)) → PROPER(X)
PROPER(U31(X1, X2, X3, X4)) → PROPER(X4)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(s(X)) → PROPER(X)
PROPER(take(X1, X2)) → PROPER(X1)
PROPER(U21(X)) → PROPER(X)
PROPER(take(X1, X2)) → PROPER(X2)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(U31(X1, X2, X3, X4)) → PROPER(X1)
PROPER(U31(X1, X2, X3, X4)) → PROPER(X2)
PROPER(U11(X1, X2)) → PROPER(X1)

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

PROPER(isNat(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(and(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(U11(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(length(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(U31(X1, X2, X3, X4)) → PROPER(X3)
The graph contains the following edges 1 > 1
PROPER(isNatIList(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(and(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(isNatList(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(U31(X1, X2, X3, X4)) → PROPER(X4)
The graph contains the following edges 1 > 1
PROPER(cons(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(s(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(take(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(U21(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(cons(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(take(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(U31(X1, X2, X3, X4)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(U31(X1, X2, X3, X4)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(U11(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

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

ACTIVE(U31(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(U21(X)) → ACTIVE(X)
ACTIVE(take(X1, X2)) → ACTIVE(X2)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → ACTIVE(X1)
ACTIVE(and(X1, X2)) → ACTIVE(X1)
ACTIVE(length(X)) → ACTIVE(X)
ACTIVE(U11(X1, X2)) → ACTIVE(X1)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

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

ACTIVE(U21(X)) → ACTIVE(X)
ACTIVE(U31(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → ACTIVE(X2)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → ACTIVE(X1)
ACTIVE(and(X1, X2)) → ACTIVE(X1)
ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(length(X)) → ACTIVE(X)

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

ACTIVE(U31(X1, X2, X3, X4)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(U21(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
ACTIVE(take(X1, X2)) → ACTIVE(X2)
The graph contains the following edges 1 > 1
ACTIVE(s(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(take(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(and(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(length(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
ACTIVE(U11(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

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

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

No dependency pairs are removed.

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = 2·x₁
POL(U11(x₁, x₂)) = x₁ + x₂
POL(U21(x₁)) = 2·x₁
POL(U31(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + 2·x₄
POL(active(x₁)) = 2·x₁
POL(and(x₁, x₂)) = 2·x₁ + x₂
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(ok(x₁)) = 2·x₁
POL(proper(x₁)) = x₁
POL(s(x₁)) = 2·x₁
POL(take(x₁, x₂)) = x₁ + x₂
POL(tt) = 0
POL(zeros) = 0

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

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

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

The TRS R consists of the following rules:

proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)

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

TOP(mark(zeros)) → TOP(ok(zeros))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U31(x0, x1, x2, x3))) → TOP(U31(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(0)) → TOP(ok(0))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(mark(tt)) → TOP(ok(tt))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(nil)) → TOP(ok(nil))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

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

TOP(mark(zeros)) → TOP(ok(zeros))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(U31(x0, x1, x2, x3))) → TOP(U31(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(0)) → TOP(ok(0))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(mark(tt)) → TOP(ok(tt))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(mark(nil)) → TOP(ok(nil))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(X)) → TOP(active(X))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))

The TRS R consists of the following rules:

proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)

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

TOP(ok(isNat(length(x0)))) → TOP(mark(isNatList(x0)))
TOP(ok(take(0, x0))) → TOP(mark(U21(isNatIList(x0))))
TOP(ok(U31(x0, x1, x2, x3))) → TOP(U31(active(x0), x1, x2, x3))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(ok(zeros)) → TOP(mark(cons(0, zeros)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U31(and(isNatIList(x2), and(isNat(x0), isNat(x1))), x2, x0, x1)))
TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(and(isNat(x0), isNatIList(x1))))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(ok(isNatIList(x0))) → TOP(mark(isNatList(x0)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(ok(isNatList(nil))) → TOP(mark(tt))
TOP(ok(isNat(0))) → TOP(mark(tt))
TOP(ok(U11(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(U31(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(length(nil))) → TOP(mark(0))
TOP(ok(isNatIList(zeros))) → TOP(mark(tt))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(isNat(s(x0)))) → TOP(mark(isNat(x0)))
TOP(ok(U21(tt))) → TOP(mark(nil))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(and(isNat(x0), isNatList(x1))))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(and(isNat(x0), isNatIList(x1))))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U11(and(isNatList(x1), isNat(x0)), x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

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

TOP(ok(isNat(length(x0)))) → TOP(mark(isNatList(x0)))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(take(0, x0))) → TOP(mark(U21(isNatIList(x0))))
TOP(ok(U31(x0, x1, x2, x3))) → TOP(U31(active(x0), x1, x2, x3))
TOP(mark(U31(x0, x1, x2, x3))) → TOP(U31(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(tt)) → TOP(ok(tt))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(ok(zeros)) → TOP(mark(cons(0, zeros)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(nil)) → TOP(ok(nil))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U31(and(isNatIList(x2), and(isNat(x0), isNat(x1))), x2, x0, x1)))
TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(and(isNat(x0), isNatIList(x1))))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(ok(isNatIList(x0))) → TOP(mark(isNatList(x0)))
TOP(mark(zeros)) → TOP(ok(zeros))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(ok(isNatList(nil))) → TOP(mark(tt))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(isNat(0))) → TOP(mark(tt))
TOP(ok(U11(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(length(nil))) → TOP(mark(0))
TOP(ok(U31(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(mark(0)) → TOP(ok(0))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(isNatIList(zeros))) → TOP(mark(tt))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U21(tt))) → TOP(mark(nil))
TOP(ok(isNat(s(x0)))) → TOP(mark(isNat(x0)))
TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(and(isNat(x0), isNatList(x1))))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(and(isNat(x0), isNatIList(x1))))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U11(and(isNatList(x1), isNat(x0)), x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

The TRS R consists of the following rules:

proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)

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 8 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

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

TOP(ok(isNat(length(x0)))) → TOP(mark(isNatList(x0)))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(take(0, x0))) → TOP(mark(U21(isNatIList(x0))))
TOP(ok(U31(x0, x1, x2, x3))) → TOP(U31(active(x0), x1, x2, x3))
TOP(mark(U31(x0, x1, x2, x3))) → TOP(U31(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(ok(zeros)) → TOP(mark(cons(0, zeros)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U31(and(isNatIList(x2), and(isNat(x0), isNat(x1))), x2, x0, x1)))
TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(and(isNat(x0), isNatIList(x1))))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(ok(isNatIList(x0))) → TOP(mark(isNatList(x0)))
TOP(mark(zeros)) → TOP(ok(zeros))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(U31(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(isNat(s(x0)))) → TOP(mark(isNat(x0)))
TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(and(isNat(x0), isNatList(x1))))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(and(isNat(x0), isNatIList(x1))))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U11(and(isNatList(x1), isNat(x0)), x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

The TRS R consists of the following rules:

proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)

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

The following pairs can be oriented strictly and are deleted.

TOP(ok(isNat(length(x0)))) → TOP(mark(isNatList(x0)))

The remaining pairs can at least be oriented weakly.

TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(take(0, x0))) → TOP(mark(U21(isNatIList(x0))))
TOP(ok(U31(x0, x1, x2, x3))) → TOP(U31(active(x0), x1, x2, x3))
TOP(mark(U31(x0, x1, x2, x3))) → TOP(U31(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(ok(zeros)) → TOP(mark(cons(0, zeros)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U31(and(isNatIList(x2), and(isNat(x0), isNat(x1))), x2, x0, x1)))
TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(and(isNat(x0), isNatIList(x1))))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(ok(isNatIList(x0))) → TOP(mark(isNatList(x0)))
TOP(mark(zeros)) → TOP(ok(zeros))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(U31(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(isNat(s(x0)))) → TOP(mark(isNat(x0)))
TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(and(isNat(x0), isNatList(x1))))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(and(isNat(x0), isNatIList(x1))))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U11(and(isNatList(x1), isNat(x0)), x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U21(x₁)) = 0
POL(U31(x₁, x₂, x₃, x₄)) = 0
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 1
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 1
POL(ok(x₁)) = x₁
POL(proper(x₁)) = x₁
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 0
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(length(X)) → length(proper(X))
proper(s(X)) → s(proper(X))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
length(ok(X)) → ok(length(X))
length(mark(X)) → mark(length(X))
U21(ok(X)) → ok(U21(X))
U21(mark(X)) → mark(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
isNat(ok(X)) → ok(isNat(X))
and(mark(X1), X2) → mark(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

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

TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(take(0, x0))) → TOP(mark(U21(isNatIList(x0))))
TOP(ok(U31(x0, x1, x2, x3))) → TOP(U31(active(x0), x1, x2, x3))
TOP(mark(U31(x0, x1, x2, x3))) → TOP(U31(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(ok(zeros)) → TOP(mark(cons(0, zeros)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U31(and(isNatIList(x2), and(isNat(x0), isNat(x1))), x2, x0, x1)))
TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(and(isNat(x0), isNatIList(x1))))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(ok(isNatIList(x0))) → TOP(mark(isNatList(x0)))
TOP(mark(zeros)) → TOP(ok(zeros))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(U31(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(isNat(s(x0)))) → TOP(mark(isNat(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(and(isNat(x0), isNatList(x1))))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(and(isNat(x0), isNatIList(x1))))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U11(and(isNatList(x1), isNat(x0)), x1)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

The TRS R consists of the following rules:

proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)

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

The following pairs can be oriented strictly and are deleted.

TOP(ok(take(0, x0))) → TOP(mark(U21(isNatIList(x0))))
TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(and(isNat(x0), isNatIList(x1))))

The remaining pairs can at least be oriented weakly.

TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(U31(x0, x1, x2, x3))) → TOP(U31(active(x0), x1, x2, x3))
TOP(mark(U31(x0, x1, x2, x3))) → TOP(U31(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(ok(zeros)) → TOP(mark(cons(0, zeros)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U31(and(isNatIList(x2), and(isNat(x0), isNat(x1))), x2, x0, x1)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(ok(isNatIList(x0))) → TOP(mark(isNatList(x0)))
TOP(mark(zeros)) → TOP(ok(zeros))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(U31(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(isNat(s(x0)))) → TOP(mark(isNat(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(and(isNat(x0), isNatList(x1))))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(and(isNat(x0), isNatIList(x1))))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U11(and(isNatList(x1), isNat(x0)), x1)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁, x₂)) = x₂
POL(U21(x₁)) = 0
POL(U31(x₁, x₂, x₃, x₄)) = 1 + x₂
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = x₂
POL(isNat(x₁)) = 1 + x₁
POL(isNatIList(x₁)) = 1 + x₁
POL(isNatList(x₁)) = 1 + x₁
POL(length(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = x₁
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = 1 + x₂
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(length(X)) → length(proper(X))
proper(s(X)) → s(proper(X))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
length(ok(X)) → ok(length(X))
length(mark(X)) → mark(length(X))
U21(ok(X)) → ok(U21(X))
U21(mark(X)) → mark(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt)) → mark(nil)
active(U11(tt, L)) → mark(s(length(L)))
active(zeros) → mark(cons(0, zeros))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
isNat(ok(X)) → ok(isNat(X))
and(mark(X1), X2) → mark(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
active(take(X1, X2)) → take(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(length(X)) → length(active(X))
active(s(X)) → s(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(U21(X)) → U21(active(X))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

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

TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(U31(x0, x1, x2, x3))) → TOP(U31(active(x0), x1, x2, x3))
TOP(mark(U31(x0, x1, x2, x3))) → TOP(U31(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(ok(zeros)) → TOP(mark(cons(0, zeros)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U31(and(isNatIList(x2), and(isNat(x0), isNat(x1))), x2, x0, x1)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(ok(isNatIList(x0))) → TOP(mark(isNatList(x0)))
TOP(mark(zeros)) → TOP(ok(zeros))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(U31(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(isNat(s(x0)))) → TOP(mark(isNat(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(and(isNat(x0), isNatList(x1))))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(and(isNat(x0), isNatIList(x1))))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U11(and(isNatList(x1), isNat(x0)), x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

The TRS R consists of the following rules:

proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)

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

The following pairs can be oriented strictly and are deleted.

TOP(ok(U31(tt, x0, x1, x2))) → TOP(mark(cons(x2, take(x1, x0))))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U11(and(isNatList(x1), isNat(x0)), x1)))

The remaining pairs can at least be oriented weakly.

TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(U31(x0, x1, x2, x3))) → TOP(U31(active(x0), x1, x2, x3))
TOP(mark(U31(x0, x1, x2, x3))) → TOP(U31(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(ok(zeros)) → TOP(mark(cons(0, zeros)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U31(and(isNatIList(x2), and(isNat(x0), isNat(x1))), x2, x0, x1)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(ok(isNatIList(x0))) → TOP(mark(isNatList(x0)))
TOP(mark(zeros)) → TOP(ok(zeros))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(s(length(x0))))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(isNat(s(x0)))) → TOP(mark(isNat(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(and(isNat(x0), isNatList(x1))))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(and(isNat(x0), isNatIList(x1))))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁, x₂)) = x₁
POL(U21(x₁)) = 0
POL(U31(x₁, x₂, x₃, x₄)) = 1 + x₁
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = 1
POL(isNat(x₁)) = 1
POL(isNatIList(x₁)) = 1
POL(isNatList(x₁)) = 1
POL(length(x₁)) = 1 + x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = x₁
POL(s(x₁)) = 1
POL(take(x₁, x₂)) = 1 + x₁
POL(tt) = 1
POL(zeros) = 1

The following usable rules [17] were oriented:

proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(length(X)) → length(proper(X))
proper(s(X)) → s(proper(X))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
length(ok(X)) → ok(length(X))
length(mark(X)) → mark(length(X))
U21(ok(X)) → ok(U21(X))
U21(mark(X)) → mark(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt)) → mark(nil)
active(U11(tt, L)) → mark(s(length(L)))
active(zeros) → mark(cons(0, zeros))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
isNat(ok(X)) → ok(isNat(X))
and(mark(X1), X2) → mark(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
active(take(X1, X2)) → take(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(length(X)) → length(active(X))
active(s(X)) → s(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(U21(X)) → U21(active(X))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

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

TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(U31(x0, x1, x2, x3))) → TOP(U31(active(x0), x1, x2, x3))
TOP(mark(U31(x0, x1, x2, x3))) → TOP(U31(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(ok(zeros)) → TOP(mark(cons(0, zeros)))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U31(and(isNatIList(x2), and(isNat(x0), isNat(x1))), x2, x0, x1)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(ok(isNatIList(x0))) → TOP(mark(isNatList(x0)))
TOP(mark(zeros)) → TOP(ok(zeros))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(s(length(x0))))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(isNat(s(x0)))) → TOP(mark(isNat(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(and(isNat(x0), isNatList(x1))))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(and(isNat(x0), isNatIList(x1))))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

The TRS R consists of the following rules:

proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)

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

The following pairs can be oriented strictly and are deleted.

TOP(ok(zeros)) → TOP(mark(cons(0, zeros)))

The remaining pairs can at least be oriented weakly.

TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(U31(x0, x1, x2, x3))) → TOP(U31(active(x0), x1, x2, x3))
TOP(mark(U31(x0, x1, x2, x3))) → TOP(U31(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U31(and(isNatIList(x2), and(isNat(x0), isNat(x1))), x2, x0, x1)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(ok(isNatIList(x0))) → TOP(mark(isNatList(x0)))
TOP(mark(zeros)) → TOP(ok(zeros))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(s(length(x0))))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(isNat(s(x0)))) → TOP(mark(isNat(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(and(isNat(x0), isNatList(x1))))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(and(isNat(x0), isNatIList(x1))))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U21(x₁)) = 0
POL(U31(x₁, x₂, x₃, x₄)) = 0
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = x₁
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 0
POL(tt) = 0
POL(zeros) = 1

The following usable rules [17] were oriented:

proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(length(X)) → length(proper(X))
proper(s(X)) → s(proper(X))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
length(ok(X)) → ok(length(X))
length(mark(X)) → mark(length(X))
U21(ok(X)) → ok(U21(X))
U21(mark(X)) → mark(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
isNat(ok(X)) → ok(isNat(X))
and(mark(X1), X2) → mark(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ DependencyGraphProof

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

TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(U31(x0, x1, x2, x3))) → TOP(U31(active(x0), x1, x2, x3))
TOP(mark(U31(x0, x1, x2, x3))) → TOP(U31(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U31(and(isNatIList(x2), and(isNat(x0), isNat(x1))), x2, x0, x1)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(ok(isNatIList(x0))) → TOP(mark(isNatList(x0)))
TOP(mark(zeros)) → TOP(ok(zeros))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(s(length(x0))))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(isNat(s(x0)))) → TOP(mark(isNat(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(and(isNat(x0), isNatList(x1))))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(and(isNat(x0), isNatIList(x1))))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

The TRS R consists of the following rules:

proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

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

TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(U31(x0, x1, x2, x3))) → TOP(U31(active(x0), x1, x2, x3))
TOP(mark(U31(x0, x1, x2, x3))) → TOP(U31(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U31(and(isNatIList(x2), and(isNat(x0), isNat(x1))), x2, x0, x1)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(ok(isNatIList(x0))) → TOP(mark(isNatList(x0)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(s(length(x0))))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(isNat(s(x0)))) → TOP(mark(isNat(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(and(isNat(x0), isNatList(x1))))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(and(isNat(x0), isNatIList(x1))))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

The TRS R consists of the following rules:

proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)

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

The following pairs can be oriented strictly and are deleted.

TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(and(isNat(x0), isNatList(x1))))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(and(isNat(x0), isNatIList(x1))))

The remaining pairs can at least be oriented weakly.

TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(U31(x0, x1, x2, x3))) → TOP(U31(active(x0), x1, x2, x3))
TOP(mark(U31(x0, x1, x2, x3))) → TOP(U31(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U31(and(isNatIList(x2), and(isNat(x0), isNat(x1))), x2, x0, x1)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(ok(isNatIList(x0))) → TOP(mark(isNatList(x0)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(s(length(x0))))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(isNat(s(x0)))) → TOP(mark(isNat(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U21(x₁)) = 0
POL(U31(x₁, x₂, x₃, x₄)) = 0
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = 1 + x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 1 + x₁
POL(isNatList(x₁)) = 1 + x₁
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = x₁
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 0
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(length(X)) → length(proper(X))
proper(s(X)) → s(proper(X))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
length(ok(X)) → ok(length(X))
length(mark(X)) → mark(length(X))
U21(ok(X)) → ok(U21(X))
U21(mark(X)) → mark(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
isNat(ok(X)) → ok(isNat(X))
and(mark(X1), X2) → mark(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

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

TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(U31(x0, x1, x2, x3))) → TOP(U31(active(x0), x1, x2, x3))
TOP(mark(U31(x0, x1, x2, x3))) → TOP(U31(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U31(and(isNatIList(x2), and(isNat(x0), isNat(x1))), x2, x0, x1)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(ok(isNatIList(x0))) → TOP(mark(isNatList(x0)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(s(length(x0))))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(isNat(s(x0)))) → TOP(mark(isNat(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

The TRS R consists of the following rules:

proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)

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

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.

TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(U31(x0, x1, x2, x3))) → TOP(U31(active(x0), x1, x2, x3))
TOP(mark(U31(x0, x1, x2, x3))) → TOP(U31(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U31(and(isNatIList(x2), and(isNat(x0), isNat(x1))), x2, x0, x1)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(ok(isNatIList(x0))) → TOP(mark(isNatList(x0)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(s(length(x0))))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(isNat(s(x0)))) → TOP(mark(isNat(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁, x₂)) = 1
POL(U21(x₁)) = 1
POL(U31(x₁, x₂, x₃, x₄)) = 1
POL(active(x₁)) = 1 + x₁
POL(and(x₁, x₂)) = 1
POL(cons(x₁, x₂)) = 1
POL(isNat(x₁)) = 1
POL(isNatIList(x₁)) = 1
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 1
POL(mark(x₁)) = 1
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(s(x₁)) = 1
POL(take(x₁, x₂)) = 1
POL(tt) = 0
POL(zeros) = 1

The following usable rules [17] were oriented:

s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
length(ok(X)) → ok(length(X))
length(mark(X)) → mark(length(X))
U21(ok(X)) → ok(U21(X))
U21(mark(X)) → mark(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
isNat(ok(X)) → ok(isNat(X))
and(mark(X1), X2) → mark(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ Narrowing

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

TOP(ok(isNatIList(x0))) → TOP(mark(isNatList(x0)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(U31(x0, x1, x2, x3))) → TOP(U31(active(x0), x1, x2, x3))
TOP(ok(U11(tt, x0))) → TOP(mark(s(length(x0))))
TOP(mark(U31(x0, x1, x2, x3))) → TOP(U31(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(isNat(s(x0)))) → TOP(mark(isNat(x0)))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U31(and(isNatIList(x2), and(isNat(x0), isNat(x1))), x2, x0, x1)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))

The TRS R consists of the following rules:

proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule TOP(ok(isNatIList(x0))) → TOP(mark(isNatList(x0))) at position [0,0] we obtained the following new rules:

TOP(ok(isNatIList(ok(x0)))) → TOP(mark(ok(isNatList(x0))))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

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

TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(U31(x0, x1, x2, x3))) → TOP(U31(active(x0), x1, x2, x3))
TOP(mark(U31(x0, x1, x2, x3))) → TOP(U31(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(isNatIList(ok(x0)))) → TOP(mark(ok(isNatList(x0))))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U31(and(isNatIList(x2), and(isNat(x0), isNat(x1))), x2, x0, x1)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(s(length(x0))))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(isNat(s(x0)))) → TOP(mark(isNat(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

The TRS R consists of the following rules:

proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

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

TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(U31(x0, x1, x2, x3))) → TOP(U31(active(x0), x1, x2, x3))
TOP(ok(U11(tt, x0))) → TOP(mark(s(length(x0))))
TOP(mark(U31(x0, x1, x2, x3))) → TOP(U31(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(isNat(s(x0)))) → TOP(mark(isNat(x0)))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U31(and(isNatIList(x2), and(isNat(x0), isNat(x1))), x2, x0, x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

The TRS R consists of the following rules:

proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)

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

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.

TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(U31(x0, x1, x2, x3))) → TOP(U31(active(x0), x1, x2, x3))
TOP(ok(U11(tt, x0))) → TOP(mark(s(length(x0))))
TOP(mark(U31(x0, x1, x2, x3))) → TOP(U31(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(isNat(s(x0)))) → TOP(mark(isNat(x0)))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U31(and(isNatIList(x2), and(isNat(x0), isNat(x1))), x2, x0, x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁, x₂)) = 1
POL(U21(x₁)) = 1
POL(U31(x₁, x₂, x₃, x₄)) = 1
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = 1
POL(cons(x₁, x₂)) = 1
POL(isNat(x₁)) = 1
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 1
POL(mark(x₁)) = 1
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(s(x₁)) = 1
POL(take(x₁, x₂)) = 1
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
length(ok(X)) → ok(length(X))
length(mark(X)) → mark(length(X))
U21(ok(X)) → ok(U21(X))
U21(mark(X)) → mark(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
isNat(ok(X)) → ok(isNat(X))
and(mark(X1), X2) → mark(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

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

TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(U31(x0, x1, x2, x3))) → TOP(U31(active(x0), x1, x2, x3))
TOP(ok(U11(tt, x0))) → TOP(mark(s(length(x0))))
TOP(mark(U31(x0, x1, x2, x3))) → TOP(U31(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(isNat(s(x0)))) → TOP(mark(isNat(x0)))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U31(and(isNatIList(x2), and(isNat(x0), isNat(x1))), x2, x0, x1)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

The TRS R consists of the following rules:

proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)

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

The following pairs can be oriented strictly and are deleted.

TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(U31(and(isNatIList(x2), and(isNat(x0), isNat(x1))), x2, x0, x1)))

The remaining pairs can at least be oriented weakly.

TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(U31(x0, x1, x2, x3))) → TOP(U31(active(x0), x1, x2, x3))
TOP(ok(U11(tt, x0))) → TOP(mark(s(length(x0))))
TOP(mark(U31(x0, x1, x2, x3))) → TOP(U31(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(isNat(s(x0)))) → TOP(mark(isNat(x0)))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 1
POL(TOP(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U21(x₁)) = 0
POL(U31(x₁, x₂, x₃, x₄)) = 0
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = x₁
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 1
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(length(X)) → length(proper(X))
proper(s(X)) → s(proper(X))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
length(ok(X)) → ok(length(X))
length(mark(X)) → mark(length(X))
U21(ok(X)) → ok(U21(X))
U21(mark(X)) → mark(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
isNat(ok(X)) → ok(isNat(X))
and(mark(X1), X2) → mark(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

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

TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(U31(x0, x1, x2, x3))) → TOP(U31(active(x0), x1, x2, x3))
TOP(ok(U11(tt, x0))) → TOP(mark(s(length(x0))))
TOP(mark(U31(x0, x1, x2, x3))) → TOP(U31(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(isNat(s(x0)))) → TOP(mark(isNat(x0)))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

The TRS R consists of the following rules:

proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)

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

The following pairs can be oriented strictly and are deleted.

TOP(ok(U11(tt, x0))) → TOP(mark(s(length(x0))))

The remaining pairs can at least be oriented weakly.

TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(U31(x0, x1, x2, x3))) → TOP(U31(active(x0), x1, x2, x3))
TOP(mark(U31(x0, x1, x2, x3))) → TOP(U31(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(isNat(s(x0)))) → TOP(mark(isNat(x0)))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁, x₂)) = 1
POL(U21(x₁)) = 0
POL(U31(x₁, x₂, x₃, x₄)) = 0
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = x₁
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 0
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(length(X)) → length(proper(X))
proper(s(X)) → s(proper(X))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
length(ok(X)) → ok(length(X))
length(mark(X)) → mark(length(X))
U21(ok(X)) → ok(U21(X))
U21(mark(X)) → mark(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
isNat(ok(X)) → ok(isNat(X))
and(mark(X1), X2) → mark(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

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

TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(U31(x0, x1, x2, x3))) → TOP(U31(active(x0), x1, x2, x3))
TOP(mark(U31(x0, x1, x2, x3))) → TOP(U31(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(isNat(s(x0)))) → TOP(mark(isNat(x0)))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

The TRS R consists of the following rules:

proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)

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

The following pairs can be oriented strictly and are deleted.

TOP(ok(and(tt, x0))) → TOP(mark(x0))

The remaining pairs can at least be oriented weakly.

TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(U31(x0, x1, x2, x3))) → TOP(U31(active(x0), x1, x2, x3))
TOP(mark(U31(x0, x1, x2, x3))) → TOP(U31(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(isNat(s(x0)))) → TOP(mark(isNat(x0)))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U21(x₁)) = 0
POL(U31(x₁, x₂, x₃, x₄)) = 0
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = 1 + x₂
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = x₁
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 0
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(length(X)) → length(proper(X))
proper(s(X)) → s(proper(X))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
length(ok(X)) → ok(length(X))
length(mark(X)) → mark(length(X))
U21(ok(X)) → ok(U21(X))
U21(mark(X)) → mark(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
isNat(ok(X)) → ok(isNat(X))
and(mark(X1), X2) → mark(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ 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:

TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(U31(x0, x1, x2, x3))) → TOP(U31(active(x0), x1, x2, x3))
TOP(mark(U31(x0, x1, x2, x3))) → TOP(U31(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(isNat(s(x0)))) → TOP(mark(isNat(x0)))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

The TRS R consists of the following rules:

proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)

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

The following pairs can be oriented strictly and are deleted.

TOP(ok(isNat(s(x0)))) → TOP(mark(isNat(x0)))

The remaining pairs can at least be oriented weakly.

TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(U31(x0, x1, x2, x3))) → TOP(U31(active(x0), x1, x2, x3))
TOP(mark(U31(x0, x1, x2, x3))) → TOP(U31(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

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

M( U31(x₁, ..., x₄) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

/	0	0	\
\	0	0	/

x₄

M( ok(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( mark(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	1	\
\	0	0	/

x₂

M( U11(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( take(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( 0 ) =

/	0	\
\	0	/

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( active(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	1	1	/

x₁

M( tt ) =

/	0	\
\	0	/

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( isNatIList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( s(x₁) ) =

/	1	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( proper(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( length(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( nil ) =

/	0	\
\	0	/

M( U21(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

Tuple symbols:

M( TOP(x₁) ) =

[

]

x₁

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

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

proper(zeros) → ok(zeros)
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(length(X)) → length(proper(X))
proper(s(X)) → s(proper(X))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
length(ok(X)) → ok(length(X))
length(mark(X)) → mark(length(X))
U21(ok(X)) → ok(U21(X))
U21(mark(X)) → mark(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt)) → mark(nil)
active(U11(tt, L)) → mark(s(length(L)))
active(zeros) → mark(cons(0, zeros))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
isNat(ok(X)) → ok(isNat(X))
and(mark(X1), X2) → mark(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
active(take(X1, X2)) → take(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(length(X)) → length(active(X))
active(s(X)) → s(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(U21(X)) → U21(active(X))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ 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:

TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(U31(x0, x1, x2, x3))) → TOP(U31(active(x0), x1, x2, x3))
TOP(mark(U31(x0, x1, x2, x3))) → TOP(U31(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

The TRS R consists of the following rules:

proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)

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

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.

TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(U31(x0, x1, x2, x3))) → TOP(U31(active(x0), x1, x2, x3))
TOP(mark(U31(x0, x1, x2, x3))) → TOP(U31(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁, x₂)) = 1
POL(U21(x₁)) = 1
POL(U31(x₁, x₂, x₃, x₄)) = 1
POL(active(x₁)) = 1 + x₁
POL(and(x₁, x₂)) = 1
POL(cons(x₁, x₂)) = 1
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 1
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 1
POL(mark(x₁)) = 1
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(s(x₁)) = 1
POL(take(x₁, x₂)) = 1
POL(tt) = 0
POL(zeros) = 1

The following usable rules [17] were oriented:

s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
length(ok(X)) → ok(length(X))
length(mark(X)) → mark(length(X))
U21(ok(X)) → ok(U21(X))
U21(mark(X)) → mark(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
isNat(ok(X)) → ok(isNat(X))
and(mark(X1), X2) → mark(and(X1, X2))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

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

TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(U31(x0, x1, x2, x3))) → TOP(U31(active(x0), x1, x2, x3))
TOP(mark(U31(x0, x1, x2, x3))) → TOP(U31(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))

The TRS R consists of the following rules:

proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
and(mark(X1), X2) → mark(and(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)

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