Termination w.r.t. Q proof of /tmp/tmp0VA7Nz/LengthOfFiniteLists_nokinds-noand

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:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → 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:

ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATILIST(V) → ACTIVATE(V)
U61¹(tt, L, N) → ISNAT(activate(N))
ISNAT(n__length(V1)) → ISNATLIST(activate(V1))
U51¹(tt, V2) → ISNATLIST(activate(V2))
U51¹(tt, V2) → U52¹(isNatList(activate(V2)))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U51¹(tt, V2) → ACTIVATE(V2)
ISNATILIST(V) → U31¹(isNatList(activate(V)))
ACTIVATE(n__s(X)) → ACTIVATE(X)
U61¹(tt, L, N) → U62¹(isNat(activate(N)), activate(L))
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__0) → 0¹
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(V) → ISNATLIST(activate(V))
U41¹(tt, V2) → ISNATILIST(activate(V2))
U62¹(tt, L) → LENGTH(activate(L))
ISNAT(n__s(V1)) → U21¹(isNat(activate(V1)))
U62¹(tt, L) → S(length(activate(L)))
ISNAT(n__length(V1)) → U11¹(isNatList(activate(V1)))
ACTIVATE(n__zeros) → ZEROS
LENGTH(nil) → 0¹
U41¹(tt, V2) → U42¹(isNatIList(activate(V2)))
U41¹(tt, V2) → ACTIVATE(V2)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → ACTIVATE(L)
LENGTH(cons(N, L)) → U61¹(isNatList(activate(L)), activate(L), N)
ACTIVATE(n__length(X)) → LENGTH(activate(X))
ACTIVATE(n__nil) → NIL
ACTIVATE(n__s(X)) → S(activate(X))
U61¹(tt, L, N) → ACTIVATE(L)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNATILIST(n__cons(V1, V2)) → U41¹(isNat(activate(V1)), activate(V2))
ACTIVATE(n__cons(X1, X2)) → CONS(activate(X1), X2)
ISNAT(n__s(V1)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
U61¹(tt, L, N) → ACTIVATE(N)
ZEROS → CONS(0, n__zeros)
U62¹(tt, L) → ACTIVATE(L)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__cons(V1, V2)) → U51¹(isNat(activate(V1)), activate(V2))
ZEROS → 0¹
ISNAT(n__length(V1)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → 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:

ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATILIST(V) → ACTIVATE(V)
U61¹(tt, L, N) → ISNAT(activate(N))
ISNAT(n__length(V1)) → ISNATLIST(activate(V1))
U51¹(tt, V2) → ISNATLIST(activate(V2))
U51¹(tt, V2) → U52¹(isNatList(activate(V2)))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U51¹(tt, V2) → ACTIVATE(V2)
ISNATILIST(V) → U31¹(isNatList(activate(V)))
ACTIVATE(n__s(X)) → ACTIVATE(X)
U61¹(tt, L, N) → U62¹(isNat(activate(N)), activate(L))
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__0) → 0¹
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(V) → ISNATLIST(activate(V))
U41¹(tt, V2) → ISNATILIST(activate(V2))
U62¹(tt, L) → LENGTH(activate(L))
ISNAT(n__s(V1)) → U21¹(isNat(activate(V1)))
U62¹(tt, L) → S(length(activate(L)))
ISNAT(n__length(V1)) → U11¹(isNatList(activate(V1)))
ACTIVATE(n__zeros) → ZEROS
LENGTH(nil) → 0¹
U41¹(tt, V2) → U42¹(isNatIList(activate(V2)))
U41¹(tt, V2) → ACTIVATE(V2)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → ACTIVATE(L)
LENGTH(cons(N, L)) → U61¹(isNatList(activate(L)), activate(L), N)
ACTIVATE(n__length(X)) → LENGTH(activate(X))
ACTIVATE(n__nil) → NIL
ACTIVATE(n__s(X)) → S(activate(X))
U61¹(tt, L, N) → ACTIVATE(L)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNATILIST(n__cons(V1, V2)) → U41¹(isNat(activate(V1)), activate(V2))
ACTIVATE(n__cons(X1, X2)) → CONS(activate(X1), X2)
ISNAT(n__s(V1)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
U61¹(tt, L, N) → ACTIVATE(N)
ZEROS → CONS(0, n__zeros)
U62¹(tt, L) → ACTIVATE(L)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__cons(V1, V2)) → U51¹(isNat(activate(V1)), activate(V2))
ZEROS → 0¹
ISNAT(n__length(V1)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 20 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

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

U61¹(tt, L, N) → ISNAT(activate(N))
ISNAT(n__length(V1)) → ISNATLIST(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → ACTIVATE(L)
U51¹(tt, V2) → ISNATLIST(activate(V2))
LENGTH(cons(N, L)) → U61¹(isNatList(activate(L)), activate(L), N)
ACTIVATE(n__length(X)) → LENGTH(activate(X))
U61¹(tt, L, N) → ACTIVATE(L)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U51¹(tt, V2) → ACTIVATE(V2)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ACTIVATE(n__s(X)) → ACTIVATE(X)
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
U61¹(tt, L, N) → U62¹(isNat(activate(N)), activate(L))
U61¹(tt, L, N) → ACTIVATE(N)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
U62¹(tt, L) → LENGTH(activate(L))
U62¹(tt, L) → ACTIVATE(L)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__cons(V1, V2)) → U51¹(isNat(activate(V1)), activate(V2))
ISNAT(n__length(V1)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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.

ISNAT(n__length(V1)) → ISNATLIST(activate(V1))
ACTIVATE(n__length(X)) → LENGTH(activate(X))
ISNAT(n__length(V1)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → ACTIVATE(X)

The remaining pairs can at least be oriented weakly.

U61¹(tt, L, N) → ISNAT(activate(N))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → ACTIVATE(L)
U51¹(tt, V2) → ISNATLIST(activate(V2))
LENGTH(cons(N, L)) → U61¹(isNatList(activate(L)), activate(L), N)
U61¹(tt, L, N) → ACTIVATE(L)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U51¹(tt, V2) → ACTIVATE(V2)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ACTIVATE(n__s(X)) → ACTIVATE(X)
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
U61¹(tt, L, N) → U62¹(isNat(activate(N)), activate(L))
U61¹(tt, L, N) → ACTIVATE(N)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
U62¹(tt, L) → LENGTH(activate(L))
U62¹(tt, L) → ACTIVATE(L)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__cons(V1, V2)) → U51¹(isNat(activate(V1)), activate(V2))

Used ordering: Polynomial interpretation [25,35]:

POL(U51¹(x₁, x₂)) = (1/2)x_2
POL(U62(x₁, x₂)) = 2 + (4)x_2
POL(U61(x₁, x₂, x₃)) = 2 + (4)x_2
POL(activate(x₁)) = x_1
POL(n__nil) = 0
POL(n__s(x₁)) = x_1
POL(U51(x₁, x₂)) = 0
POL(ISNATLIST(x₁)) = (1/2)x_1
POL(tt) = 0
POL(U61¹(x₁, x₂, x₃)) = (1/2)x_2 + (2)x_3
POL(isNatList(x₁)) = 0
POL(zeros) = 0
POL(U52(x₁)) = 0
POL(s(x₁)) = x_1
POL(U11(x₁)) = 0
POL(isNat(x₁)) = 0
POL(ACTIVATE(x₁)) = (1/2)x_1
POL(nil) = 0
POL(LENGTH(x₁)) = (1/2)x_1
POL(n__length(x₁)) = 2 + (2)x_1
POL(n__zeros) = 0
POL(n__cons(x₁, x₂)) = (4)x_1 + (4)x_2
POL(U62¹(x₁, x₂)) = (1/2)x_2
POL(0) = 0
POL(ISNAT(x₁)) = (2)x_1
POL(cons(x₁, x₂)) = (4)x_1 + (4)x_2
POL(n__0) = 0
POL(length(x₁)) = 2 + (2)x_1
POL(U21(x₁)) = 0

The value of delta used in the strict ordering is 1.
The following usable rules [17] were oriented:

length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
length(nil) → 0
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U62(tt, L) → s(length(activate(L)))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U52(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

          ↳ QDP

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

U61¹(tt, L, N) → ISNAT(activate(N))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → ACTIVATE(L)
U51¹(tt, V2) → ISNATLIST(activate(V2))
LENGTH(cons(N, L)) → U61¹(isNatList(activate(L)), activate(L), N)
U61¹(tt, L, N) → ACTIVATE(L)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U51¹(tt, V2) → ACTIVATE(V2)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ACTIVATE(n__s(X)) → ACTIVATE(X)
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
U61¹(tt, L, N) → U62¹(isNat(activate(N)), activate(L))
U61¹(tt, L, N) → ACTIVATE(N)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
U62¹(tt, L) → LENGTH(activate(L))
U62¹(tt, L) → ACTIVATE(L)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__cons(V1, V2)) → U51¹(isNat(activate(V1)), activate(V2))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 4 SCCs with 11 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                      ↳ QDPOrderProof

                    ↳ QDP

                    ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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.

ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(n__cons(x₁, x₂)) = 1/2 + (4)x_1
POL(n__s(x₁)) = 4 + (4)x_1
POL(ACTIVATE(x₁)) = (4)x_1

The value of delta used in the strict ordering is 2.
The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ PisEmptyProof

                    ↳ QDP

                    ↳ QDP

                    ↳ QDP

          ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                    ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ISNAT(n__s(V1)) → ISNAT(activate(V1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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.

ISNAT(n__s(V1)) → ISNAT(activate(V1))

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(U62(x₁, x₂)) = 1/4 + x_2
POL(n__length(x₁)) = (1/2)x_1
POL(U61(x₁, x₂, x₃)) = (1/4)x_1 + x_2
POL(n__zeros) = 0
POL(n__cons(x₁, x₂)) = (4)x_2
POL(activate(x₁)) = x_1
POL(n__s(x₁)) = 1/4 + (2)x_1
POL(n__nil) = 1/2
POL(0) = 0
POL(U51(x₁, x₂)) = (4)x_2
POL(ISNAT(x₁)) = (1/4)x_1
POL(cons(x₁, x₂)) = (4)x_2
POL(tt) = 1
POL(isNatList(x₁)) = (2)x_1
POL(zeros) = 0
POL(n__0) = 0
POL(U52(x₁)) = (2)x_1
POL(s(x₁)) = 1/4 + (2)x_1
POL(U11(x₁)) = x_1
POL(length(x₁)) = (1/2)x_1
POL(isNat(x₁)) = (4)x_1
POL(nil) = 1/2
POL(U21(x₁)) = 1

The value of delta used in the strict ordering is 1/16.
The following usable rules [17] were oriented:

length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
length(nil) → 0
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U62(tt, L) → s(length(activate(L)))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U52(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ PisEmptyProof

                    ↳ QDP

                    ↳ QDP

          ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                    ↳ QDP

                    ↳ QDP

          ↳ QDP

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

U51¹(tt, V2) → ISNATLIST(activate(V2))
ISNATLIST(n__cons(V1, V2)) → U51¹(isNat(activate(V1)), activate(V2))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                    ↳ QDP

                    ↳ QDP

          ↳ QDP

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

U61¹(tt, L, N) → U62¹(isNat(activate(N)), activate(L))
LENGTH(cons(N, L)) → U61¹(isNatList(activate(L)), activate(L), N)
U62¹(tt, L) → LENGTH(activate(L))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

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

U41¹(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(n__cons(V1, V2)) → U41¹(isNat(activate(V1)), activate(V2))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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