Termination w.r.t. Q proof of /tmp/tmp2Atnth/LengthOfFiniteLists_nokinds

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:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

Q is empty.

↳ QTRS

  ↳ RRRPoloQTRSProof

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

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

Q is empty.

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

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
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, x₂)) = x₁ + 2·x₂
POL(activate(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = 2·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__isNat(x₁)) = x₁
POL(n__isNatIList(x₁)) = 2·x₁
POL(n__isNatList(x₁)) = x₁
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:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

Q is empty.

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

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
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)) → isNatList(activate(V1))

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U11(x₁, x₂)) = 1 + x₁ + x₂
POL(activate(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 1 + x₁
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = x₁ + 2·x₂
POL(n__isNat(x₁)) = x₁
POL(n__isNatIList(x₁)) = x₁
POL(n__isNatList(x₁)) = x₁
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

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

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

Q is empty.

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

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
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) → isNatList(activate(V))
isNatIList(n__zeros) → tt

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U11(x₁, x₂)) = x₁ + 2·x₂
POL(activate(x₁)) = x₁
POL(and(x₁, x₂)) = 2·x₁ + x₂
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = 2 + 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__isNat(x₁)) = x₁
POL(n__isNatIList(x₁)) = 2 + x₁
POL(n__isNatList(x₁)) = x₁
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

              ↳ DependencyPairsProof

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

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

Q is empty.

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

ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__0) → 0¹
ACTIVATE(n__cons(X1, X2)) → CONS(X1, X2)
LENGTH(cons(N, L)) → U11¹(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U11¹(tt, L) → ACTIVATE(L)
ACTIVATE(n__zeros) → ZEROS
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → LENGTH(X)
LENGTH(cons(N, L)) → ACTIVATE(L)
ACTIVATE(n__nil) → NIL
ISNAT(n__s(V1)) → ISNAT(activate(V1))
U11¹(tt, L) → LENGTH(activate(L))
ACTIVATE(n__s(X)) → S(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ACTIVATE(n__isNat(X)) → ISNAT(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
U11¹(tt, L) → S(length(activate(L)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ZEROS → CONS(0, n__zeros)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ZEROS → 0¹
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNat(N))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

↳ 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))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__0) → 0¹
ACTIVATE(n__cons(X1, X2)) → CONS(X1, X2)
LENGTH(cons(N, L)) → U11¹(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U11¹(tt, L) → ACTIVATE(L)
ACTIVATE(n__zeros) → ZEROS
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → LENGTH(X)
LENGTH(cons(N, L)) → ACTIVATE(L)
ACTIVATE(n__nil) → NIL
ISNAT(n__s(V1)) → ISNAT(activate(V1))
U11¹(tt, L) → LENGTH(activate(L))
ACTIVATE(n__s(X)) → S(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ACTIVATE(n__isNat(X)) → ISNAT(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
U11¹(tt, L) → S(length(activate(L)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ZEROS → CONS(0, n__zeros)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ZEROS → 0¹
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNat(N))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
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 1 SCC with 8 less nodes.

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ RuleRemovalProof

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

ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → LENGTH(X)
LENGTH(cons(N, L)) → ACTIVATE(L)
U11¹(tt, L) → LENGTH(activate(L))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
AND(tt, X) → ACTIVATE(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
LENGTH(cons(N, L)) → U11¹(and(isNatList(activate(L)), n__isNat(N)), activate(L))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
U11¹(tt, L) → ACTIVATE(L)
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNat(N))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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(V1, V2)) → ISNAT(activate(V1))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)

Used ordering: POLO with Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(AND(x₁, x₂)) = x₁ + x₂
POL(ISNAT(x₁)) = x₁
POL(ISNATILIST(x₁)) = 2 + 2·x₁
POL(ISNATLIST(x₁)) = x₁
POL(LENGTH(x₁)) = 2·x₁
POL(U11(x₁, x₂)) = x₁ + 2·x₂
POL(U11¹(x₁, x₂)) = x₁ + 2·x₂
POL(activate(x₁)) = x₁
POL(and(x₁, x₂)) = 2·x₁ + x₂
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = 2 + 2·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__isNat(x₁)) = x₁
POL(n__isNatIList(x₁)) = 2 + 2·x₁
POL(n__isNatList(x₁)) = x₁
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

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ RuleRemovalProof

                        ↳ QDP

                          ↳ RuleRemovalProof

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

ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → LENGTH(X)
LENGTH(cons(N, L)) → ACTIVATE(L)
U11¹(tt, L) → LENGTH(activate(L))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
AND(tt, X) → ACTIVATE(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
LENGTH(cons(N, L)) → U11¹(and(isNatList(activate(L)), n__isNat(N)), activate(L))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
U11¹(tt, L) → ACTIVATE(L)
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNat(N))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

ACTIVATE(n__length(X)) → LENGTH(X)

Used ordering: POLO with Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(AND(x₁, x₂)) = x₁ + x₂
POL(ISNAT(x₁)) = x₁
POL(ISNATILIST(x₁)) = x₁
POL(ISNATLIST(x₁)) = x₁
POL(LENGTH(x₁)) = 2·x₁
POL(U11(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(U11¹(x₁, x₂)) = x₁ + 2·x₂
POL(activate(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = 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)) = x₁ + 2·x₂
POL(n__isNat(x₁)) = x₁
POL(n__isNatIList(x₁)) = x₁
POL(n__isNatList(x₁)) = x₁
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

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ RuleRemovalProof

                        ↳ QDP

                          ↳ RuleRemovalProof

                            ↳ QDP

                              ↳ DependencyGraphProof

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

ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → ACTIVATE(L)
U11¹(tt, L) → LENGTH(activate(L))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
AND(tt, X) → ACTIVATE(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
LENGTH(cons(N, L)) → U11¹(and(isNatList(activate(L)), n__isNat(N)), activate(L))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
U11¹(tt, L) → ACTIVATE(L)
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNat(N))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
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 4 less nodes.

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ RuleRemovalProof

                        ↳ QDP

                          ↳ RuleRemovalProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ AND

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                  ↳ QDP

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

ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

The following pairs can be oriented strictly and are deleted.

ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))

The remaining pairs can at least be oriented weakly.

ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(AND(x₁, x₂)) = x₂
POL(ISNAT(x₁)) = x₁
POL(ISNATILIST(x₁)) = 0
POL(ISNATLIST(x₁)) = 1 + x₁
POL(U11(x₁, x₂)) = 0
POL(activate(x₁)) = x₁
POL(and(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 1 + x₁
POL(length(x₁)) = 0
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = x₁ + x₂
POL(n__isNat(x₁)) = x₁
POL(n__isNatIList(x₁)) = 0
POL(n__isNatList(x₁)) = 1 + x₁
POL(n__length(x₁)) = 0
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

The following usable rules [17] were oriented:

isNatList(X) → n__isNatList(X)
nil → n__nil
activate(n__zeros) → zeros
isNat(X) → n__isNat(X)
s(X) → n__s(X)
length(X) → n__length(X)
isNatIList(X) → n__isNatIList(X)
cons(X1, X2) → n__cons(X1, X2)
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
0 → n__0
zeros → n__zeros
isNat(n__0) → tt
isNat(n__s(V1)) → isNat(activate(V1))
zeros → cons(0, n__zeros)
activate(X) → X
U11(tt, L) → s(length(activate(L)))
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
activate(n__isNatIList(X)) → isNatIList(X)
and(tt, X) → activate(X)
activate(n__isNatList(X)) → isNatList(X)
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
activate(n__nil) → nil
activate(n__isNat(X)) → isNat(X)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ RuleRemovalProof

                        ↳ QDP

                          ↳ RuleRemovalProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ AND

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                  ↳ QDP

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

ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
AND(tt, X) → ACTIVATE(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

The following pairs can be oriented strictly and are deleted.

ACTIVATE(n__isNat(X)) → ISNAT(X)

The remaining pairs can at least be oriented weakly.

ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
AND(tt, X) → ACTIVATE(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(AND(x₁, x₂)) = x₂
POL(ISNAT(x₁)) = x₁
POL(ISNATILIST(x₁)) = 0
POL(ISNATLIST(x₁)) = 0
POL(U11(x₁, x₂)) = 0
POL(activate(x₁)) = x₁
POL(and(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 1 + x₁
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = 0
POL(n__isNat(x₁)) = 1 + x₁
POL(n__isNatIList(x₁)) = 0
POL(n__isNatList(x₁)) = 0
POL(n__length(x₁)) = 0
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

The following usable rules [17] were oriented:

isNatList(X) → n__isNatList(X)
nil → n__nil
activate(n__zeros) → zeros
isNat(X) → n__isNat(X)
s(X) → n__s(X)
length(X) → n__length(X)
isNatIList(X) → n__isNatIList(X)
cons(X1, X2) → n__cons(X1, X2)
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
0 → n__0
zeros → n__zeros
isNat(n__0) → tt
isNat(n__s(V1)) → isNat(activate(V1))
zeros → cons(0, n__zeros)
activate(X) → X
U11(tt, L) → s(length(activate(L)))
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
activate(n__isNatIList(X)) → isNatIList(X)
and(tt, X) → activate(X)
activate(n__isNatList(X)) → isNatList(X)
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
activate(n__nil) → nil
activate(n__isNat(X)) → isNat(X)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ RuleRemovalProof

                        ↳ QDP

                          ↳ RuleRemovalProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ AND

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                  ↳ QDP

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

ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
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 1 less node.

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ RuleRemovalProof

                        ↳ QDP

                          ↳ RuleRemovalProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ AND

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ AND

                                                ↳ QDP

                                                  ↳ Narrowing

                                                ↳ QDP

                                  ↳ QDP

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

ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
AND(tt, X) → ACTIVATE(X)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__length(x0), y1)) → AND(isNat(length(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatList(activate(y1)))
ISNATLIST(n__cons(y0, y1)) → AND(n__isNat(activate(y0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatList(activate(y1)))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ RuleRemovalProof

                        ↳ QDP

                          ↳ RuleRemovalProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ AND

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ AND

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                ↳ QDP

                                  ↳ QDP

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

ISNATLIST(n__cons(n__length(x0), y1)) → AND(isNat(length(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(y0, y1)) → AND(n__isNat(activate(y0)), n__isNatList(activate(y1)))
AND(tt, X) → ACTIVATE(X)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatList(activate(y1)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatList(activate(y1)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATLIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatList(activate(y1)))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
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 1 SCC with 1 less node.

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ RuleRemovalProof

                        ↳ QDP

                          ↳ RuleRemovalProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ AND

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ AND

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                ↳ QDP

                                  ↳ QDP

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

ISNATLIST(n__cons(n__length(x0), y1)) → AND(isNat(length(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
AND(tt, X) → ACTIVATE(X)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatList(activate(y1)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatList(activate(y1)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATLIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatList(activate(y1)))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

ISNATILIST(n__cons(y0, y1)) → AND(n__isNat(activate(y0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__length(x0), y1)) → AND(isNat(length(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatIList(activate(y1)))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ RuleRemovalProof

                        ↳ QDP

                          ↳ RuleRemovalProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ AND

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ AND

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                  ↳ QDP

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

ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))
ISNATILIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatIList(activate(y1)))
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
ISNATILIST(n__cons(y0, y1)) → AND(n__isNat(activate(y0)), n__isNatIList(activate(y1)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatList(activate(y1)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatIList(activate(y1)))
ISNATLIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__length(x0), y1)) → AND(isNat(length(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatList(activate(y1)))
AND(tt, X) → ACTIVATE(X)
ISNATILIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatIList(activate(y1)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatList(activate(y1)))
ISNATILIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__length(x0), y1)) → AND(isNat(length(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatIList(activate(y1)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATLIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatList(activate(y1)))
ISNATILIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatIList(activate(y1)))
ISNATLIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatList(activate(y1)))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
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 1 SCC with 1 less node.

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ RuleRemovalProof

                        ↳ QDP

                          ↳ RuleRemovalProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ AND

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ AND

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Instantiation

                                                ↳ QDP

                                  ↳ QDP

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

ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__length(x0), y1)) → AND(isNat(length(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))
ISNATILIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatIList(activate(y1)))
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
AND(tt, X) → ACTIVATE(X)
ISNATILIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatIList(activate(y1)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatList(activate(y1)))
ISNATILIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__length(x0), y1)) → AND(isNat(length(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatIList(activate(y1)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))
ISNATILIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatIList(activate(y1)))
ISNATLIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatList(activate(y1)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATILIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatIList(activate(y1)))
ISNATLIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatList(activate(y1)))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule AND(tt, X) → ACTIVATE(X) we obtained the following new rules:

AND(tt, n__isNatIList(y_3)) → ACTIVATE(n__isNatIList(y_3))
AND(tt, n__isNatList(y_5)) → ACTIVATE(n__isNatList(y_5))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ RuleRemovalProof

                        ↳ QDP

                          ↳ RuleRemovalProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ AND

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ AND

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Instantiation

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                ↳ QDP

                                  ↳ QDP

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

ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))
ISNATILIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatIList(activate(y1)))
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatIList(activate(y1)))
ISNATLIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__length(x0), y1)) → AND(isNat(length(x0)), n__isNatList(activate(y1)))
ISNATILIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatIList(activate(y1)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatList(activate(y1)))
ISNATILIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__length(x0), y1)) → AND(isNat(length(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatIList(activate(y1)))
AND(tt, n__isNatIList(y_3)) → ACTIVATE(n__isNatIList(y_3))
ISNATLIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatList(activate(y1)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATILIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatIList(activate(y1)))
ISNATLIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatList(activate(y1)))
AND(tt, n__isNatList(y_5)) → ACTIVATE(n__isNatList(y_5))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
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.

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ RuleRemovalProof

                        ↳ QDP

                          ↳ RuleRemovalProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ AND

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ AND

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Instantiation

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ AND

                                                                          ↳ QDP

                                                                          ↳ QDP

                                                ↳ QDP

                                  ↳ QDP

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

ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__length(x0), y1)) → AND(isNat(length(x0)), n__isNatList(activate(y1)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatList(activate(y1)))
AND(tt, n__isNatList(y_5)) → ACTIVATE(n__isNatList(y_5))
ISNATLIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatList(activate(y1)))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ RuleRemovalProof

                        ↳ QDP

                          ↳ RuleRemovalProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ AND

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ AND

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Instantiation

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ AND

                                                                          ↳ QDP

                                                                          ↳ QDP

                                                                            ↳ RuleRemovalProof

                                                ↳ QDP

                                  ↳ QDP

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

ISNATILIST(n__cons(n__length(x0), y1)) → AND(isNat(length(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatIList(activate(y1)))
AND(tt, n__isNatIList(y_3)) → ACTIVATE(n__isNatIList(y_3))
ISNATILIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatIList(activate(y1)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatIList(activate(y1)))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

ISNATILIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatIList(activate(y1)))

Used ordering: POLO with Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = 2·x₁
POL(AND(x₁, x₂)) = 2·x₁ + 2·x₂
POL(ISNATILIST(x₁)) = 2·x₁
POL(U11(x₁, x₂)) = x₁ + x₂
POL(activate(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = x₁
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(n__isNat(x₁)) = x₁
POL(n__isNatIList(x₁)) = x₁
POL(n__isNatList(x₁)) = x₁
POL(n__length(x₁)) = x₁
POL(n__nil) = 2
POL(n__s(x₁)) = x₁
POL(n__zeros) = 0
POL(nil) = 2
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ RuleRemovalProof

                        ↳ QDP

                          ↳ RuleRemovalProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ AND

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ AND

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Instantiation

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ AND

                                                                          ↳ QDP

                                                                          ↳ QDP

                                                                            ↳ RuleRemovalProof

                                                                              ↳ QDP

                                                                                ↳ RuleRemovalProof

                                                ↳ QDP

                                  ↳ QDP

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

ISNATILIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__length(x0), y1)) → AND(isNat(length(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatIList(activate(y1)))
AND(tt, n__isNatIList(y_3)) → ACTIVATE(n__isNatIList(y_3))
ISNATILIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatIList(activate(y1)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATILIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatIList(activate(y1)))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

Used ordering: POLO with Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(AND(x₁, x₂)) = 2·x₁ + 2·x₂
POL(ISNATILIST(x₁)) = 2·x₁
POL(U11(x₁, x₂)) = 2 + 2·x₁ + 2·x₂
POL(activate(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = 2·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__isNat(x₁)) = x₁
POL(n__isNatIList(x₁)) = 2·x₁
POL(n__isNatList(x₁)) = x₁
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

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ RuleRemovalProof

                        ↳ QDP

                          ↳ RuleRemovalProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ AND

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ AND

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Instantiation

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ AND

                                                                          ↳ QDP

                                                                          ↳ QDP

                                                                            ↳ RuleRemovalProof

                                                                              ↳ QDP

                                                                                ↳ RuleRemovalProof

                                                                                  ↳ QDP

                                                                                    ↳ RuleRemovalProof

                                                ↳ QDP

                                  ↳ QDP

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

ISNATILIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatIList(activate(y1)))
AND(tt, n__isNatIList(y_3)) → ACTIVATE(n__isNatIList(y_3))
ISNATILIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatIList(activate(y1)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatIList(activate(y1)))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

ISNATILIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatIList(activate(y1)))

Used ordering: POLO with Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(AND(x₁, x₂)) = x₁ + x₂
POL(ISNATILIST(x₁)) = 2 + 2·x₁
POL(U11(x₁, x₂)) = 2·x₁ + 2·x₂
POL(activate(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = 2·x₁
POL(isNatIList(x₁)) = 2 + 2·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__isNat(x₁)) = 2·x₁
POL(n__isNatIList(x₁)) = 2 + 2·x₁
POL(n__isNatList(x₁)) = x₁
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

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ RuleRemovalProof

                        ↳ QDP

                          ↳ RuleRemovalProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ AND

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ AND

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Instantiation

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ AND

                                                                          ↳ QDP

                                                                          ↳ QDP

                                                                            ↳ RuleRemovalProof

                                                                              ↳ QDP

                                                                                ↳ RuleRemovalProof

                                                                                  ↳ QDP

                                                                                    ↳ RuleRemovalProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                ↳ QDP

                                  ↳ QDP

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

ISNATILIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatIList(activate(y1)))
AND(tt, n__isNatIList(y_3)) → ACTIVATE(n__isNatIList(y_3))
ISNATILIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatIList(activate(y1)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATILIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatIList(activate(y1)))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

The following pairs can be oriented strictly and are deleted.

ISNATILIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatIList(activate(y1)))

The remaining pairs can at least be oriented weakly.

ISNATILIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatIList(activate(y1)))
AND(tt, n__isNatIList(y_3)) → ACTIVATE(n__isNatIList(y_3))
ISNATILIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatIList(activate(y1)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATILIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatIList(activate(y1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(AND(x₁, x₂)) = x₂
POL(ISNATILIST(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(activate(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = x₁
POL(isNatList(x₁)) = 1 + x₁
POL(length(x₁)) = 0
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = x₁ + x₂
POL(n__isNat(x₁)) = x₁
POL(n__isNatIList(x₁)) = x₁
POL(n__isNatList(x₁)) = 1 + x₁
POL(n__length(x₁)) = 0
POL(n__nil) = 0
POL(n__s(x₁)) = x₁
POL(n__zeros) = 1
POL(nil) = 0
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 1

The following usable rules [17] were oriented:

isNatList(X) → n__isNatList(X)
nil → n__nil
activate(n__zeros) → zeros
isNat(X) → n__isNat(X)
s(X) → n__s(X)
length(X) → n__length(X)
isNatIList(X) → n__isNatIList(X)
cons(X1, X2) → n__cons(X1, X2)
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
0 → n__0
zeros → n__zeros
isNat(n__0) → tt
isNat(n__s(V1)) → isNat(activate(V1))
activate(X) → X
zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
activate(n__isNatIList(X)) → isNatIList(X)
and(tt, X) → activate(X)
activate(n__isNatList(X)) → isNatList(X)
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
activate(n__nil) → nil
activate(n__isNat(X)) → isNat(X)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ RuleRemovalProof

                        ↳ QDP

                          ↳ RuleRemovalProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ AND

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ AND

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Instantiation

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ AND

                                                                          ↳ QDP

                                                                          ↳ QDP

                                                                            ↳ RuleRemovalProof

                                                                              ↳ QDP

                                                                                ↳ RuleRemovalProof

                                                                                  ↳ QDP

                                                                                    ↳ RuleRemovalProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ QDPOrderProof

                                                ↳ QDP

                                  ↳ QDP

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

ISNATILIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatIList(activate(y1)))
AND(tt, n__isNatIList(y_3)) → ACTIVATE(n__isNatIList(y_3))
ISNATILIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatIList(activate(y1)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatIList(activate(y1)))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

The following pairs can be oriented strictly and are deleted.

ISNATILIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatIList(activate(y1)))

The remaining pairs can at least be oriented weakly.

ISNATILIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatIList(activate(y1)))
AND(tt, n__isNatIList(y_3)) → ACTIVATE(n__isNatIList(y_3))
ISNATILIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatIList(activate(y1)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatIList(activate(y1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(AND(x₁, x₂)) = x₂
POL(ISNATILIST(x₁)) = x₁
POL(U11(x₁, x₂)) = x₂
POL(activate(x₁)) = x₁
POL(and(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 1
POL(isNatIList(x₁)) = x₁
POL(isNatList(x₁)) = 0
POL(length(x₁)) = x₁
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = x₁ + x₂
POL(n__isNat(x₁)) = 1
POL(n__isNatIList(x₁)) = x₁
POL(n__isNatList(x₁)) = 0
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

The following usable rules [17] were oriented:

isNatList(X) → n__isNatList(X)
nil → n__nil
activate(n__zeros) → zeros
isNat(X) → n__isNat(X)
s(X) → n__s(X)
length(X) → n__length(X)
isNatIList(X) → n__isNatIList(X)
cons(X1, X2) → n__cons(X1, X2)
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
0 → n__0
zeros → n__zeros
isNat(n__0) → tt
isNat(n__s(V1)) → isNat(activate(V1))
zeros → cons(0, n__zeros)
activate(X) → X
U11(tt, L) → s(length(activate(L)))
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
activate(n__isNatIList(X)) → isNatIList(X)
and(tt, X) → activate(X)
activate(n__isNatList(X)) → isNatList(X)
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
activate(n__nil) → nil
activate(n__isNat(X)) → isNat(X)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ RuleRemovalProof

                        ↳ QDP

                          ↳ RuleRemovalProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ AND

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ AND

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Instantiation

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ AND

                                                                          ↳ QDP

                                                                          ↳ QDP

                                                                            ↳ RuleRemovalProof

                                                                              ↳ QDP

                                                                                ↳ RuleRemovalProof

                                                                                  ↳ QDP

                                                                                    ↳ RuleRemovalProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ QDPOrderProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                ↳ QDP

                                  ↳ QDP

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

ISNATILIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatIList(activate(y1)))
AND(tt, n__isNatIList(y_3)) → ACTIVATE(n__isNatIList(y_3))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATILIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatIList(activate(y1)))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

The following pairs can be oriented strictly and are deleted.

ISNATILIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatIList(activate(y1)))

The remaining pairs can at least be oriented weakly.

ISNATILIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatIList(activate(y1)))
AND(tt, n__isNatIList(y_3)) → ACTIVATE(n__isNatIList(y_3))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATILIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatIList(activate(y1)))

Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(AND(x₁, x₂)) = x₂
POL(ISNATILIST(x₁)) = x₁
POL(U11(x₁, x₂)) = 1
POL(activate(x₁)) = x₁
POL(and(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = x₁
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 1 + x₁
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = x₁ + x₂
POL(n__isNat(x₁)) = 0
POL(n__isNatIList(x₁)) = x₁
POL(n__isNatList(x₁)) = 0
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:

isNatList(X) → n__isNatList(X)
nil → n__nil
activate(n__zeros) → zeros
isNat(X) → n__isNat(X)
s(X) → n__s(X)
length(X) → n__length(X)
isNatIList(X) → n__isNatIList(X)
cons(X1, X2) → n__cons(X1, X2)
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
0 → n__0
zeros → n__zeros
isNat(n__0) → tt
isNat(n__s(V1)) → isNat(activate(V1))
zeros → cons(0, n__zeros)
activate(X) → X
U11(tt, L) → s(length(activate(L)))
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
activate(n__isNatIList(X)) → isNatIList(X)
and(tt, X) → activate(X)
activate(n__isNatList(X)) → isNatList(X)
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
activate(n__nil) → nil
activate(n__isNat(X)) → isNat(X)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ RuleRemovalProof

                        ↳ QDP

                          ↳ RuleRemovalProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ AND

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ AND

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Instantiation

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ AND

                                                                          ↳ QDP

                                                                          ↳ QDP

                                                                            ↳ RuleRemovalProof

                                                                              ↳ QDP

                                                                                ↳ RuleRemovalProof

                                                                                  ↳ QDP

                                                                                    ↳ RuleRemovalProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ QDPOrderProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                ↳ QDP

                                  ↳ QDP

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

ISNATILIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatIList(activate(y1)))
AND(tt, n__isNatIList(y_3)) → ACTIVATE(n__isNatIList(y_3))
ISNATILIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatIList(activate(y1)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatIList(activate(y1)))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

The following pairs can be oriented strictly and are deleted.

ISNATILIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatIList(activate(y1)))

The remaining pairs can at least be oriented weakly.

ISNATILIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatIList(activate(y1)))
AND(tt, n__isNatIList(y_3)) → ACTIVATE(n__isNatIList(y_3))
ISNATILIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatIList(activate(y1)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)

Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(AND(x₁, x₂)) = x₂
POL(ISNATILIST(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(activate(x₁)) = 1 + x₁
POL(and(x₁, x₂)) = 1 + x₂
POL(cons(x₁, x₂)) = 1 + x₁ + x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 1 + x₁
POL(isNatList(x₁)) = 1 + x₁
POL(length(x₁)) = 0
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = 1 + x₁ + x₂
POL(n__isNat(x₁)) = 0
POL(n__isNatIList(x₁)) = x₁
POL(n__isNatList(x₁)) = x₁
POL(n__length(x₁)) = 0
POL(n__nil) = 0
POL(n__s(x₁)) = 0
POL(n__zeros) = 0
POL(nil) = 1
POL(s(x₁)) = 0
POL(tt) = 0
POL(zeros) = 1

The following usable rules [17] were oriented:

isNatList(X) → n__isNatList(X)
nil → n__nil
activate(n__zeros) → zeros
isNat(X) → n__isNat(X)
s(X) → n__s(X)
length(X) → n__length(X)
isNatIList(X) → n__isNatIList(X)
cons(X1, X2) → n__cons(X1, X2)
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
0 → n__0
zeros → n__zeros
isNat(n__0) → tt
isNat(n__s(V1)) → isNat(activate(V1))
zeros → cons(0, n__zeros)
activate(X) → X
U11(tt, L) → s(length(activate(L)))
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
activate(n__isNatIList(X)) → isNatIList(X)
and(tt, X) → activate(X)
activate(n__isNatList(X)) → isNatList(X)
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
activate(n__nil) → nil
activate(n__isNat(X)) → isNat(X)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ RuleRemovalProof

                        ↳ QDP

                          ↳ RuleRemovalProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ AND

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ AND

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Instantiation

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ AND

                                                                          ↳ QDP

                                                                          ↳ QDP

                                                                            ↳ RuleRemovalProof

                                                                              ↳ QDP

                                                                                ↳ RuleRemovalProof

                                                                                  ↳ QDP

                                                                                    ↳ RuleRemovalProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ QDPOrderProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ NonTerminationProof

                                                ↳ QDP

                                  ↳ QDP

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

ISNATILIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatIList(activate(y1)))
AND(tt, n__isNatIList(y_3)) → ACTIVATE(n__isNatIList(y_3))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATILIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatIList(activate(y1)))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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:

ISNATILIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatIList(activate(y1)))
AND(tt, n__isNatIList(y_3)) → ACTIVATE(n__isNatIList(y_3))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATILIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatIList(activate(y1)))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

s = AND(isNat(0), n__isNatIList(activate(n__zeros))) evaluates to t =AND(isNat(0), n__isNatIList(activate(n__zeros)))

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

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

AND(isNat(0), n__isNatIList(activate(n__zeros))) → AND(isNat(0), n__isNatIList(zeros))
with rule activate(n__zeros) → zeros at position [1,0] and matcher [ ]

AND(isNat(0), n__isNatIList(zeros)) → AND(isNat(0), n__isNatIList(cons(0, n__zeros)))
with rule zeros → cons(0, n__zeros) at position [1,0] and matcher [ ]

AND(isNat(0), n__isNatIList(cons(0, n__zeros))) → AND(isNat(0), n__isNatIList(n__cons(0, n__zeros)))
with rule cons(X1, X2) → n__cons(X1, X2) at position [1,0] and matcher [X2 / n__zeros, X1 / 0]

AND(isNat(0), n__isNatIList(n__cons(0, n__zeros))) → AND(isNat(n__0), n__isNatIList(n__cons(0, n__zeros)))
with rule 0 → n__0 at position [0,0] and matcher [ ]

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

AND(tt, n__isNatIList(n__cons(0, n__zeros))) → ACTIVATE(n__isNatIList(n__cons(0, n__zeros)))
with rule AND(tt, n__isNatIList(y_3)) → ACTIVATE(n__isNatIList(y_3)) at position [] and matcher [y_3 / n__cons(0, n__zeros)]

ACTIVATE(n__isNatIList(n__cons(0, n__zeros))) → ISNATILIST(n__cons(0, n__zeros))
with rule ACTIVATE(n__isNatIList(X)) → ISNATILIST(X) at position [] and matcher [X / n__cons(0, n__zeros)]

ISNATILIST(n__cons(0, n__zeros)) → AND(isNat(0), n__isNatIList(activate(n__zeros)))
with rule ISNATILIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatIList(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

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ RuleRemovalProof

                        ↳ QDP

                          ↳ RuleRemovalProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ AND

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ AND

                                                ↳ QDP

                                                ↳ QDP

                                  ↳ QDP

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

ISNAT(n__s(V1)) → ISNAT(activate(V1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ RuleRemovalProof

                        ↳ QDP

                          ↳ RuleRemovalProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ AND

                                  ↳ QDP

                                  ↳ QDP

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

U11¹(tt, L) → LENGTH(activate(L))
LENGTH(cons(N, L)) → U11¹(and(isNatList(activate(L)), n__isNat(N)), activate(L))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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