Termination w.r.t. Q of the following Term Rewriting System could be disproven:

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

zeroscons(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)
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
niln__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
  ↳ RRRPoloQTRSProof

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

zeroscons(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)
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
niln__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.

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

zeroscons(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)
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
niln__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.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

isNatList(n__nil) → tt
length(nil) → 0
Used ordering:
Polynomial interpretation [25]:

POL(0) = 0   
POL(U11(x1)) = 2·x1   
POL(U21(x1)) = x1   
POL(U31(x1)) = x1   
POL(U41(x1, x2)) = x1 + 2·x2   
POL(U42(x1)) = 2·x1   
POL(U51(x1, x2)) = x1 + 2·x2   
POL(U52(x1)) = 2·x1   
POL(U61(x1, x2, x3)) = 2·x1 + 2·x2 + 2·x3   
POL(U62(x1, x2)) = x1 + 2·x2   
POL(activate(x1)) = x1   
POL(cons(x1, x2)) = 2·x1 + 2·x2   
POL(isNat(x1)) = 2·x1   
POL(isNatIList(x1)) = x1   
POL(isNatList(x1)) = x1   
POL(length(x1)) = 2·x1   
POL(n__0) = 0   
POL(n__cons(x1, x2)) = 2·x1 + 2·x2   
POL(n__length(x1)) = 2·x1   
POL(n__nil) = 1   
POL(n__s(x1)) = x1   
POL(n__zeros) = 0   
POL(nil) = 1   
POL(s(x1)) = x1   
POL(tt) = 0   
POL(zeros) = 0   




↳ QTRS
  ↳ RRRPoloQTRSProof
QTRS
      ↳ RRRPoloQTRSProof

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

zeroscons(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__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
niln__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.

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

zeroscons(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__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
niln__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.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

U11(tt) → tt
Used ordering:
Polynomial interpretation [25]:

POL(0) = 0   
POL(U11(x1)) = 1 + x1   
POL(U21(x1)) = x1   
POL(U31(x1)) = x1   
POL(U41(x1, x2)) = 2·x1 + 2·x2   
POL(U42(x1)) = 2·x1   
POL(U51(x1, x2)) = 2·x1 + 2·x2   
POL(U52(x1)) = x1   
POL(U61(x1, x2, x3)) = 1 + 2·x1 + 2·x2 + x3   
POL(U62(x1, x2)) = 1 + x1 + 2·x2   
POL(activate(x1)) = x1   
POL(cons(x1, x2)) = 2·x1 + 2·x2   
POL(isNat(x1)) = x1   
POL(isNatIList(x1)) = x1   
POL(isNatList(x1)) = x1   
POL(length(x1)) = 1 + 2·x1   
POL(n__0) = 0   
POL(n__cons(x1, x2)) = 2·x1 + 2·x2   
POL(n__length(x1)) = 1 + 2·x1   
POL(n__nil) = 0   
POL(n__s(x1)) = x1   
POL(n__zeros) = 0   
POL(nil) = 0   
POL(s(x1)) = x1   
POL(tt) = 0   
POL(zeros) = 0   




↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ RRRPoloQTRSProof
QTRS
          ↳ RRRPoloQTRSProof

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

zeroscons(0, n__zeros)
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__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
niln__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.

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

zeroscons(0, n__zeros)
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__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
niln__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.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
Used ordering:
Polynomial interpretation [25]:

POL(0) = 0   
POL(U11(x1)) = x1   
POL(U21(x1)) = x1   
POL(U31(x1)) = x1   
POL(U41(x1, x2)) = 1 + 2·x1 + x2   
POL(U42(x1)) = x1   
POL(U51(x1, x2)) = x1 + x2   
POL(U52(x1)) = x1   
POL(U61(x1, x2, x3)) = x1 + x2 + 2·x3   
POL(U62(x1, x2)) = 2·x1 + x2   
POL(activate(x1)) = x1   
POL(cons(x1, x2)) = 2·x1 + 2·x2   
POL(isNat(x1)) = x1   
POL(isNatIList(x1)) = 1 + x1   
POL(isNatList(x1)) = x1   
POL(length(x1)) = x1   
POL(n__0) = 0   
POL(n__cons(x1, x2)) = 2·x1 + 2·x2   
POL(n__length(x1)) = x1   
POL(n__nil) = 0   
POL(n__s(x1)) = x1   
POL(n__zeros) = 0   
POL(nil) = 0   
POL(s(x1)) = x1   
POL(tt) = 0   
POL(zeros) = 0   




↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ RRRPoloQTRSProof
        ↳ QTRS
          ↳ RRRPoloQTRSProof
QTRS
              ↳ RRRPoloQTRSProof

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

zeroscons(0, n__zeros)
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(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
niln__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.

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

zeroscons(0, n__zeros)
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(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
niln__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.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

U31(tt) → tt
Used ordering:
Polynomial interpretation [25]:

POL(0) = 0   
POL(U11(x1)) = x1   
POL(U21(x1)) = x1   
POL(U31(x1)) = 1 + x1   
POL(U41(x1, x2)) = x1 + 2·x2   
POL(U42(x1)) = 2·x1   
POL(U51(x1, x2)) = x1 + x2   
POL(U52(x1)) = x1   
POL(U61(x1, x2, x3)) = 2·x1 + 2·x2 + x3   
POL(U62(x1, x2)) = x1 + 2·x2   
POL(activate(x1)) = x1   
POL(cons(x1, x2)) = 2·x1 + 2·x2   
POL(isNat(x1)) = x1   
POL(isNatIList(x1)) = x1   
POL(isNatList(x1)) = x1   
POL(length(x1)) = 2·x1   
POL(n__0) = 0   
POL(n__cons(x1, x2)) = 2·x1 + 2·x2   
POL(n__length(x1)) = 2·x1   
POL(n__nil) = 0   
POL(n__s(x1)) = x1   
POL(n__zeros) = 0   
POL(nil) = 0   
POL(s(x1)) = x1   
POL(tt) = 0   
POL(zeros) = 0   




↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ RRRPoloQTRSProof
        ↳ QTRS
          ↳ RRRPoloQTRSProof
            ↳ QTRS
              ↳ RRRPoloQTRSProof
QTRS
                  ↳ RRRPoloQTRSProof

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

zeroscons(0, n__zeros)
U21(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(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
niln__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.

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

zeroscons(0, n__zeros)
U21(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(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
niln__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.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

isNat(n__length(V1)) → U11(isNatList(activate(V1)))
Used ordering:
Polynomial interpretation [25]:

POL(0) = 0   
POL(U11(x1)) = 1 + x1   
POL(U21(x1)) = x1   
POL(U41(x1, x2)) = 2·x1 + 2·x2   
POL(U42(x1)) = x1   
POL(U51(x1, x2)) = 2·x1 + x2   
POL(U52(x1)) = x1   
POL(U61(x1, x2, x3)) = 2 + 2·x1 + 2·x2 + 2·x3   
POL(U62(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(activate(x1)) = x1   
POL(cons(x1, x2)) = 2·x1 + 2·x2   
POL(isNat(x1)) = x1   
POL(isNatIList(x1)) = x1   
POL(isNatList(x1)) = x1   
POL(length(x1)) = 2 + 2·x1   
POL(n__0) = 0   
POL(n__cons(x1, x2)) = 2·x1 + 2·x2   
POL(n__length(x1)) = 2 + 2·x1   
POL(n__nil) = 0   
POL(n__s(x1)) = x1   
POL(n__zeros) = 0   
POL(nil) = 0   
POL(s(x1)) = x1   
POL(tt) = 0   
POL(zeros) = 0   




↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ RRRPoloQTRSProof
        ↳ QTRS
          ↳ RRRPoloQTRSProof
            ↳ QTRS
              ↳ RRRPoloQTRSProof
                ↳ QTRS
                  ↳ RRRPoloQTRSProof
QTRS
                      ↳ DependencyPairsProof

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

zeroscons(0, n__zeros)
U21(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__s(V1)) → U21(isNat(activate(V1)))
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
niln__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))
U611(tt, L, N) → ISNAT(activate(N))
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)
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)
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)))
ACTIVATE(n__zeros) → ZEROS
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)
ZEROSCONS(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))
ZEROS01
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

zeroscons(0, n__zeros)
U21(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__s(V1)) → U21(isNat(activate(V1)))
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
niln__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
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ RRRPoloQTRSProof
        ↳ QTRS
          ↳ RRRPoloQTRSProof
            ↳ QTRS
              ↳ RRRPoloQTRSProof
                ↳ QTRS
                  ↳ RRRPoloQTRSProof
                    ↳ QTRS
                      ↳ DependencyPairsProof
QDP
                          ↳ DependencyGraphProof

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

ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
U611(tt, L, N) → ISNAT(activate(N))
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)
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)
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)))
ACTIVATE(n__zeros) → ZEROS
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)
ZEROSCONS(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))
ZEROS01
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

zeroscons(0, n__zeros)
U21(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__s(V1)) → U21(isNat(activate(V1)))
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
niln__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 15 less nodes.

↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ RRRPoloQTRSProof
        ↳ QTRS
          ↳ RRRPoloQTRSProof
            ↳ QTRS
              ↳ RRRPoloQTRSProof
                ↳ QTRS
                  ↳ RRRPoloQTRSProof
                    ↳ QTRS
                      ↳ DependencyPairsProof
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ AND
QDP
                                ↳ UsableRulesReductionPairsProof
                              ↳ 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)
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))
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

zeroscons(0, n__zeros)
U21(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__s(V1)) → U21(isNat(activate(V1)))
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
niln__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.
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(ACTIVATE(x1)) = 2·x1   
POL(ISNAT(x1)) = 2·x1   
POL(ISNATLIST(x1)) = x1   
POL(LENGTH(x1)) = 2·x1   
POL(U21(x1)) = x1   
POL(U51(x1, x2)) = 2·x1 + 2·x2   
POL(U511(x1, x2)) = x1 + 2·x2   
POL(U52(x1)) = x1   
POL(U61(x1, x2, x3)) = x1 + 2·x2 + 2·x3   
POL(U611(x1, x2, x3)) = x1 + 2·x2 + 2·x3   
POL(U62(x1, x2)) = x1 + 2·x2   
POL(U621(x1, x2)) = x1 + 2·x2   
POL(activate(x1)) = x1   
POL(cons(x1, x2)) = 2·x1 + 2·x2   
POL(isNat(x1)) = 2·x1   
POL(isNatList(x1)) = 2·x1   
POL(length(x1)) = 2·x1   
POL(n__0) = 0   
POL(n__cons(x1, x2)) = 2·x1 + 2·x2   
POL(n__length(x1)) = 2·x1   
POL(n__nil) = 0   
POL(n__s(x1)) = x1   
POL(n__zeros) = 0   
POL(nil) = 0   
POL(s(x1)) = x1   
POL(tt) = 0   
POL(zeros) = 0   



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ RRRPoloQTRSProof
        ↳ QTRS
          ↳ RRRPoloQTRSProof
            ↳ QTRS
              ↳ RRRPoloQTRSProof
                ↳ QTRS
                  ↳ RRRPoloQTRSProof
                    ↳ QTRS
                      ↳ DependencyPairsProof
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ AND
                              ↳ QDP
                                ↳ UsableRulesReductionPairsProof
QDP
                                    ↳ RuleRemovalProof
                              ↳ 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)
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))
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

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

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

ACTIVATE(n__length(X)) → LENGTH(activate(X))
ACTIVATE(n__length(X)) → ACTIVATE(X)


Used ordering: POLO with Polynomial interpretation [25]:

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



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

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

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
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ RRRPoloQTRSProof
        ↳ QTRS
          ↳ RRRPoloQTRSProof
            ↳ QTRS
              ↳ RRRPoloQTRSProof
                ↳ QTRS
                  ↳ RRRPoloQTRSProof
                    ↳ QTRS
                      ↳ DependencyPairsProof
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ AND
                              ↳ QDP
                                ↳ UsableRulesReductionPairsProof
                                  ↳ QDP
                                    ↳ RuleRemovalProof
                                      ↳ QDP
                                        ↳ DependencyGraphProof
                                          ↳ AND
QDP
                                              ↳ UsableRulesProof
                                            ↳ 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:

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

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

↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ RRRPoloQTRSProof
        ↳ QTRS
          ↳ RRRPoloQTRSProof
            ↳ QTRS
              ↳ RRRPoloQTRSProof
                ↳ QTRS
                  ↳ RRRPoloQTRSProof
                    ↳ QTRS
                      ↳ DependencyPairsProof
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ AND
                              ↳ QDP
                                ↳ UsableRulesReductionPairsProof
                                  ↳ QDP
                                    ↳ RuleRemovalProof
                                      ↳ QDP
                                        ↳ DependencyGraphProof
                                          ↳ AND
                                            ↳ QDP
                                              ↳ UsableRulesProof
QDP
                                                  ↳ QDPSizeChangeProof
                                            ↳ 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)

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

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



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

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

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)) = 3   
POL(n__length(x1)) = 0   
POL(U61(x1, x2, x3)) = (4)x_1   
POL(n__zeros) = 3   
POL(n__cons(x1, x2)) = 3   
POL(activate(x1)) = x_1   
POL(n__s(x1)) = 2 + (4)x_1   
POL(n__nil) = 1   
POL(0) = 2   
POL(U51(x1, x2)) = 0   
POL(ISNAT(x1)) = x_1   
POL(cons(x1, x2)) = 3   
POL(tt) = 1   
POL(isNatList(x1)) = 0   
POL(n__0) = 2   
POL(zeros) = 3   
POL(U52(x1)) = (4)x_1   
POL(s(x1)) = 2 + (4)x_1   
POL(length(x1)) = 0   
POL(isNat(x1)) = 2   
POL(nil) = 1   
POL(U21(x1)) = 2   
The value of delta used in the strict ordering is 2.
The following usable rules [17] were oriented:

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



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

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

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

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATLIST(n__cons(V1, V2)) → U511(isNat(activate(V1)), activate(V2)) at position [0] we obtained the following new rules:

ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
ISNATLIST(n__cons(n__nil, y1)) → U511(isNat(nil), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
ISNATLIST(n__cons(n__zeros, y1)) → U511(isNat(zeros), activate(y1))
ISNATLIST(n__cons(n__0, y1)) → U511(isNat(0), activate(y1))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))



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

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

ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
ISNATLIST(n__cons(n__nil, y1)) → U511(isNat(nil), activate(y1))
ISNATLIST(n__cons(n__zeros, y1)) → U511(isNat(zeros), activate(y1))
U511(tt, V2) → ISNATLIST(activate(V2))
ISNATLIST(n__cons(n__0, y1)) → U511(isNat(0), activate(y1))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U511(tt, V2) → ISNATLIST(activate(V2)) at position [0] we obtained the following new rules:

U511(tt, n__nil) → ISNATLIST(nil)
U511(tt, n__zeros) → ISNATLIST(zeros)
U511(tt, n__0) → ISNATLIST(0)
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
U511(tt, x0) → ISNATLIST(x0)
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))



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

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

ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
ISNATLIST(n__cons(n__nil, y1)) → U511(isNat(nil), activate(y1))
ISNATLIST(n__cons(n__0, y1)) → U511(isNat(0), activate(y1))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
U511(tt, n__nil) → ISNATLIST(nil)
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
ISNATLIST(n__cons(n__zeros, y1)) → U511(isNat(zeros), activate(y1))
U511(tt, n__zeros) → ISNATLIST(zeros)
U511(tt, n__0) → ISNATLIST(0)
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
U511(tt, x0) → ISNATLIST(x0)
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATLIST(n__cons(n__nil, y1)) → U511(isNat(nil), activate(y1)) at position [0] we obtained the following new rules:

ISNATLIST(n__cons(n__nil, y0)) → U511(isNat(n__nil), activate(y0))



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

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

ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
ISNATLIST(n__cons(n__0, y1)) → U511(isNat(0), activate(y1))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
U511(tt, n__nil) → ISNATLIST(nil)
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__0) → ISNATLIST(0)
U511(tt, n__zeros) → ISNATLIST(zeros)
ISNATLIST(n__cons(n__zeros, y1)) → U511(isNat(zeros), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__nil, y0)) → U511(isNat(n__nil), activate(y0))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))

The TRS R consists of the following rules:

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

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

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

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

ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
ISNATLIST(n__cons(n__0, y1)) → U511(isNat(0), activate(y1))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
U511(tt, n__nil) → ISNATLIST(nil)
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__zeros) → ISNATLIST(zeros)
U511(tt, n__0) → ISNATLIST(0)
ISNATLIST(n__cons(n__zeros, y1)) → U511(isNat(zeros), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
U511(tt, x0) → ISNATLIST(x0)
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U511(tt, n__nil) → ISNATLIST(nil) at position [0] we obtained the following new rules:

U511(tt, n__nil) → ISNATLIST(n__nil)



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

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

U511(tt, n__nil) → ISNATLIST(n__nil)
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
ISNATLIST(n__cons(n__0, y1)) → U511(isNat(0), activate(y1))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
ISNATLIST(n__cons(n__zeros, y1)) → U511(isNat(zeros), activate(y1))
U511(tt, n__0) → ISNATLIST(0)
U511(tt, n__zeros) → ISNATLIST(zeros)
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
U511(tt, x0) → ISNATLIST(x0)
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))

The TRS R consists of the following rules:

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

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

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

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

ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
ISNATLIST(n__cons(n__zeros, y1)) → U511(isNat(zeros), activate(y1))
U511(tt, n__0) → ISNATLIST(0)
U511(tt, n__zeros) → ISNATLIST(zeros)
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__0, y1)) → U511(isNat(0), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U511(tt, n__zeros) → ISNATLIST(zeros) at position [0] we obtained the following new rules:

U511(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
U511(tt, n__zeros) → ISNATLIST(n__zeros)



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

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

ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
ISNATLIST(n__cons(n__0, y1)) → U511(isNat(0), activate(y1))
U511(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__0) → ISNATLIST(0)
ISNATLIST(n__cons(n__zeros, y1)) → U511(isNat(zeros), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
U511(tt, x0) → ISNATLIST(x0)
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
U511(tt, n__zeros) → ISNATLIST(n__zeros)

The TRS R consists of the following rules:

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

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

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

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

ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
ISNATLIST(n__cons(n__zeros, y1)) → U511(isNat(zeros), activate(y1))
U511(tt, n__0) → ISNATLIST(0)
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__0, y1)) → U511(isNat(0), activate(y1))
U511(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U511(tt, n__0) → ISNATLIST(0) at position [0] we obtained the following new rules:

U511(tt, n__0) → ISNATLIST(n__0)



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

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

ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
ISNATLIST(n__cons(n__0, y1)) → U511(isNat(0), activate(y1))
U511(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
ISNATLIST(n__cons(n__zeros, y1)) → U511(isNat(zeros), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
U511(tt, x0) → ISNATLIST(x0)
U511(tt, n__0) → ISNATLIST(n__0)
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))

The TRS R consists of the following rules:

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

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

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

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

ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
ISNATLIST(n__cons(n__zeros, y1)) → U511(isNat(zeros), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__0, y1)) → U511(isNat(0), activate(y1))
U511(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATLIST(n__cons(n__zeros, y1)) → U511(isNat(zeros), activate(y1)) at position [0] we obtained the following new rules:

ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(0, n__zeros)), activate(y0))
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(n__zeros), activate(y0))



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

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

ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
ISNATLIST(n__cons(n__0, y1)) → U511(isNat(0), activate(y1))
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(0, n__zeros)), activate(y0))
U511(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(n__zeros), activate(y0))
U511(tt, x0) → ISNATLIST(x0)
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))

The TRS R consists of the following rules:

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

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

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

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

ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__0, y1)) → U511(isNat(0), activate(y1))
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(0, n__zeros)), activate(y0))
U511(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATLIST(n__cons(n__0, y1)) → U511(isNat(0), activate(y1)) at position [0] we obtained the following new rules:

ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))



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

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

ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(0, n__zeros)), activate(y0))
U511(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U511(tt, n__zeros) → ISNATLIST(cons(0, n__zeros)) at position [0] we obtained the following new rules:

U511(tt, n__zeros) → ISNATLIST(cons(n__0, n__zeros))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))



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

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

ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(0, n__zeros)), activate(y0))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
U511(tt, n__zeros) → ISNATLIST(cons(n__0, n__zeros))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(0, n__zeros)), activate(y0)) at position [0] we obtained the following new rules:

ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(n__0, n__zeros)), activate(y0))
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(n__cons(0, n__zeros)), activate(y0))



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

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

ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(n__cons(0, n__zeros)), activate(y0))
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(n__0, n__zeros)), activate(y0))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
U511(tt, n__zeros) → ISNATLIST(cons(n__0, n__zeros))

The TRS R consists of the following rules:

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

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

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

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

ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(n__0, n__zeros)), activate(y0))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))
U511(tt, n__zeros) → ISNATLIST(cons(n__0, n__zeros))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U511(tt, n__zeros) → ISNATLIST(cons(n__0, n__zeros)) at position [0] we obtained the following new rules:

U511(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))



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

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

ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
U511(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(n__0, n__zeros)), activate(y0))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(n__0, n__zeros)), activate(y0)) at position [0] we obtained the following new rules:

ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(n__cons(n__0, n__zeros)), activate(y0))



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

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

ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
U511(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(n__cons(n__0, n__zeros)), activate(y0))
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))

The TRS R consists of the following rules:

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

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

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

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

ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))

The TRS R consists of the following rules:

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

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

U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))


Used ordering: POLO with Polynomial interpretation [25]:

POL(0) = 0   
POL(ISNATLIST(x1)) = x1   
POL(U21(x1)) = x1   
POL(U51(x1, x2)) = 2·x1 + 2·x2   
POL(U511(x1, x2)) = 2·x1 + 2·x2   
POL(U52(x1)) = x1   
POL(U61(x1, x2, x3)) = 1 + x1 + 2·x2 + x3   
POL(U62(x1, x2)) = 1 + x1 + 2·x2   
POL(activate(x1)) = x1   
POL(cons(x1, x2)) = 2·x1 + 2·x2   
POL(isNat(x1)) = x1   
POL(isNatList(x1)) = x1   
POL(length(x1)) = 1 + 2·x1   
POL(n__0) = 0   
POL(n__cons(x1, x2)) = 2·x1 + 2·x2   
POL(n__length(x1)) = 1 + 2·x1   
POL(n__nil) = 0   
POL(n__s(x1)) = x1   
POL(n__zeros) = 0   
POL(nil) = 0   
POL(s(x1)) = x1   
POL(tt) = 0   
POL(zeros) = 0   



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

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

ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U511(tt, x0) → ISNATLIST(x0)
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))

The TRS R consists of the following rules:

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

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

ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))


Used ordering: POLO with Polynomial interpretation [25]:

POL(0) = 0   
POL(ISNATLIST(x1)) = 2·x1   
POL(U21(x1)) = x1   
POL(U51(x1, x2)) = 2·x1 + 2·x2   
POL(U511(x1, x2)) = 2·x1 + 2·x2   
POL(U52(x1)) = 2·x1   
POL(U61(x1, x2, x3)) = 1 + x1 + x2 + 2·x3   
POL(U62(x1, x2)) = 1 + x1 + x2   
POL(activate(x1)) = x1   
POL(cons(x1, x2)) = 2·x1 + 2·x2   
POL(isNat(x1)) = x1   
POL(isNatList(x1)) = x1   
POL(length(x1)) = 1 + x1   
POL(n__0) = 0   
POL(n__cons(x1, x2)) = 2·x1 + 2·x2   
POL(n__length(x1)) = 1 + x1   
POL(n__nil) = 0   
POL(n__s(x1)) = x1   
POL(n__zeros) = 0   
POL(nil) = 0   
POL(s(x1)) = x1   
POL(tt) = 0   
POL(zeros) = 0   



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

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

ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))

The TRS R consists of the following rules:

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

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.


ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
The remaining pairs can at least be oriented weakly.

ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(ISNATLIST(x1)) = x1   
POL(U21(x1)) = 0   
POL(U51(x1, x2)) = x2   
POL(U511(x1, x2)) = x2   
POL(U52(x1)) = x1   
POL(U61(x1, x2, x3)) = 1   
POL(U62(x1, x2)) = 1   
POL(activate(x1)) = x1   
POL(cons(x1, x2)) = x1 + x2   
POL(isNat(x1)) = 0   
POL(isNatList(x1)) = x1   
POL(length(x1)) = 1 + x1   
POL(n__0) = 0   
POL(n__cons(x1, x2)) = x1 + x2   
POL(n__length(x1)) = 1 + x1   
POL(n__nil) = 0   
POL(n__s(x1)) = 1   
POL(n__zeros) = 0   
POL(nil) = 0   
POL(s(x1)) = 1   
POL(tt) = 0   
POL(zeros) = 0   

The following usable rules [17] were oriented:

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



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

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

ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U511(tt, x0) → ISNATLIST(x0)
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))

The TRS R consists of the following rules:

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

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.


U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
The remaining pairs can at least be oriented weakly.

ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U511(tt, x0) → ISNATLIST(x0)
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
Used ordering: Polynomial interpretation [25,35]:

POL(U62(x1, x2)) = 1   
POL(U511(x1, x2)) = (4)x_2   
POL(n__length(x1)) = 4 + (3)x_1   
POL(U61(x1, x2, x3)) = 1 + (2)x_1 + (3)x_2   
POL(n__zeros) = 0   
POL(n__cons(x1, x2)) = (4)x_2   
POL(activate(x1)) = x_1   
POL(n__s(x1)) = 1   
POL(n__nil) = 0   
POL(0) = 0   
POL(U51(x1, x2)) = x_2   
POL(ISNATLIST(x1)) = x_1   
POL(cons(x1, x2)) = (4)x_2   
POL(tt) = 0   
POL(isNatList(x1)) = (2)x_1   
POL(n__0) = 0   
POL(zeros) = 0   
POL(U52(x1)) = 0   
POL(s(x1)) = 1   
POL(isNat(x1)) = 0   
POL(length(x1)) = 4 + (3)x_1   
POL(nil) = 0   
POL(U21(x1)) = 0   
The value of delta used in the strict ordering is 3.
The following usable rules [17] were oriented:

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



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

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

ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))

The TRS R consists of the following rules:

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

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.


ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
The remaining pairs can at least be oriented weakly.

ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( U62(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

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

M( U61(x1, ..., x3) ) =
/0\
\0/
+
/01\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3

M( n__zeros ) =
/1\
\0/

M( n__cons(x1, x2) ) =
/1\
\0/
+
/00\
\10/
·x1+
/00\
\01/
·x2

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

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

M( n__nil ) =
/0\
\0/

M( 0 ) =
/0\
\0/

M( U51(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\01/
·x2

M( cons(x1, x2) ) =
/1\
\0/
+
/00\
\10/
·x1+
/00\
\01/
·x2

M( tt ) =
/0\
\0/

M( isNatList(x1) ) =
/1\
\0/
+
/00\
\01/
·x1

M( n__0 ) =
/0\
\0/

M( zeros ) =
/1\
\0/

M( U52(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( s(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

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

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

M( nil ) =
/0\
\0/

M( U21(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

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

M( U511(x1, x2) ) = 0+
[0,0]
·x1+
[0,1]
·x2


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:

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



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

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

ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U511(tt, x0) → ISNATLIST(x0)
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

The TRS P consists of the following rules:

ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U511(tt, x0) → ISNATLIST(x0)
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))

The TRS R consists of the following rules:

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


s = U511(isNat(n__0), activate(n__zeros)) evaluates to t =U511(isNat(n__0), activate(n__zeros))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:




Rewriting sequence

U511(isNat(n__0), activate(n__zeros))U511(isNat(n__0), n__zeros)
with rule activate(X) → X at position [1] and matcher [X / n__zeros]

U511(isNat(n__0), n__zeros)U511(tt, n__zeros)
with rule isNat(n__0) → tt at position [0] and matcher [ ]

U511(tt, n__zeros)ISNATLIST(n__cons(n__0, n__zeros))
with rule U511(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros)) at position [] and matcher [ ]

ISNATLIST(n__cons(n__0, n__zeros))U511(isNat(n__0), activate(n__zeros))
with rule ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.





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

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

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

↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ RRRPoloQTRSProof
        ↳ QTRS
          ↳ RRRPoloQTRSProof
            ↳ QTRS
              ↳ RRRPoloQTRSProof
                ↳ QTRS
                  ↳ RRRPoloQTRSProof
                    ↳ QTRS
                      ↳ DependencyPairsProof
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ AND
                              ↳ QDP
QDP
                                ↳ UsableRulesReductionPairsProof

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:

zeroscons(0, n__zeros)
U21(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__s(V1)) → U21(isNat(activate(V1)))
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
niln__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.
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(ISNATILIST(x1)) = 2·x1   
POL(U21(x1)) = x1   
POL(U411(x1, x2)) = x1 + 2·x2   
POL(U51(x1, x2)) = x1 + 2·x2   
POL(U52(x1)) = x1   
POL(U61(x1, x2, x3)) = 1 + x1 + 2·x2 + 2·x3   
POL(U62(x1, x2)) = 1 + 2·x1 + 2·x2   
POL(activate(x1)) = x1   
POL(cons(x1, x2)) = x1 + 2·x2   
POL(isNat(x1)) = x1   
POL(isNatList(x1)) = 2·x1   
POL(length(x1)) = 1 + 2·x1   
POL(n__0) = 0   
POL(n__cons(x1, x2)) = x1 + 2·x2   
POL(n__length(x1)) = 1 + 2·x1   
POL(n__nil) = 1   
POL(n__s(x1)) = x1   
POL(n__zeros) = 0   
POL(nil) = 1   
POL(s(x1)) = x1   
POL(tt) = 0   
POL(zeros) = 0   



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

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:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U411(tt, V2) → ISNATILIST(activate(V2)) at position [0] we obtained the following new rules:

U411(tt, n__zeros) → ISNATILIST(zeros)
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
U411(tt, n__0) → ISNATILIST(0)
U411(tt, n__nil) → ISNATILIST(nil)
U411(tt, x0) → ISNATILIST(x0)
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))



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

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

U411(tt, n__zeros) → ISNATILIST(zeros)
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
U411(tt, n__nil) → ISNATILIST(nil)
U411(tt, n__0) → ISNATILIST(0)
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(V1, V2)) → U411(isNat(activate(V1)), activate(V2))
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATILIST(n__cons(V1, V2)) → U411(isNat(activate(V1)), activate(V2)) at position [0] we obtained the following new rules:

ISNATILIST(n__cons(n__0, y1)) → U411(isNat(0), activate(y1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
ISNATILIST(n__cons(n__nil, y1)) → U411(isNat(nil), activate(y1))
ISNATILIST(n__cons(n__zeros, y1)) → U411(isNat(zeros), activate(y1))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(activate(x0))), activate(y1))



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

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

ISNATILIST(n__cons(n__0, y1)) → U411(isNat(0), activate(y1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
U411(tt, n__0) → ISNATILIST(0)
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(activate(x0))), activate(y1))
ISNATILIST(n__cons(n__zeros, y1)) → U411(isNat(zeros), activate(y1))
U411(tt, n__zeros) → ISNATILIST(zeros)
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U411(tt, n__nil) → ISNATILIST(nil)
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__nil, y1)) → U411(isNat(nil), activate(y1))
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U411(tt, n__zeros) → ISNATILIST(zeros) at position [0] we obtained the following new rules:

U411(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
U411(tt, n__zeros) → ISNATILIST(n__zeros)



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

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

U411(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(n__0, y1)) → U411(isNat(0), activate(y1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
U411(tt, n__0) → ISNATILIST(0)
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
ISNATILIST(n__cons(n__zeros, y1)) → U411(isNat(zeros), activate(y1))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(activate(x0))), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U411(tt, n__zeros) → ISNATILIST(n__zeros)
U411(tt, n__nil) → ISNATILIST(nil)
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__nil, y1)) → U411(isNat(nil), activate(y1))
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))

The TRS R consists of the following rules:

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

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

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

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

U411(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(n__0, y1)) → U411(isNat(0), activate(y1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
U411(tt, n__0) → ISNATILIST(0)
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
ISNATILIST(n__cons(n__zeros, y1)) → U411(isNat(zeros), activate(y1))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(activate(x0))), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U411(tt, n__nil) → ISNATILIST(nil)
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__nil, y1)) → U411(isNat(nil), activate(y1))
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATILIST(n__cons(n__0, y1)) → U411(isNat(0), activate(y1)) at position [0] we obtained the following new rules:

ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))



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

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

U411(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
U411(tt, n__0) → ISNATILIST(0)
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(activate(x0))), activate(y1))
ISNATILIST(n__cons(n__zeros, y1)) → U411(isNat(zeros), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
U411(tt, n__nil) → ISNATILIST(nil)
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__nil, y1)) → U411(isNat(nil), activate(y1))
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U411(tt, n__0) → ISNATILIST(0) at position [0] we obtained the following new rules:

U411(tt, n__0) → ISNATILIST(n__0)



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

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

U411(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
U411(tt, n__0) → ISNATILIST(n__0)
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
ISNATILIST(n__cons(n__zeros, y1)) → U411(isNat(zeros), activate(y1))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(activate(x0))), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
U411(tt, n__nil) → ISNATILIST(nil)
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__nil, y1)) → U411(isNat(nil), activate(y1))
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))

The TRS R consists of the following rules:

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

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

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

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

U411(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
ISNATILIST(n__cons(n__zeros, y1)) → U411(isNat(zeros), activate(y1))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(activate(x0))), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
U411(tt, n__nil) → ISNATILIST(nil)
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__nil, y1)) → U411(isNat(nil), activate(y1))
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U411(tt, n__nil) → ISNATILIST(nil) at position [0] we obtained the following new rules:

U411(tt, n__nil) → ISNATILIST(n__nil)



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

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

U411(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
U411(tt, n__nil) → ISNATILIST(n__nil)
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(activate(x0))), activate(y1))
ISNATILIST(n__cons(n__zeros, y1)) → U411(isNat(zeros), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__nil, y1)) → U411(isNat(nil), activate(y1))
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))

The TRS R consists of the following rules:

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

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

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

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

U411(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__nil, y1)) → U411(isNat(nil), activate(y1))
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(activate(x0))), activate(y1))
ISNATILIST(n__cons(n__zeros, y1)) → U411(isNat(zeros), activate(y1))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATILIST(n__cons(n__nil, y1)) → U411(isNat(nil), activate(y1)) at position [0] we obtained the following new rules:

ISNATILIST(n__cons(n__nil, y0)) → U411(isNat(n__nil), activate(y0))



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

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

U411(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
ISNATILIST(n__cons(n__zeros, y1)) → U411(isNat(zeros), activate(y1))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(activate(x0))), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
U411(tt, x0) → ISNATILIST(x0)
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__nil, y0)) → U411(isNat(n__nil), activate(y0))

The TRS R consists of the following rules:

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

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

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

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

U411(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(activate(x0))), activate(y1))
ISNATILIST(n__cons(n__zeros, y1)) → U411(isNat(zeros), activate(y1))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATILIST(n__cons(n__zeros, y1)) → U411(isNat(zeros), activate(y1)) at position [0] we obtained the following new rules:

ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(n__zeros), activate(y0))
ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(cons(0, n__zeros)), activate(y0))



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

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

U411(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(cons(0, n__zeros)), activate(y0))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(activate(x0))), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(n__zeros), activate(y0))
U411(tt, x0) → ISNATILIST(x0)
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))

The TRS R consists of the following rules:

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

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

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

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

U411(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(cons(0, n__zeros)), activate(y0))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
U411(tt, x0) → ISNATILIST(x0)
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(activate(x0))), activate(y1))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U411(tt, n__zeros) → ISNATILIST(cons(0, n__zeros)) at position [0] we obtained the following new rules:

U411(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U411(tt, n__zeros) → ISNATILIST(cons(n__0, n__zeros))



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

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

ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(cons(0, n__zeros)), activate(y0))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
U411(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(activate(x0))), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
U411(tt, n__zeros) → ISNATILIST(cons(n__0, n__zeros))
U411(tt, x0) → ISNATILIST(x0)
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(cons(0, n__zeros)), activate(y0)) at position [0] we obtained the following new rules:

ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(n__cons(0, n__zeros)), activate(y0))
ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(cons(n__0, n__zeros)), activate(y0))



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

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

ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(n__cons(0, n__zeros)), activate(y0))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
U411(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(activate(x0))), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
U411(tt, n__zeros) → ISNATILIST(cons(n__0, n__zeros))
U411(tt, x0) → ISNATILIST(x0)
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(cons(n__0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

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

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

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

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

U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
U411(tt, n__zeros) → ISNATILIST(cons(n__0, n__zeros))
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
U411(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(cons(n__0, n__zeros)), activate(y0))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(activate(x0))), activate(y1))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(cons(n__0, n__zeros)), activate(y0)) at position [0] we obtained the following new rules:

ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(n__cons(n__0, n__zeros)), activate(y0))



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

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

ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(n__cons(n__0, n__zeros)), activate(y0))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
U411(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(activate(x0))), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
U411(tt, n__zeros) → ISNATILIST(cons(n__0, n__zeros))
U411(tt, x0) → ISNATILIST(x0)
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))

The TRS R consists of the following rules:

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

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

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

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

U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
U411(tt, n__zeros) → ISNATILIST(cons(n__0, n__zeros))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
U411(tt, x0) → ISNATILIST(x0)
U411(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(activate(x0))), activate(y1))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U411(tt, n__zeros) → ISNATILIST(cons(n__0, n__zeros)) at position [0] we obtained the following new rules:

U411(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))



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

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

U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
U411(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U411(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(activate(x0))), activate(y1))

The TRS R consists of the following rules:

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

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

U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))


Used ordering: POLO with Polynomial interpretation [25]:

POL(0) = 0   
POL(ISNATILIST(x1)) = x1   
POL(U21(x1)) = x1   
POL(U411(x1, x2)) = 2·x1 + 2·x2   
POL(U51(x1, x2)) = 2·x1 + x2   
POL(U52(x1)) = x1   
POL(U61(x1, x2, x3)) = 1 + x1 + x2 + 2·x3   
POL(U62(x1, x2)) = 1 + x1 + x2   
POL(activate(x1)) = x1   
POL(cons(x1, x2)) = 2·x1 + 2·x2   
POL(isNat(x1)) = x1   
POL(isNatList(x1)) = x1   
POL(length(x1)) = 1 + x1   
POL(n__0) = 0   
POL(n__cons(x1, x2)) = 2·x1 + 2·x2   
POL(n__length(x1)) = 1 + x1   
POL(n__nil) = 0   
POL(n__s(x1)) = x1   
POL(n__zeros) = 0   
POL(nil) = 0   
POL(s(x1)) = x1   
POL(tt) = 0   
POL(zeros) = 0   



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

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

U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
U411(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
U411(tt, x0) → ISNATILIST(x0)
U411(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(activate(x0))), activate(y1))

The TRS R consists of the following rules:

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

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

ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(activate(x0))), activate(y1))


Used ordering: POLO with Polynomial interpretation [25]:

POL(0) = 0   
POL(ISNATILIST(x1)) = 2·x1   
POL(U21(x1)) = x1   
POL(U411(x1, x2)) = 2·x1 + 2·x2   
POL(U51(x1, x2)) = x1 + 2·x2   
POL(U52(x1)) = 2·x1   
POL(U61(x1, x2, x3)) = 1 + 2·x1 + 2·x2 + x3   
POL(U62(x1, x2)) = 1 + x1 + 2·x2   
POL(activate(x1)) = x1   
POL(cons(x1, x2)) = 2·x1 + 2·x2   
POL(isNat(x1)) = x1   
POL(isNatList(x1)) = x1   
POL(length(x1)) = 1 + 2·x1   
POL(n__0) = 0   
POL(n__cons(x1, x2)) = 2·x1 + 2·x2   
POL(n__length(x1)) = 1 + 2·x1   
POL(n__nil) = 0   
POL(n__s(x1)) = x1   
POL(n__zeros) = 0   
POL(nil) = 0   
POL(s(x1)) = x1   
POL(tt) = 0   
POL(zeros) = 0   



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

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

U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
U411(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U411(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))

The TRS R consists of the following rules:

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

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.


ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
The remaining pairs can at least be oriented weakly.

U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
U411(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
U411(tt, x0) → ISNATILIST(x0)
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U411(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
Used ordering: Polynomial interpretation [25,35]:

POL(U62(x1, x2)) = 4   
POL(n__length(x1)) = (3)x_1   
POL(U61(x1, x2, x3)) = (4)x_1 + x_2 + x_3   
POL(ISNATILIST(x1)) = (2)x_1   
POL(n__zeros) = 0   
POL(n__cons(x1, x2)) = (2)x_1 + x_2   
POL(activate(x1)) = x_1   
POL(n__nil) = 4   
POL(n__s(x1)) = 1   
POL(0) = 0   
POL(U51(x1, x2)) = 0   
POL(cons(x1, x2)) = (2)x_1 + x_2   
POL(tt) = 1   
POL(U411(x1, x2)) = (2)x_2   
POL(isNatList(x1)) = 0   
POL(zeros) = 0   
POL(n__0) = 0   
POL(U52(x1)) = x_1   
POL(s(x1)) = 1   
POL(isNat(x1)) = x_1   
POL(length(x1)) = (3)x_1   
POL(nil) = 4   
POL(U21(x1)) = 1   
The value of delta used in the strict ordering is 4.
The following usable rules [17] were oriented:

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



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

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

U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
U411(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
U411(tt, x0) → ISNATILIST(x0)
U411(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))

The TRS R consists of the following rules:

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

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.


ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
The remaining pairs can at least be oriented weakly.

U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
U411(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
U411(tt, x0) → ISNATILIST(x0)
U411(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( U62(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

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

M( U61(x1, ..., x3) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\01/
·x2+
/00\
\10/
·x3

M( n__zeros ) =
/1\
\0/

M( n__cons(x1, x2) ) =
/1\
\0/
+
/00\
\10/
·x1+
/00\
\01/
·x2

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

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

M( n__nil ) =
/0\
\0/

M( 0 ) =
/0\
\0/

M( U51(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( cons(x1, x2) ) =
/1\
\0/
+
/00\
\10/
·x1+
/00\
\01/
·x2

M( tt ) =
/0\
\0/

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

M( n__0 ) =
/0\
\0/

M( zeros ) =
/1\
\0/

M( U52(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( s(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

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

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

M( nil ) =
/0\
\0/

M( U21(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

Tuple symbols:
M( U411(x1, x2) ) = 0+
[0,0]
·x1+
[0,1]
·x2

M( ISNATILIST(x1) ) = 0+
[0,1]
·x1


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


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

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



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

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

U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
U411(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
U411(tt, x0) → ISNATILIST(x0)
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U411(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))

The TRS R consists of the following rules:

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

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.


U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
The remaining pairs can at least be oriented weakly.

U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
U411(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
U411(tt, x0) → ISNATILIST(x0)
U411(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
Used ordering: Polynomial interpretation [25,35]:

POL(U62(x1, x2)) = 4 + x_2   
POL(n__length(x1)) = (2)x_1   
POL(U61(x1, x2, x3)) = (4)x_1 + x_2 + (3)x_3   
POL(ISNATILIST(x1)) = x_1   
POL(n__zeros) = 0   
POL(n__cons(x1, x2)) = (2)x_1 + (4)x_2   
POL(activate(x1)) = x_1   
POL(n__nil) = 1   
POL(n__s(x1)) = 4   
POL(0) = 0   
POL(U51(x1, x2)) = 0   
POL(cons(x1, x2)) = (2)x_1 + (4)x_2   
POL(tt) = 4   
POL(U411(x1, x2)) = (2)x_2   
POL(isNatList(x1)) = 0   
POL(zeros) = 0   
POL(n__0) = 0   
POL(U52(x1)) = (2)x_1   
POL(s(x1)) = 4   
POL(isNat(x1)) = (4)x_1   
POL(length(x1)) = (2)x_1   
POL(nil) = 1   
POL(U21(x1)) = 4   
The value of delta used in the strict ordering is 4.
The following usable rules [17] were oriented:

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



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

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

U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
U411(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
U411(tt, x0) → ISNATILIST(x0)
U411(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

The TRS P consists of the following rules:

U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
U411(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
U411(tt, x0) → ISNATILIST(x0)
U411(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))

The TRS R consists of the following rules:

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


s = U411(isNat(n__0), activate(n__zeros)) evaluates to t =U411(isNat(n__0), activate(n__zeros))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:




Rewriting sequence

U411(isNat(n__0), activate(n__zeros))U411(isNat(n__0), n__zeros)
with rule activate(X) → X at position [1] and matcher [X / n__zeros]

U411(isNat(n__0), n__zeros)U411(tt, n__zeros)
with rule isNat(n__0) → tt at position [0] and matcher [ ]

U411(tt, n__zeros)ISNATILIST(n__cons(n__0, n__zeros))
with rule U411(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros)) at position [] and matcher [ ]

ISNATILIST(n__cons(n__0, n__zeros))U411(isNat(n__0), activate(n__zeros))
with rule ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.




We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
The remaining pairs can at least be oriented weakly.

U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
U411(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
U411(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( U62(x1, x2) ) =
/0\
\1/
+
/00\
\00/
·x1+
/01\
\00/
·x2

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

M( U61(x1, ..., x3) ) =
/0\
\0/
+
/00\
\01/
·x1+
/11\
\00/
·x2+
/00\
\00/
·x3

M( n__zeros ) =
/0\
\0/

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

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

M( n__s(x1) ) =
/0\
\1/
+
/01\
\00/
·x1

M( n__nil ) =
/0\
\0/

M( 0 ) =
/0\
\0/

M( U51(x1, x2) ) =
/1\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

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

M( tt ) =
/0\
\1/

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

M( n__0 ) =
/0\
\0/

M( zeros ) =
/0\
\0/

M( U52(x1) ) =
/0\
\0/
+
/00\
\01/
·x1

M( s(x1) ) =
/0\
\1/
+
/01\
\00/
·x1

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

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

M( nil ) =
/0\
\0/

M( U21(x1) ) =
/0\
\1/
+
/00\
\00/
·x1

Tuple symbols:
M( U411(x1, x2) ) = 0+
[0,0]
·x1+
[1,1]
·x2

M( ISNATILIST(x1) ) = 0+
[1,0]
·x1


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


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

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



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

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

U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
U411(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
U411(tt, x0) → ISNATILIST(x0)
U411(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))

The TRS R consists of the following rules:

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

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