Termination w.r.t. Q proof of Transformed_CSR_04_LengthOfFiniteLists_nokinds

Termination w.r.t. Q of the given QTRS could be disproven:

0 QTRS

↳1 QTRSRRRProof (⇔, 78 ms)

↳2 QTRS

↳3 QTRSRRRProof (⇔, 0 ms)

↳4 QTRS

↳5 DependencyPairsProof (⇔, 0 ms)

↳6 QDP

↳7 DependencyGraphProof (⇔, 0 ms)

↳8 QDP

↳9 MRRProof (⇔, 10 ms)

↳10 QDP

↳11 DependencyGraphProof (⇔, 0 ms)

↳12 AND

    ↳13 QDP

        ↳14 QDPOrderProof (⇔, 38 ms)

        ↳15 QDP

        ↳16 DependencyGraphProof (⇔, 0 ms)

        ↳17 TRUE

    ↳18 QDP

        ↳19 QDPOrderProof (⇔, 33 ms)

        ↳20 QDP

        ↳21 QDPOrderProof (⇔, 86 ms)

        ↳22 QDP

        ↳23 DependencyGraphProof (⇔, 0 ms)

        ↳24 QDP

        ↳25 QDPOrderProof (⇔, 22 ms)

        ↳26 QDP

        ↳27 TransformationProof (⇔, 0 ms)

        ↳28 QDP

        ↳29 DependencyGraphProof (⇔, 0 ms)

        ↳30 QDP

        ↳31 TransformationProof (⇔, 0 ms)

        ↳32 QDP

        ↳33 DependencyGraphProof (⇔, 0 ms)

        ↳34 QDP

        ↳35 TransformationProof (⇔, 0 ms)

        ↳36 QDP

        ↳37 DependencyGraphProof (⇔, 0 ms)

        ↳38 AND

            ↳39 QDP

                ↳40 MRRProof (⇔, 20 ms)

                ↳41 QDP

                ↳42 MRRProof (⇔, 25 ms)

                ↳43 QDP

                ↳44 QDPOrderProof (⇔, 44 ms)

                ↳45 QDP

                ↳46 QDPOrderProof (⇔, 68 ms)

                ↳47 QDP

                ↳48 NonTerminationLoopProof (⇒, 0 ms)

                ↳49 NO

            ↳50 QDP

                ↳51 MRRProof (⇔, 16 ms)

                ↳52 QDP

                ↳53 MRRProof (⇔, 29 ms)

                ↳54 QDP

                ↳55 QDPOrderProof (⇔, 59 ms)

                ↳56 QDP

                ↳57 QDPOrderProof (⇔, 14 ms)

                ↳58 QDP

(0) Obligation:

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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(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.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

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)) = 2·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)) = 2·x₁
POL(n__isNatList(x₁)) = x₁
POL(n__length(x₁)) = 1 + 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

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

isNat(n__length(V1)) → isNatList(activate(V1))
isNatList(n__nil) → tt
length(nil) → 0

(2) Obligation:

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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(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.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(U11(x₁, x₂)) = 1 + 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)) = 2 + 2·x₁
POL(isNatList(x₁)) = 2·x₁
POL(length(x₁)) = 1 + 2·x₁
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = x₁ + 2·x₂
POL(n__isNat(x₁)) = x₁
POL(n__isNatIList(x₁)) = 2 + 2·x₁
POL(n__isNatList(x₁)) = 2·x₁
POL(n__length(x₁)) = 1 + 2·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

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt

(4) Obligation:

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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(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.

(5) DependencyPairsProof (EQUIVALENT transformation)

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

(6) Obligation:

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

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

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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(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.

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 8 less nodes.

(8) Obligation:

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

ACTIVATE(n__length(X)) → LENGTH(activate(X))
LENGTH(cons(N, L)) → U11¹(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U11¹(tt, L) → LENGTH(activate(L))
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNat(N))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__length(X)) → ACTIVATE(X)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
LENGTH(cons(N, L)) → ACTIVATE(L)
U11¹(tt, L) → 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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(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.

(9) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

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

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(ACTIVATE(x₁)) = 1 + x₁
POL(AND(x₁, x₂)) = 1 + x₁ + x₂
POL(ISNAT(x₁)) = 1 + x₁
POL(ISNATILIST(x₁)) = 2 + x₁
POL(ISNATLIST(x₁)) = 1 + x₁
POL(LENGTH(x₁)) = 2 + 2·x₁
POL(U11(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(U11¹(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)) = 2·x₁
POL(isNatIList(x₁)) = 1 + x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 1 + 2·x₁
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(n__isNat(x₁)) = 2·x₁
POL(n__isNatIList(x₁)) = 1 + 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

(10) Obligation:

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

ACTIVATE(n__length(X)) → LENGTH(activate(X))
LENGTH(cons(N, L)) → U11¹(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U11¹(tt, L) → LENGTH(activate(L))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ISNATLIST(n__cons(V1, V2)) → 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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(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.

(11) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.

(12) Complex Obligation (AND)

(13) Obligation:

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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(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.

(14) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( LENGTH(x₁) ) = 2x₁ + 2

POL( U11¹(x₁, x₂) ) = x₁ + 2x₂ + 1

POL( activate(x₁) ) = x₁

POL( n__zeros ) = 2

POL( zeros ) = 2

POL( n__0 ) = 1

POL( 0 ) = 1

POL( n__length(x₁) ) = 2

POL( length(x₁) ) = 2

POL( n__s(x₁) ) = 0

POL( s(x₁) ) = max{0, -1}

POL( n__cons(x₁, x₂) ) = x₂

POL( cons(x₁, x₂) ) = x₂

POL( n__isNatIList(x₁) ) = 2

POL( isNatIList(x₁) ) = 2

POL( and(x₁, x₂) ) = max{0, 2x₁ + x₂ - 2}

POL( isNat(x₁) ) = 1

POL( tt ) = 1

POL( n__isNatList(x₁) ) = max{0, -2}

POL( isNatList(x₁) ) = max{0, -2}

POL( n__nil ) = 0

POL( nil ) = 0

POL( n__isNat(x₁) ) = 1

POL( U11(x₁, x₂) ) = max{0, x₁ - 2}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

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

(15) Obligation:

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

U11¹(tt, L) → LENGTH(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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(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.

(16) DependencyGraphProof (EQUIVALENT transformation)

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

(17) TRUE

(18) Obligation:

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

ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → 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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(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.

(19) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

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.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( AND(x₁, x₂) ) = max{0, 2x₁ + 2x₂ - 2}

POL( ISNAT(x₁) ) = 2x₁ + 2

POL( n__isNatIList(x₁) ) = x₁

POL( n__isNatList(x₁) ) = x₁

POL( activate(x₁) ) = x₁

POL( n__zeros ) = 2

POL( zeros ) = 2

POL( n__0 ) = 0

POL( 0 ) = 0

POL( n__length(x₁) ) = 0

POL( length(x₁) ) = max{0, -2}

POL( n__s(x₁) ) = 2x₁

POL( s(x₁) ) = 2x₁

POL( n__cons(x₁, x₂) ) = x₁ + x₂

POL( cons(x₁, x₂) ) = x₁ + x₂

POL( isNatIList(x₁) ) = x₁

POL( and(x₁, x₂) ) = max{0, x₁ + x₂ - 2}

POL( isNat(x₁) ) = x₁ + 2

POL( tt ) = 2

POL( isNatList(x₁) ) = x₁

POL( n__nil ) = 0

POL( nil ) = 0

POL( n__isNat(x₁) ) = x₁ + 2

POL( U11(x₁, x₂) ) = max{0, -2}

POL( ACTIVATE(x₁) ) = 2x₁ + 2

POL( ISNATILIST(x₁) ) = 2x₁ + 2

POL( ISNATLIST(x₁) ) = 2x₁ + 2

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

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

(20) Obligation:

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

ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → 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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(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.

(21) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( AND(x₁, x₂) ) = 2x₂ + 2

POL( ISNAT(x₁) ) = max{0, 2x₁ - 2}

POL( n__isNatIList(x₁) ) = 0

POL( n__isNatList(x₁) ) = x₁

POL( activate(x₁) ) = x₁

POL( n__zeros ) = 0

POL( zeros ) = 0

POL( n__0 ) = 0

POL( 0 ) = 0

POL( n__length(x₁) ) = 2x₁

POL( length(x₁) ) = 2x₁

POL( n__s(x₁) ) = x₁ + 2

POL( s(x₁) ) = x₁ + 2

POL( n__cons(x₁, x₂) ) = 2x₁ + 2x₂

POL( cons(x₁, x₂) ) = 2x₁ + 2x₂

POL( isNatIList(x₁) ) = 0

POL( and(x₁, x₂) ) = max{0, x₁ + x₂ - 1}

POL( isNat(x₁) ) = 1

POL( tt ) = 1

POL( isNatList(x₁) ) = x₁

POL( n__nil ) = 0

POL( nil ) = 0

POL( n__isNat(x₁) ) = 1

POL( U11(x₁, x₂) ) = 2x₁ + 2x₂

POL( ACTIVATE(x₁) ) = 2x₁ + 2

POL( ISNATILIST(x₁) ) = 2

POL( ISNATLIST(x₁) ) = x₁ + 2

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

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

(22) Obligation:

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

ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → 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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(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.

(23) DependencyGraphProof (EQUIVALENT transformation)

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

(24) Obligation:

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

ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → 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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(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.

(25) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( AND(x₁, x₂) ) = 2x₁ + x₂

POL( n__isNatIList(x₁) ) = 0

POL( n__isNatList(x₁) ) = max{0, 2x₁ - 1}

POL( activate(x₁) ) = x₁ + 2

POL( n__zeros ) = 0

POL( zeros ) = 2

POL( n__0 ) = 0

POL( 0 ) = 0

POL( n__length(x₁) ) = x₁ + 2

POL( length(x₁) ) = x₁ + 2

POL( n__s(x₁) ) = 2

POL( s(x₁) ) = 2

POL( n__cons(x₁, x₂) ) = x₁ + 2x₂ + 2

POL( cons(x₁, x₂) ) = x₁ + 2x₂ + 2

POL( isNatIList(x₁) ) = 2

POL( and(x₁, x₂) ) = x₂ + 2

POL( isNat(x₁) ) = 1

POL( tt ) = 1

POL( isNatList(x₁) ) = 2x₁ + 1

POL( n__nil ) = 0

POL( nil ) = 2

POL( n__isNat(x₁) ) = 1

POL( U11(x₁, x₂) ) = 2

POL( ACTIVATE(x₁) ) = x₁ + 2

POL( ISNATILIST(x₁) ) = 2

POL( ISNATLIST(x₁) ) = 2x₁ + 1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

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

(26) Obligation:

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

ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(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.

(27) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2))) at position [0] we obtained the following new rules [LPAR04]:

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

(28) Obligation:

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

ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
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__0, y1)) → AND(isNat(0), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__length(x0), y1)) → AND(isNat(length(activate(x0))), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__s(x0), y1)) → AND(isNat(s(activate(x0))), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(activate(x0), x1)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(x0, y1)) → AND(isNat(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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(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.

(29) DependencyGraphProof (EQUIVALENT transformation)

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

(30) Obligation:

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

ISNATILIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatIList(activate(y1)))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATILIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__length(x0), y1)) → AND(isNat(length(activate(x0))), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__s(x0), y1)) → AND(isNat(s(activate(x0))), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(activate(x0), x1)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(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(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(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.

(31) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2))) at position [0] we obtained the following new rules [LPAR04]:

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

(32) Obligation:

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

ISNATILIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatIList(activate(y1)))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATILIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__length(x0), y1)) → AND(isNat(length(activate(x0))), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__s(x0), y1)) → AND(isNat(s(activate(x0))), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(activate(x0), x1)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(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(y0, y1)) → AND(n__isNat(activate(y0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__length(x0), y1)) → AND(isNat(length(activate(x0))), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(activate(x0))), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(activate(x0), x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(x0, y1)) → AND(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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(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.

(33) DependencyGraphProof (EQUIVALENT transformation)

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

(34) Obligation:

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

AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATILIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__length(x0), y1)) → AND(isNat(length(activate(x0))), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__s(x0), y1)) → AND(isNat(s(activate(x0))), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(activate(x0), x1)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(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__zeros, y1)) → AND(isNat(zeros), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__length(x0), y1)) → AND(isNat(length(activate(x0))), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(activate(x0))), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(activate(x0), x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(x0, y1)) → AND(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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(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.

(35) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule AND(tt, X) → ACTIVATE(X) we obtained the following new rules [LPAR04]:

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

(36) Obligation:

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

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

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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(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.

(37) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.

(38) Complex Obligation (AND)

(39) Obligation:

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

ISNATLIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatList(activate(y1)))
AND(tt, n__isNatList(y_3)) → ACTIVATE(n__isNatList(y_3))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__length(x0), y1)) → AND(isNat(length(activate(x0))), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(activate(x0))), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(activate(x0), x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(x0, y1)) → AND(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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(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.

(40) MRRProof (EQUIVALENT transformation)

ISNATLIST(n__cons(n__length(x0), y1)) → AND(isNat(length(activate(x0))), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatList(activate(y1)))

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(ACTIVATE(x₁)) = 1 + 2·x₁
POL(AND(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(ISNATLIST(x₁)) = 1 + 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)) = x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 2 + 2·x₁
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = x₁ + 2·x₂
POL(n__isNat(x₁)) = x₁
POL(n__isNatIList(x₁)) = x₁
POL(n__isNatList(x₁)) = x₁
POL(n__length(x₁)) = 2 + 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

(41) Obligation:

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

ISNATLIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatList(activate(y1)))
AND(tt, n__isNatList(y_3)) → ACTIVATE(n__isNatList(y_3))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(activate(x0))), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(activate(x0), x1)), n__isNatList(activate(y1)))
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(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(x0, y1)) → AND(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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(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.

(42) MRRProof (EQUIVALENT transformation)

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

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(ACTIVATE(x₁)) = 1 + 2·x₁
POL(AND(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(ISNATLIST(x₁)) = 1 + 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) = 2
POL(n__s(x₁)) = x₁
POL(n__zeros) = 0
POL(nil) = 2
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

(43) Obligation:

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

ISNATLIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatList(activate(y1)))
AND(tt, n__isNatList(y_3)) → ACTIVATE(n__isNatList(y_3))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(activate(x0))), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(activate(x0), x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(x0, y1)) → AND(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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(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.

(44) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

ISNATLIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(activate(x0))), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( AND(x₁, x₂) ) = max{0, x₁ + x₂ - 1}

POL( n__isNatList(x₁) ) = 2x₁ + 1

POL( zeros ) = 1

POL( cons(x₁, x₂) ) = x₁ + x₂

POL( 0 ) = 0

POL( n__zeros ) = 1

POL( isNat(x₁) ) = 0

POL( n__0 ) = 0

POL( tt ) = 0

POL( n__s(x₁) ) = 2

POL( activate(x₁) ) = x₁

POL( n__isNat(x₁) ) = 0

POL( n__length(x₁) ) = 2

POL( length(x₁) ) = 2

POL( s(x₁) ) = 2

POL( n__cons(x₁, x₂) ) = x₁ + x₂

POL( n__isNatIList(x₁) ) = 2x₁ + 2

POL( isNatIList(x₁) ) = 2x₁ + 2

POL( and(x₁, x₂) ) = x₂

POL( isNatList(x₁) ) = 2x₁ + 1

POL( n__nil ) = 0

POL( nil ) = 0

POL( U11(x₁, x₂) ) = 2

POL( ISNATLIST(x₁) ) = 2x₁

POL( ACTIVATE(x₁) ) = max{0, x₁ - 1}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

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

(45) Obligation:

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

AND(tt, n__isNatList(y_3)) → ACTIVATE(n__isNatList(y_3))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(activate(x0), x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(x0, y1)) → AND(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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(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.

(46) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(activate(x0), x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatList(activate(y1)))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( AND(x₁, x₂) ) = x₂ + 2

POL( n__isNatList(x₁) ) = 2x₁

POL( 0 ) = 0

POL( n__0 ) = 0

POL( isNat(x₁) ) = 2

POL( tt ) = 0

POL( n__s(x₁) ) = 0

POL( activate(x₁) ) = x₁ + 2

POL( n__isNat(x₁) ) = 2

POL( n__zeros ) = 0

POL( zeros ) = 2

POL( n__length(x₁) ) = 2

POL( length(x₁) ) = 2

POL( s(x₁) ) = max{0, -2}

POL( n__cons(x₁, x₂) ) = x₁ + 2x₂ + 2

POL( cons(x₁, x₂) ) = x₁ + 2x₂ + 2

POL( n__isNatIList(x₁) ) = 0

POL( isNatIList(x₁) ) = 2

POL( and(x₁, x₂) ) = x₂ + 2

POL( isNatList(x₁) ) = 2x₁ + 2

POL( n__nil ) = 0

POL( nil ) = 0

POL( U11(x₁, x₂) ) = max{0, x₁ - 2}

POL( ACTIVATE(x₁) ) = x₁ + 2

POL( ISNATLIST(x₁) ) = 2x₁ + 2

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

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

(47) Obligation:

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

AND(tt, n__isNatList(y_3)) → ACTIVATE(n__isNatList(y_3))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(x0, y1)) → AND(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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(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.

(48) NonTerminationLoopProof (COMPLETE transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = ACTIVATE(n__isNatList(activate(n__zeros))) evaluates to t =ACTIVATE(n__isNatList(activate(n__zeros)))

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

Matcher: [ ]
Semiunifier: [ ]

Rewriting sequence

ACTIVATE(n__isNatList(activate(n__zeros))) → ACTIVATE(n__isNatList(zeros))
with rule activate(n__zeros) → zeros at position [0,0] and matcher [ ]

ACTIVATE(n__isNatList(zeros)) → ACTIVATE(n__isNatList(cons(0, n__zeros)))
with rule zeros → cons(0, n__zeros) at position [0,0] and matcher [ ]

ACTIVATE(n__isNatList(cons(0, n__zeros))) → ACTIVATE(n__isNatList(cons(n__0, n__zeros)))
with rule 0 → n__0 at position [0,0,0] and matcher [ ]

ACTIVATE(n__isNatList(cons(n__0, n__zeros))) → ACTIVATE(n__isNatList(n__cons(n__0, n__zeros)))
with rule cons(X1, X2) → n__cons(X1, X2) at position [0,0] and matcher [X1 / n__0, X2 / n__zeros]

ACTIVATE(n__isNatList(n__cons(n__0, n__zeros))) → ISNATLIST(n__cons(n__0, n__zeros))
with rule ACTIVATE(n__isNatList(X)) → ISNATLIST(X) at position [] and matcher [X / n__cons(n__0, n__zeros)]

ISNATLIST(n__cons(n__0, n__zeros)) → AND(isNat(n__0), n__isNatList(activate(n__zeros)))
with rule ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1))) at position [] and matcher [x0 / n__0, y1 / n__zeros]

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

AND(tt, n__isNatList(activate(n__zeros))) → ACTIVATE(n__isNatList(activate(n__zeros)))
with rule AND(tt, n__isNatList(y_3)) → ACTIVATE(n__isNatList(y_3))

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.

(49) NO

(50) Obligation:

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

ISNATILIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), 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__length(x0), y1)) → AND(isNat(length(activate(x0))), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__s(x0), y1)) → AND(isNat(s(activate(x0))), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(activate(x0), x1)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(x0, y1)) → AND(isNat(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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(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.

(51) MRRProof (EQUIVALENT transformation)

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

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(ACTIVATE(x₁)) = 2·x₁
POL(AND(x₁, x₂)) = x₁ + 2·x₂
POL(ISNATILIST(x₁)) = 2 + 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)) = 2·x₁
POL(isNatIList(x₁)) = 1 + x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = x₁
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(n__isNat(x₁)) = 2·x₁
POL(n__isNatIList(x₁)) = 1 + 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

(52) Obligation:

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

ISNATILIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), 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__length(x0), y1)) → AND(isNat(length(activate(x0))), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__s(x0), y1)) → AND(isNat(s(activate(x0))), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(activate(x0), x1)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(x0, y1)) → AND(isNat(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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(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.

(53) MRRProof (EQUIVALENT transformation)

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

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(AND(x₁, x₂)) = 2·x₁ + x₂
POL(ISNATILIST(x₁)) = 2 + 2·x₁
POL(U11(x₁, x₂)) = 1 + 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)) = 2 + 2·x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 1 + x₁
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(n__isNat(x₁)) = x₁
POL(n__isNatIList(x₁)) = 2 + 2·x₁
POL(n__isNatList(x₁)) = x₁
POL(n__length(x₁)) = 1 + 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

(54) Obligation:

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

ISNATILIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), 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(activate(x0))), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(activate(x0), x1)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(x0, y1)) → AND(isNat(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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(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.

(55) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

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__cons(x0, x1), y1)) → AND(isNat(cons(activate(x0), x1)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatIList(activate(y1)))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( AND(x₁, x₂) ) = max{0, x₁ + x₂ - 1}

POL( n__isNatIList(x₁) ) = x₁ + 1

POL( zeros ) = 2

POL( cons(x₁, x₂) ) = x₁ + x₂ + 1

POL( 0 ) = 0

POL( n__zeros ) = 1

POL( isNat(x₁) ) = 2

POL( n__0 ) = 0

POL( tt ) = 2

POL( n__s(x₁) ) = 0

POL( activate(x₁) ) = x₁ + 1

POL( n__isNat(x₁) ) = 2

POL( n__length(x₁) ) = x₁ + 1

POL( length(x₁) ) = x₁ + 1

POL( s(x₁) ) = 0

POL( n__cons(x₁, x₂) ) = x₁ + x₂ + 1

POL( isNatIList(x₁) ) = x₁ + 2

POL( and(x₁, x₂) ) = max{0, x₁ + x₂ - 1}

POL( n__isNatList(x₁) ) = x₁ + 1

POL( isNatList(x₁) ) = x₁ + 2

POL( n__nil ) = 0

POL( nil ) = 0

POL( U11(x₁, x₂) ) = max{0, -2}

POL( ISNATILIST(x₁) ) = x₁ + 2

POL( ACTIVATE(x₁) ) = x₁ + 1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

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

(56) Obligation:

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

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(activate(x0))), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(x0, y1)) → AND(isNat(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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(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.

(57) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( AND(x₁, x₂) ) = max{0, x₁ + x₂ - 1}

POL( n__isNatIList(x₁) ) = 2x₁ + 1

POL( 0 ) = 0

POL( n__0 ) = 0

POL( isNat(x₁) ) = 1

POL( tt ) = 1

POL( n__s(x₁) ) = 2

POL( activate(x₁) ) = x₁

POL( n__isNat(x₁) ) = 1

POL( n__zeros ) = 2

POL( zeros ) = 2

POL( n__length(x₁) ) = x₁ + 2

POL( length(x₁) ) = x₁ + 2

POL( s(x₁) ) = 2

POL( n__cons(x₁, x₂) ) = x₁ + x₂

POL( cons(x₁, x₂) ) = x₁ + x₂

POL( isNatIList(x₁) ) = 2x₁ + 1

POL( and(x₁, x₂) ) = max{0, x₁ + x₂ - 1}

POL( n__isNatList(x₁) ) = 2

POL( isNatList(x₁) ) = 2

POL( n__nil ) = 0

POL( nil ) = 0

POL( U11(x₁, x₂) ) = 2

POL( ACTIVATE(x₁) ) = x₁

POL( ISNATILIST(x₁) ) = 2x₁ + 1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

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

(58) Obligation:

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

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(x0, y1)) → AND(isNat(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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(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.