Termination w.r.t. Q proof of /tmp/tmpuZs3bn/LengthOfFiniteLists_nokinds_noand

Termination w.r.t. Q of the given QTRS could not be shown:

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 AND

  ↳5 QDP

  ↳6 QDP

    ↳7 QDPOrderProof (⇔)

    ↳8 QDP

    ↳9 DependencyGraphProof (⇔)

    ↳10 TRUE

  ↳11 QDP

(0) Obligation:

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

a__zeros → cons(0, zeros)
a__U11(tt) → tt
a__U21(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatList(V2))
a__U52(tt) → tt
a__U61(tt, L, N) → a__U62(a__isNat(N), L)
a__U62(tt, L) → s(a__length(mark(L)))
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatList(V1))
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatIList(V) → a__U31(a__isNatList(V))
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, N)
mark(zeros) → a__zeros
mark(U11(X)) → a__U11(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X) → U11(X)
a__U21(X) → U21(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__isNatList(X) → isNatList(X)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(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:

A__U41(tt, V2) → A__U42(a__isNatIList(V2))
A__U41(tt, V2) → A__ISNATILIST(V2)
A__U51(tt, V2) → A__U52(a__isNatList(V2))
A__U51(tt, V2) → A__ISNATLIST(V2)
A__U61(tt, L, N) → A__U62(a__isNat(N), L)
A__U61(tt, L, N) → A__ISNAT(N)
A__U62(tt, L) → A__LENGTH(mark(L))
A__U62(tt, L) → MARK(L)
A__ISNAT(length(V1)) → A__U11(a__isNatList(V1))
A__ISNAT(length(V1)) → A__ISNATLIST(V1)
A__ISNAT(s(V1)) → A__U21(a__isNat(V1))
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__ISNATILIST(V) → A__U31(a__isNatList(V))
A__ISNATILIST(V) → A__ISNATLIST(V)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__isNat(V1), V2)
A__ISNATILIST(cons(V1, V2)) → A__ISNAT(V1)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__isNat(V1), V2)
A__ISNATLIST(cons(V1, V2)) → A__ISNAT(V1)
A__LENGTH(cons(N, L)) → A__U61(a__isNatList(L), L, N)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
MARK(zeros) → A__ZEROS
MARK(U11(X)) → A__U11(mark(X))
MARK(U11(X)) → MARK(X)
MARK(U21(X)) → A__U21(mark(X))
MARK(U21(X)) → MARK(X)
MARK(U31(X)) → A__U31(mark(X))
MARK(U31(X)) → MARK(X)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → A__U42(mark(X))
MARK(U42(X)) → MARK(X)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(U51(X1, X2)) → A__U51(mark(X1), X2)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X)) → A__U52(mark(X))
MARK(U52(X)) → MARK(X)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(U61(X1, X2, X3)) → A__U61(mark(X1), X2, X3)
MARK(U61(X1, X2, X3)) → MARK(X1)
MARK(U62(X1, X2)) → A__U62(mark(X1), X2)
MARK(U62(X1, X2)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(length(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt) → tt
a__U21(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatList(V2))
a__U52(tt) → tt
a__U61(tt, L, N) → a__U62(a__isNat(N), L)
a__U62(tt, L) → s(a__length(mark(L)))
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatList(V1))
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatIList(V) → a__U31(a__isNatList(V))
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, N)
mark(zeros) → a__zeros
mark(U11(X)) → a__U11(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X) → U11(X)
a__U21(X) → U21(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__isNatList(X) → isNatList(X)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(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 3 SCCs with 20 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

A__U51(tt, V2) → A__ISNATLIST(V2)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__isNat(V1), V2)
A__ISNATLIST(cons(V1, V2)) → A__ISNAT(V1)
A__ISNAT(length(V1)) → A__ISNATLIST(V1)
A__ISNAT(s(V1)) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt) → tt
a__U21(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatList(V2))
a__U52(tt) → tt
a__U61(tt, L, N) → a__U62(a__isNat(N), L)
a__U62(tt, L) → s(a__length(mark(L)))
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatList(V1))
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatIList(V) → a__U31(a__isNatList(V))
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, N)
mark(zeros) → a__zeros
mark(U11(X)) → a__U11(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X) → U11(X)
a__U21(X) → U21(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__isNatList(X) → isNatList(X)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(X)

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

(6) Obligation:

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

A__U41(tt, V2) → A__ISNATILIST(V2)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__isNat(V1), V2)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt) → tt
a__U21(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatList(V2))
a__U52(tt) → tt
a__U61(tt, L, N) → a__U62(a__isNat(N), L)
a__U62(tt, L) → s(a__length(mark(L)))
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatList(V1))
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatIList(V) → a__U31(a__isNatList(V))
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, N)
mark(zeros) → a__zeros
mark(U11(X)) → a__U11(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X) → U11(X)
a__U21(X) → U21(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__isNatList(X) → isNatList(X)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(X)

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

(7) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__ISNATILIST(cons(V1, V2)) → A__U41(a__isNat(V1), V2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__U41(x1, x2) = x2
tt = tt
A__ISNATILIST(x1) = x1
cons(x1, x2) = cons(x2)
a__isNat(x1) = a__isNat
a__zeros = a__zeros
0 = 0
zeros = zeros
a__U11(x1) = a__U11
a__U21(x1) = x1
a__U31(x1) = a__U31(x1)
a__U41(x1, x2) = a__U41
a__U42(x1) = a__U42
a__isNatIList(x1) = a__isNatIList(x1)
a__U51(x1, x2) = a__U51(x1, x2)
a__U52(x1) = a__U52(x1)
a__isNatList(x1) = x1
a__U61(x1, x2, x3) = a__U61
a__U62(x1, x2) = x1
s(x1) = s
a__length(x1) = a__length
mark(x1) = mark(x1)
length(x1) = length
nil = nil
U11(x1) = U11
U21(x1) = x1
U31(x1) = U31(x1)
U41(x1, x2) = U41
U42(x1) = U42
isNatIList(x1) = isNatIList(x1)
U51(x1, x2) = U51(x1, x2)
U52(x1) = U52(x1)
isNatList(x1) = x1
U61(x1, x2, x3) = U61
U62(x1, x2) = x1
isNat(x1) = isNat

Recursive path order with status [RPO].
Precedence:

mark₁ > a_{_zeros} > cons₁ > a_{_isNat} > a_{_U11} > tt > s > U11
mark₁ > a_{_zeros} > cons₁ > a_{_isNat} > isNat > U11
mark₁ > a_{_zeros} > cons₁ > a_{_U51₂} > a_{_U52₁} > tt > s > U11
mark₁ > a_{_zeros} > cons₁ > a_{_U51₂} > a_{_U52₁} > U52₁ > U11
mark₁ > a_{_zeros} > cons₁ > a_{_U51₂} > U51₂ > U11
mark₁ > a_{_zeros} > 0 > U11
mark₁ > a_{_zeros} > zeros > tt > s > U11
mark₁ > a_{_{isNatIList₁}} > a_{_isNat} > a_{_U11} > tt > s > U11
mark₁ > a_{_{isNatIList₁}} > a_{_isNat} > isNat > U11
mark₁ > a_{_{isNatIList₁}} > a_{_U31₁} > U31₁ > U11
mark₁ > a_{_{isNatIList₁}} > a_{_U41} > a_{_U42} > tt > s > U11
mark₁ > a_{_{isNatIList₁}} > a_{_U41} > a_{_U42} > U42 > U11
mark₁ > a_{_{isNatIList₁}} > a_{_U41} > U41 > U11
mark₁ > a_{_{isNatIList₁}} > isNatIList₁ > U11
mark₁ > a_{_length} > 0 > U11
mark₁ > a_{_length} > a_{_U61} > a_{_isNat} > a_{_U11} > tt > s > U11
mark₁ > a_{_length} > a_{_U61} > a_{_isNat} > isNat > U11
mark₁ > a_{_length} > a_{_U61} > U61 > U11
mark₁ > a_{_length} > length > a_{_U11} > tt > s > U11
mark₁ > nil > tt > s > U11
mark₁ > nil > 0 > U11

Status:

tt: multiset
cons₁: multiset
a_{_isNat}: multiset
a_{_zeros}: multiset
0: multiset
zeros: multiset
a_{_U11}: multiset
a_{_U31₁}: multiset
a_{_U41}: multiset
a_{_U42}: multiset
a_{_{isNatIList₁}}: multiset
a_{_U51₂}: multiset
a_{_U52₁}: multiset
a_{_U61}: multiset
s: multiset
a_{_length}: multiset
mark₁: [1]
length: multiset
nil: multiset
U11: multiset
U31₁: multiset
U41: multiset
U42: multiset
isNatIList₁: multiset
U51₂: multiset
U52₁: multiset
U61: multiset
isNat: multiset

The following usable rules [FROCOS05] were oriented:

a__zeros → cons(0, zeros)
a__U11(tt) → tt
a__U21(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatList(V2))
a__U52(tt) → tt
a__U61(tt, L, N) → a__U62(a__isNat(N), L)
a__U62(tt, L) → s(a__length(mark(L)))
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatList(V1))
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatIList(V) → a__U31(a__isNatList(V))
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, N)
mark(zeros) → a__zeros
mark(U11(X)) → a__U11(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X) → U11(X)
a__U21(X) → U21(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__isNatList(X) → isNatList(X)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(X)

(8) Obligation:

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

A__U41(tt, V2) → A__ISNATILIST(V2)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt) → tt
a__U21(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatList(V2))
a__U52(tt) → tt
a__U61(tt, L, N) → a__U62(a__isNat(N), L)
a__U62(tt, L) → s(a__length(mark(L)))
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatList(V1))
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatIList(V) → a__U31(a__isNatList(V))
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, N)
mark(zeros) → a__zeros
mark(U11(X)) → a__U11(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X) → U11(X)
a__U21(X) → U21(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__isNatList(X) → isNatList(X)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(X)

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

(9) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(10) TRUE

(11) Obligation:

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

MARK(U11(X)) → MARK(X)
MARK(U21(X)) → MARK(X)
MARK(U31(X)) → MARK(X)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(U61(X1, X2, X3)) → A__U61(mark(X1), X2, X3)
A__U61(tt, L, N) → A__U62(a__isNat(N), L)
A__U62(tt, L) → A__LENGTH(mark(L))
A__LENGTH(cons(N, L)) → A__U61(a__isNatList(L), L, N)
A__U62(tt, L) → MARK(L)
MARK(U61(X1, X2, X3)) → MARK(X1)
MARK(U62(X1, X2)) → A__U62(mark(X1), X2)
MARK(U62(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(length(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt) → tt
a__U21(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatList(V2))
a__U52(tt) → tt
a__U61(tt, L, N) → a__U62(a__isNat(N), L)
a__U62(tt, L) → s(a__length(mark(L)))
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatList(V1))
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatIList(V) → a__U31(a__isNatList(V))
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, N)
mark(zeros) → a__zeros
mark(U11(X)) → a__U11(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X) → U11(X)
a__U21(X) → U21(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__isNatList(X) → isNatList(X)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(X)

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