Termination w.r.t. Q proof of /tmp/tmp_5Nt6Z/LengthOfFiniteLists_nokinds

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

0 QTRS

↳1 QTRSRRRProof (⇔)

↳2 QTRS

↳3 QTRSRRRProof (⇔)

↳4 QTRS

↳5 QTRSRRRProof (⇔)

↳6 QTRS

↳7 DependencyPairsProof (⇔)

↳8 QDP

↳9 DependencyGraphProof (⇔)

↳10 QDP

↳11 MRRProof (⇔)

↳12 QDP

↳13 DependencyGraphProof (⇔)

↳14 AND

    ↳15 QDP

        ↳16 MRRProof (⇔)

        ↳17 QDP

        ↳18 QDPOrderProof (⇔)

        ↳19 QDP

        ↳20 QDPOrderProof (⇔)

        ↳21 QDP

        ↳22 DependencyGraphProof (⇔)

        ↳23 AND

            ↳24 QDP

                ↳25 QDPOrderProof (⇔)

                ↳26 QDP

                ↳27 PisEmptyProof (⇔)

                ↳28 YES

            ↳29 QDP

                ↳30 Narrowing (⇔)

                ↳31 QDP

                ↳32 DependencyGraphProof (⇔)

                ↳33 QDP

                ↳34 Narrowing (⇔)

                ↳35 QDP

                ↳36 DependencyGraphProof (⇔)

                ↳37 QDP

                ↳38 Instantiation (⇔)

                ↳39 QDP

                ↳40 DependencyGraphProof (⇔)

                ↳41 AND

                    ↳42 QDP

                        ↳43 MRRProof (⇔)

                        ↳44 QDP

                        ↳45 MRRProof (⇔)

                        ↳46 QDP

                        ↳47 MRRProof (⇔)

                        ↳48 QDP

                        ↳49 QDPOrderProof (⇔)

                        ↳50 QDP

                        ↳51 QDPOrderProof (⇔)

                        ↳52 QDP

                        ↳53 QDPOrderProof (⇔)

                        ↳54 QDP

                        ↳55 QDPOrderProof (⇔)

                        ↳56 QDP

                        ↳57 NonTerminationProof (⇔)

                        ↳58 NO

                    ↳59 QDP

                        ↳60 MRRProof (⇔)

                        ↳61 QDP

                        ↳62 MRRProof (⇔)

                        ↳63 QDP

                        ↳64 MRRProof (⇔)

                        ↳65 QDP

                        ↳66 QDPOrderProof (⇔)

                        ↳67 QDP

                        ↳68 QDPOrderProof (⇔)

                        ↳69 QDP

                        ↳70 QDPOrderProof (⇔)

                        ↳71 QDP

                        ↳72 QDPOrderProof (⇔)

                        ↳73 QDP

    ↳74 QDP

        ↳75 QDPOrderProof (⇔)

        ↳76 QDP

        ↳77 DependencyGraphProof (⇔)

        ↳78 TRUE

(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(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.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

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

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

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__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.

(3) 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)) = 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

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))

(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(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.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

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

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

(6) 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(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.

(7) DependencyPairsProof (EQUIVALENT transformation)

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

(8) 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(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)

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.

(9) DependencyGraphProof (EQUIVALENT transformation)

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

(10) Obligation:

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

ACTIVATE(n__length(X)) → LENGTH(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__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(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.

(11) 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:

ACTIVATE(n__length(X)) → LENGTH(X)

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(AND(x₁, x₂)) = x₁ + 2·x₂
POL(ISNAT(x₁)) = x₁
POL(ISNATILIST(x₁)) = 2·x₁
POL(ISNATLIST(x₁)) = x₁
POL(LENGTH(x₁)) = 2·x₁
POL(U11(x₁, x₂)) = 2 + 2·x₁ + 2·x₂
POL(U11¹(x₁, x₂)) = 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)) = 2·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)) = 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

(12) Obligation:

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

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__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(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.

(13) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 4 less nodes.

(14) Complex Obligation (AND)

(15) 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)))
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)
ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(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(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.

(16) MRRProof (EQUIVALENT transformation)

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

Used ordering: Polynomial interpretation [POLO]:

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

(17) 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)))
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(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.

(18) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

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) ) = 2x₂

POL( ISNAT(x₁) ) = x₁

POL( n__isNatIList(x₁) ) = x₁ + 1

POL( n__isNatList(x₁) ) = x₁

POL( activate(x₁) ) = 2x₁

POL( n__zeros ) = 0

POL( zeros ) = 0

POL( n__0 ) = 0

POL( 0 ) = 0

POL( n__length(x₁) ) = max{0, -1}

POL( length(x₁) ) = max{0, -2}

POL( n__s(x₁) ) = 2x₁

POL( s(x₁) ) = 2x₁

POL( n__cons(x₁, x₂) ) = x₁ + 2x₂

POL( cons(x₁, x₂) ) = 2x₁ + 2x₂

POL( isNatIList(x₁) ) = 2x₁ + 2

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

POL( isNat(x₁) ) = x₁ + 2

POL( tt ) = 2

POL( isNatList(x₁) ) = 2x₁

POL( n__nil ) = 2

POL( nil ) = 2

POL( n__isNat(x₁) ) = x₁ + 1

POL( U11(x₁, x₂) ) = max{0, -1}

POL( ACTIVATE(x₁) ) = 2x₁

POL( ISNATILIST(x₁) ) = 2x₁ + 2

POL( ISNATLIST(x₁) ) = 2x₁

The following usable rules [FROCOS05] were oriented:

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

(19) 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)))
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(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.

(20) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
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) ) = max{0, 2x₁ + 2x₂ - 2}

POL( ISNAT(x₁) ) = x₁ + 1

POL( n__isNatIList(x₁) ) = 1

POL( n__isNatList(x₁) ) = x₁ + 1

POL( activate(x₁) ) = x₁

POL( n__zeros ) = 0

POL( zeros ) = 0

POL( n__0 ) = 0

POL( 0 ) = 0

POL( n__length(x₁) ) = max{0, -2}

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) ) = 1

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

POL( isNat(x₁) ) = 1

POL( tt ) = 1

POL( isNatList(x₁) ) = x₁ + 1

POL( n__nil ) = 2

POL( nil ) = 2

POL( n__isNat(x₁) ) = 1

POL( U11(x₁, x₂) ) = max{0, -2}

POL( ACTIVATE(x₁) ) = 2x₁

POL( ISNATILIST(x₁) ) = 2

POL( ISNATLIST(x₁) ) = 2x₁ + 2

The following usable rules [FROCOS05] were oriented:

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

(21) 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)))
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.

(22) DependencyGraphProof (EQUIVALENT transformation)

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

(23) Complex Obligation (AND)

(24) Obligation:

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.

(25) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNAT(n__s(V1)) → ISNAT(activate(V1))

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

POL( ISNAT(x₁) ) = max{0, 2x₁ - 1}

POL( activate(x₁) ) = 2x₁

POL( n__zeros ) = 0

POL( zeros ) = 0

POL( n__0 ) = 1

POL( 0 ) = 2

POL( n__length(x₁) ) = max{0, -2}

POL( length(x₁) ) = max{0, -2}

POL( n__s(x₁) ) = 2x₁ + 1

POL( s(x₁) ) = 2x₁ + 1

POL( n__cons(x₁, x₂) ) = 0

POL( cons(x₁, x₂) ) = max{0, -2}

POL( n__isNatIList(x₁) ) = max{0, -2}

POL( isNatIList(x₁) ) = max{0, -2}

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

POL( isNat(x₁) ) = 1

POL( tt ) = 1

POL( n__isNatList(x₁) ) = max{0, -2}

POL( isNatList(x₁) ) = max{0, -1}

POL( n__nil ) = 1

POL( nil ) = 2

POL( n__isNat(x₁) ) = 1

POL( U11(x₁, x₂) ) = max{0, 2x₁ - 1}

The following usable rules [FROCOS05] were oriented:

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

(26) Obligation:

Q DP problem:
P is empty.
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.

(27) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(28) YES

(29) Obligation:

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

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

(30) Narrowing (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(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(x0)), 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)))
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)))

(31) Obligation:

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

AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__isNatIList(X)) → ISNATILIST(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(x0)), 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)))
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(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.

(32) DependencyGraphProof (EQUIVALENT transformation)

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

(33) 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)))
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(n__0, y1)) → AND(isNat(0), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__length(x0), y1)) → AND(isNat(length(x0)), 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)))
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(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.

(34) Narrowing (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(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(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(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)))

(35) 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)))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATILIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__length(x0), y1)) → AND(isNat(length(x0)), 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)))
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)))
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(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(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(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.

(36) DependencyGraphProof (EQUIVALENT transformation)

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

(37) 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(x0)), 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)))
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(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(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(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.

(38) Instantiation (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__isNatList(y_3)) → ACTIVATE(n__isNatList(y_3))

(39) 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)))
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(x0)), 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)))
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(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(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(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.

(40) DependencyGraphProof (EQUIVALENT transformation)

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

(41) Complex Obligation (AND)

(42) 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(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(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(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.

(43) MRRProof (EQUIVALENT transformation)

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

Used ordering: Polynomial interpretation [POLO]:

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

(44) 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(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(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(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.

(45) 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)) = 2·x₁
POL(AND(x₁, x₂)) = x₁ + 2·x₂
POL(ISNATLIST(x₁)) = 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 + 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 + 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

(46) 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(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(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(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.

(47) MRRProof (EQUIVALENT transformation)

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

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(ACTIVATE(x₁)) = 2 + 2·x₁
POL(AND(x₁, x₂)) = 2 + x₁ + 2·x₂
POL(ISNATLIST(x₁)) = 2 + x₁
POL(U11(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(activate(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + 2·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)) = 1 + 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)) = 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

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

(49) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

POL( zeros ) = 2

POL( cons(x₁, x₂) ) = x₁ + x₂ + 2

POL( 0 ) = 0

POL( n__zeros ) = 0

POL( isNat(x₁) ) = 1

POL( n__0 ) = 0

POL( tt ) = 1

POL( n__s(x₁) ) = max{0, -2}

POL( activate(x₁) ) = x₁ + 2

POL( n__isNat(x₁) ) = max{0, -1}

POL( n__length(x₁) ) = max{0, -1}

POL( length(x₁) ) = max{0, -1}

POL( s(x₁) ) = max{0, -2}

POL( n__cons(x₁, x₂) ) = x₁ + x₂ + 2

POL( n__isNatIList(x₁) ) = max{0, -1}

POL( isNatIList(x₁) ) = 2

POL( and(x₁, x₂) ) = x₂ + 2

POL( isNatList(x₁) ) = 2x₁ + 2

POL( n__nil ) = 0

POL( nil ) = 1

POL( U11(x₁, x₂) ) = max{0, -1}

POL( ISNATLIST(x₁) ) = 2x₁

POL( ACTIVATE(x₁) ) = x₁

The following usable rules [FROCOS05] 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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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)
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
isNatIList(X) → n__isNatIList(X)
U11(tt, L) → s(length(activate(L)))
nil → n__nil

(50) 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(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(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.

(51) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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)))

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

POL( AND(x₁, x₂) ) = max{0, x₂ - 1}

POL( n__isNatList(x₁) ) = 2x₁ + 1

POL( zeros ) = 0

POL( cons(x₁, x₂) ) = 2x₁ + 2x₂

POL( 0 ) = 0

POL( n__zeros ) = 0

POL( isNat(x₁) ) = max{0, -2}

POL( n__0 ) = 0

POL( tt ) = 0

POL( n__s(x₁) ) = 1

POL( activate(x₁) ) = x₁

POL( n__isNat(x₁) ) = 0

POL( n__length(x₁) ) = 2x₁ + 1

POL( length(x₁) ) = 2x₁ + 1

POL( s(x₁) ) = 1

POL( n__cons(x₁, x₂) ) = 2x₁ + 2x₂

POL( n__isNatIList(x₁) ) = 2

POL( isNatIList(x₁) ) = 2

POL( and(x₁, x₂) ) = x₂

POL( isNatList(x₁) ) = 2x₁ + 1

POL( n__nil ) = 0

POL( nil ) = 0

POL( U11(x₁, x₂) ) = x₂ + 1

POL( ISNATLIST(x₁) ) = x₁

POL( ACTIVATE(x₁) ) = max{0, x₁ - 1}

The following usable rules [FROCOS05] 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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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)
isNatList(X) → n__isNatList(X)
length(X) → n__length(X)
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
isNatIList(X) → n__isNatIList(X)
U11(tt, L) → s(length(activate(L)))
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil

(52) 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__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(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.

(53) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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) ) = x₁

POL( zeros ) = 0

POL( cons(x₁, x₂) ) = 2x₁ + 2x₂

POL( 0 ) = 0

POL( n__zeros ) = 0

POL( isNat(x₁) ) = 2

POL( n__0 ) = 0

POL( tt ) = 0

POL( n__s(x₁) ) = max{0, -1}

POL( activate(x₁) ) = 2x₁

POL( n__isNat(x₁) ) = 1

POL( n__length(x₁) ) = 0

POL( length(x₁) ) = max{0, -2}

POL( s(x₁) ) = max{0, -2}

POL( n__cons(x₁, x₂) ) = 2x₁ + 2x₂

POL( n__isNatIList(x₁) ) = x₁ + 1

POL( isNatIList(x₁) ) = 2x₁ + 2

POL( and(x₁, x₂) ) = 2x₂

POL( isNatList(x₁) ) = 2x₁

POL( n__nil ) = 2

POL( nil ) = 2

POL( U11(x₁, x₂) ) = max{0, -2}

POL( ISNATLIST(x₁) ) = x₁ + 2

POL( ACTIVATE(x₁) ) = x₁ + 2

The following usable rules [FROCOS05] 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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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
length(X) → n__length(X)
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
isNatList(X) → n__isNatList(X)
isNatIList(X) → n__isNatIList(X)
U11(tt, L) → s(length(activate(L)))
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil

(54) 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(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(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.

(55) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATLIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), 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, 2x₁ + 2x₂ - 2}

POL( n__isNatList(x₁) ) = x₁

POL( zeros ) = 1

POL( cons(x₁, x₂) ) = x₁ + x₂

POL( 0 ) = 0

POL( n__zeros ) = 1

POL( isNat(x₁) ) = 1

POL( n__0 ) = 0

POL( tt ) = 1

POL( n__s(x₁) ) = max{0, -1}

POL( activate(x₁) ) = x₁

POL( n__isNat(x₁) ) = 1

POL( n__length(x₁) ) = 2

POL( length(x₁) ) = 2

POL( s(x₁) ) = max{0, -2}

POL( n__cons(x₁, x₂) ) = x₁ + x₂

POL( n__isNatIList(x₁) ) = 2

POL( isNatIList(x₁) ) = 2

POL( and(x₁, x₂) ) = x₂

POL( isNatList(x₁) ) = x₁

POL( n__nil ) = 1

POL( nil ) = 1

POL( U11(x₁, x₂) ) = max{0, -2}

POL( ISNATLIST(x₁) ) = 2x₁

POL( ACTIVATE(x₁) ) = 2x₁

The following usable rules [FROCOS05] 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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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
length(X) → n__length(X)
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
isNatList(X) → n__isNatList(X)
isNatIList(X) → n__isNatIList(X)
U11(tt, L) → s(length(activate(L)))
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil

(56) 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(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.

(57) NonTerminationProof (EQUIVALENT 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.

(58) NO

(59) 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__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(x0)), 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)))
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(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.

(60) MRRProof (EQUIVALENT transformation)

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

(61) 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__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(x0)), 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)))
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(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.

(62) MRRProof (EQUIVALENT transformation)

ISNATILIST(n__cons(n__length(x0), y1)) → AND(isNat(length(x0)), 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·x₁
POL(U11(x₁, x₂)) = 1 + 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·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)) = 2·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

(63) 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__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__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)))
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(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.

(64) MRRProof (EQUIVALENT transformation)

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

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(AND(x₁, x₂)) = x₁ + x₂
POL(ISNATILIST(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)) = x₁
POL(isNatIList(x₁)) = 1 + 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)) = 1 + 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

(65) 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__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__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__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(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.

(66) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

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

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

POL( n__isNatIList(x₁) ) = 2x₁

POL( zeros ) = 0

POL( cons(x₁, x₂) ) = 2x₁ + 2x₂

POL( 0 ) = 0

POL( n__zeros ) = 0

POL( isNat(x₁) ) = 1

POL( n__0 ) = 0

POL( tt ) = 1

POL( n__s(x₁) ) = max{0, -2}

POL( activate(x₁) ) = x₁

POL( n__isNat(x₁) ) = 1

POL( n__length(x₁) ) = 2

POL( length(x₁) ) = 2

POL( s(x₁) ) = max{0, -2}

POL( n__cons(x₁, x₂) ) = 2x₁ + 2x₂

POL( isNatIList(x₁) ) = 2x₁

POL( and(x₁, x₂) ) = 2x₂

POL( n__isNatList(x₁) ) = 0

POL( isNatList(x₁) ) = max{0, -1}

POL( n__nil ) = 1

POL( nil ) = 1

POL( U11(x₁, x₂) ) = 2

POL( ACTIVATE(x₁) ) = x₁

POL( ISNATILIST(x₁) ) = 2x₁

The following usable rules [FROCOS05] 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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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)
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
isNatIList(X) → n__isNatIList(X)
U11(tt, L) → s(length(activate(L)))
nil → n__nil

(67) 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__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__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__isNatList(x0), y1)) → AND(isNat(isNatList(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(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.

(68) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

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

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

POL( n__isNatIList(x₁) ) = 2x₁

POL( zeros ) = 2

POL( cons(x₁, x₂) ) = 2x₁ + x₂ + 2

POL( 0 ) = 0

POL( n__zeros ) = 0

POL( isNat(x₁) ) = 2

POL( n__0 ) = 0

POL( tt ) = 2

POL( n__s(x₁) ) = max{0, -1}

POL( activate(x₁) ) = x₁ + 2

POL( n__isNat(x₁) ) = 2

POL( n__length(x₁) ) = x₁

POL( length(x₁) ) = x₁ + 2

POL( s(x₁) ) = max{0, -2}

POL( n__cons(x₁, x₂) ) = 2x₁ + x₂ + 2

POL( isNatIList(x₁) ) = 2x₁ + 2

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

POL( n__isNatList(x₁) ) = x₁

POL( isNatList(x₁) ) = x₁ + 2

POL( n__nil ) = 2

POL( nil ) = 2

POL( U11(x₁, x₂) ) = max{0, -2}

POL( ACTIVATE(x₁) ) = x₁

POL( ISNATILIST(x₁) ) = 2x₁

The following usable rules [FROCOS05] 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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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)
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
isNatIList(X) → n__isNatIList(X)
U11(tt, L) → s(length(activate(L)))
nil → n__nil

(69) 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__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__s(x0), y1)) → AND(isNat(s(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(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(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.

(70) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

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

POL( AND(x₁, x₂) ) = max{0, x₂ - 1}

POL( n__isNatIList(x₁) ) = x₁ + 1

POL( zeros ) = 1

POL( cons(x₁, x₂) ) = 2x₁ + x₂

POL( 0 ) = 0

POL( n__zeros ) = 1

POL( isNat(x₁) ) = 2

POL( n__0 ) = 0

POL( tt ) = 2

POL( n__s(x₁) ) = max{0, -2}

POL( activate(x₁) ) = x₁

POL( n__isNat(x₁) ) = 2

POL( n__length(x₁) ) = max{0, -2}

POL( length(x₁) ) = 0

POL( s(x₁) ) = max{0, -2}

POL( n__cons(x₁, x₂) ) = 2x₁ + x₂

POL( isNatIList(x₁) ) = x₁ + 1

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

POL( n__isNatList(x₁) ) = 1

POL( isNatList(x₁) ) = 1

POL( n__nil ) = 1

POL( nil ) = 1

POL( U11(x₁, x₂) ) = max{0, -2}

POL( ACTIVATE(x₁) ) = max{0, x₁ - 1}

POL( ISNATILIST(x₁) ) = x₁

The following usable rules [FROCOS05] 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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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)
isNatList(X) → n__isNatList(X)
length(X) → n__length(X)
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
isNatIList(X) → n__isNatIList(X)
U11(tt, L) → s(length(activate(L)))
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil

(71) 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(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(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.

(72) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

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

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

POL( n__isNatIList(x₁) ) = 2x₁

POL( 0 ) = 0

POL( n__0 ) = 0

POL( isNat(x₁) ) = 2

POL( tt ) = 2

POL( n__s(x₁) ) = 2

POL( activate(x₁) ) = x₁

POL( n__isNat(x₁) ) = 2

POL( n__zeros ) = 0

POL( zeros ) = 0

POL( n__length(x₁) ) = 0

POL( length(x₁) ) = max{0, -1}

POL( s(x₁) ) = 2

POL( n__cons(x₁, x₂) ) = x₁ + 2x₂

POL( cons(x₁, x₂) ) = x₁ + 2x₂

POL( isNatIList(x₁) ) = 2x₁

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

POL( n__isNatList(x₁) ) = max{0, -2}

POL( isNatList(x₁) ) = max{0, -1}

POL( n__nil ) = 2

POL( nil ) = 2

POL( U11(x₁, x₂) ) = x₁

POL( ACTIVATE(x₁) ) = 2x₁

POL( ISNATILIST(x₁) ) = 2x₁

The following usable rules [FROCOS05] 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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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)
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
isNatList(X) → n__isNatList(X)
isNatIList(X) → n__isNatIList(X)
U11(tt, L) → s(length(activate(L)))
zeros → cons(0, n__zeros)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil

(73) 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(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.

(74) 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(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.

(75) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(tt, L) → LENGTH(activate(L))

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

POL( LENGTH(x₁) ) = 2

POL( U11¹(x₁, x₂) ) = 2x₁ + 2

POL( activate(x₁) ) = 2x₁

POL( n__zeros ) = 2

POL( zeros ) = 2

POL( n__0 ) = 0

POL( 0 ) = 0

POL( n__length(x₁) ) = 2x₁ + 1

POL( length(x₁) ) = 2x₁ + 1

POL( n__s(x₁) ) = max{0, -1}

POL( s(x₁) ) = max{0, -2}

POL( n__cons(x₁, x₂) ) = x₁ + 2

POL( cons(x₁, x₂) ) = 2x₁ + 2

POL( n__isNatIList(x₁) ) = max{0, -2}

POL( isNatIList(x₁) ) = max{0, -2}

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

POL( isNat(x₁) ) = 2

POL( tt ) = 2

POL( n__isNatList(x₁) ) = max{0, -2}

POL( isNatList(x₁) ) = max{0, -2}

POL( n__nil ) = 1

POL( nil ) = 2

POL( n__isNat(x₁) ) = 1

POL( U11(x₁, x₂) ) = max{0, x₁ - 1}

The following usable rules [FROCOS05] were oriented:

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

(76) Obligation:

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

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.

(77) DependencyGraphProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) QTRSRRRProof (EQUIVALENT transformation)

(2) Obligation:

(3) QTRSRRRProof (EQUIVALENT transformation)

(4) Obligation:

(5) QTRSRRRProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyPairsProof (EQUIVALENT transformation)

(8) Obligation:

(9) DependencyGraphProof (EQUIVALENT transformation)

(10) Obligation:

(11) MRRProof (EQUIVALENT transformation)

(12) Obligation:

(13) DependencyGraphProof (EQUIVALENT transformation)

(14) Complex Obligation (AND)

(15) Obligation:

(16) MRRProof (EQUIVALENT transformation)

(17) Obligation:

(18) QDPOrderProof (EQUIVALENT transformation)

(19) Obligation:

(20) QDPOrderProof (EQUIVALENT transformation)

(21) Obligation:

(22) DependencyGraphProof (EQUIVALENT transformation)

(23) Complex Obligation (AND)

(24) Obligation:

(25) QDPOrderProof (EQUIVALENT transformation)

(26) Obligation:

(27) PisEmptyProof (EQUIVALENT transformation)

(28) YES

(29) Obligation:

(30) Narrowing (EQUIVALENT transformation)

(31) Obligation:

(32) DependencyGraphProof (EQUIVALENT transformation)

(33) Obligation:

(34) Narrowing (EQUIVALENT transformation)

(35) Obligation:

(36) DependencyGraphProof (EQUIVALENT transformation)

(37) Obligation:

(38) Instantiation (EQUIVALENT transformation)

(39) Obligation:

(40) DependencyGraphProof (EQUIVALENT transformation)

(41) Complex Obligation (AND)

(42) Obligation:

(43) MRRProof (EQUIVALENT transformation)

(44) Obligation:

(45) MRRProof (EQUIVALENT transformation)

(46) Obligation:

(47) MRRProof (EQUIVALENT transformation)

(48) Obligation:

(49) QDPOrderProof (EQUIVALENT transformation)

(50) Obligation:

(51) QDPOrderProof (EQUIVALENT transformation)

(52) Obligation:

(53) QDPOrderProof (EQUIVALENT transformation)

(54) Obligation:

(55) QDPOrderProof (EQUIVALENT transformation)

(56) Obligation:

(57) NonTerminationProof (EQUIVALENT transformation)

(58) NO

(59) Obligation:

(60) MRRProof (EQUIVALENT transformation)

(61) Obligation:

(62) MRRProof (EQUIVALENT transformation)

(63) Obligation:

(64) MRRProof (EQUIVALENT transformation)

(65) Obligation:

(66) QDPOrderProof (EQUIVALENT transformation)

(67) Obligation:

(68) QDPOrderProof (EQUIVALENT transformation)

(69) Obligation:

(70) QDPOrderProof (EQUIVALENT transformation)

(71) Obligation:

(72) QDPOrderProof (EQUIVALENT transformation)

(73) Obligation:

(74) Obligation:

(75) QDPOrderProof (EQUIVALENT transformation)

(76) Obligation:

(77) DependencyGraphProof (EQUIVALENT transformation)

(78) TRUE