(0) Obligation:

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

zeroscons(0, n__zeros)
U11(tt, L) → U12(tt, activate(L))
U12(tt, L) → s(length(activate(L)))
length(nil) → 0
length(cons(N, L)) → U11(tt, activate(L))
zerosn__zeros
activate(n__zeros) → zeros
activate(X) → X

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

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

U111(tt, L) → U121(tt, activate(L))
U111(tt, L) → ACTIVATE(L)
U121(tt, L) → LENGTH(activate(L))
U121(tt, L) → ACTIVATE(L)
LENGTH(cons(N, L)) → U111(tt, activate(L))
LENGTH(cons(N, L)) → ACTIVATE(L)
ACTIVATE(n__zeros) → ZEROS

The TRS R consists of the following rules:

zeroscons(0, n__zeros)
U11(tt, L) → U12(tt, activate(L))
U12(tt, L) → s(length(activate(L)))
length(nil) → 0
length(cons(N, L)) → U11(tt, activate(L))
zerosn__zeros
activate(n__zeros) → zeros
activate(X) → X

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.

(4) Obligation:

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

U121(tt, L) → LENGTH(activate(L))
LENGTH(cons(N, L)) → U111(tt, activate(L))
U111(tt, L) → U121(tt, activate(L))

The TRS R consists of the following rules:

zeroscons(0, n__zeros)
U11(tt, L) → U12(tt, activate(L))
U12(tt, L) → s(length(activate(L)))
length(nil) → 0
length(cons(N, L)) → U11(tt, activate(L))
zerosn__zeros
activate(n__zeros) → zeros
activate(X) → X

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