(0) Obligation:

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

zeroscons(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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0   
POL(U11(x1, x2)) = 1 + x1 + 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)) = 2·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)) = 2·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   
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))
length(nil) → 0


(2) Obligation:

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

zeroscons(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__nil) → tt
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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0   
POL(U11(x1, x2)) = 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) = 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


(4) Obligation:

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

zeroscons(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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

Q is empty.

(5) 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)) = 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   
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:

zeroscons(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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

Q is empty.

(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:

ZEROSCONS(0, n__zeros)
ZEROS01
U111(tt, L) → S(length(activate(L)))
U111(tt, L) → LENGTH(activate(L))
U111(tt, L) → ACTIVATE(L)
AND(tt, X) → ACTIVATE(X)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
LENGTH(cons(N, L)) → U111(and(isNatList(activate(L)), n__isNat(N)), activate(L))
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNat(N))
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
LENGTH(cons(N, L)) → ACTIVATE(L)
ACTIVATE(n__zeros) → ZEROS
ACTIVATE(n__0) → 01
ACTIVATE(n__length(X)) → LENGTH(activate(X))
ACTIVATE(n__length(X)) → ACTIVATE(X)
ACTIVATE(n__s(X)) → S(activate(X))
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → CONS(activate(X1), X2)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ACTIVATE(n__nil) → NIL
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ACTIVATE(n__isNat(X)) → ISNAT(X)

The TRS R consists of the following rules:

zeroscons(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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

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

The TRS R consists of the following rules:

zeroscons(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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(11) 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:

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)) = x1 + x2   
POL(ISNAT(x1)) = x1   
POL(ISNATILIST(x1)) = 1 + x1   
POL(ISNATLIST(x1)) = x1   
POL(LENGTH(x1)) = 2·x1   
POL(U11(x1, x2)) = x1 + 2·x2   
POL(U111(x1, x2)) = x1 + 2·x2   
POL(activate(x1)) = x1   
POL(and(x1, x2)) = 2·x1 + x2   
POL(cons(x1, x2)) = 2·x1 + 2·x2   
POL(isNat(x1)) = x1   
POL(isNatIList(x1)) = 1 + 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 + 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   

(12) Obligation:

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

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

zeroscons(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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(13) 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(activate(X))
ACTIVATE(n__length(X)) → ACTIVATE(X)


Used ordering: Polynomial interpretation [POLO]:

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

(14) 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__s(X)) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
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:

zeroscons(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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(15) DependencyGraphProof (EQUIVALENT transformation)

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

(16) Complex Obligation (AND)

(17) Obligation:

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

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

The TRS R consists of the following rules:

zeroscons(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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(18) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


ACTIVATE(n__isNat(X)) → ISNAT(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

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

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

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

(19) Obligation:

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

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

The TRS R consists of the following rules:

zeroscons(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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(20) QDPOrderProof (EQUIVALENT transformation)

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


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, x1 + 2x2 - 1}


POL( ISNAT(x1) ) = 2x1 + 1


POL( n__isNatIList(x1) ) = 0


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


POL( activate(x1) ) = x1


POL( n__zeros ) = 0


POL( zeros ) = 0


POL( n__0 ) = 0


POL( 0 ) = 0


POL( n__length(x1) ) = 0


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


POL( n__s(x1) ) = x1


POL( s(x1) ) = x1


POL( n__cons(x1, x2) ) = 2x1 + 2x2


POL( cons(x1, x2) ) = 2x1 + 2x2


POL( isNatIList(x1) ) = 0


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


POL( isNat(x1) ) = 1


POL( tt ) = 1


POL( isNatList(x1) ) = 2x1 + 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) ) = 0


POL( ISNATLIST(x1) ) = 2x1 + 2



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

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

(21) Obligation:

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

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

The TRS R consists of the following rules:

zeroscons(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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(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:

zeroscons(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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(25) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


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) ) = x1 + 2


POL( activate(x1) ) = 2x1


POL( n__zeros ) = 1


POL( zeros ) = 2


POL( n__0 ) = 0


POL( 0 ) = 0


POL( n__length(x1) ) = 0


POL( length(x1) ) = 0


POL( n__s(x1) ) = 2x1 + 1


POL( s(x1) ) = 2x1 + 1


POL( n__cons(x1, x2) ) = 2x1 + 2


POL( cons(x1, x2) ) = 2x1 + 2


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


POL( isNatIList(x1) ) = 0


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, -2}


POL( n__nil ) = 0


POL( nil ) = 0


POL( n__isNat(x1) ) = 1


POL( U11(x1, x2) ) = max{0, 2x1 - 1}



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

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

(26) Obligation:

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

zeroscons(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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(27) 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:

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

The TRS R consists of the following rules:

zeroscons(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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(30) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

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

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none

(31) Obligation:

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

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

The TRS R consists of the following rules:

zeroscons(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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(32) 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(activate(x0))), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__s(x0), y1)) → AND(isNat(s(activate(x0))), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(activate(x0), x1)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatIList(activate(y1)))

(33) Obligation:

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

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

The TRS R consists of the following rules:

zeroscons(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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(34) DependencyGraphProof (EQUIVALENT transformation)

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

(35) Obligation:

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

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

The TRS R consists of the following rules:

zeroscons(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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(36) 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(activate(x0))), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(activate(x0))), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(activate(x0), x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))

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

The TRS R consists of the following rules:

zeroscons(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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(38) DependencyGraphProof (EQUIVALENT transformation)

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

(39) Obligation:

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

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

The TRS R consists of the following rules:

zeroscons(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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(40) 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))

(41) Obligation:

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

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

The TRS R consists of the following rules:

zeroscons(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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(42) DependencyGraphProof (EQUIVALENT transformation)

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

(43) Complex Obligation (AND)

(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(activate(x0))), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(activate(x0))), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(activate(x0), x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))

The TRS R consists of the following rules:

zeroscons(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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(45) 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:

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)) = 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   

(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(activate(x0))), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(activate(x0))), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(activate(x0), x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__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:

zeroscons(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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(47) 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:

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

(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__length(x0), y1)) → AND(isNat(length(activate(x0))), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(activate(x0))), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(activate(x0), x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__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:

zeroscons(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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(49) 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:

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


Used ordering: Polynomial interpretation [POLO]:

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

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

The TRS R consists of the following rules:

zeroscons(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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(51) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


ISNATLIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatList(activate(y1)))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

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

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

zeroscons(0, n__zeros)
zerosn__zeros
isNat(n__0) → tt
isNat(n__s(V1)) → isNat(activate(V1))
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
and(tt, X) → activate(X)
activate(n__isNatList(X)) → isNatList(X)
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
activate(n__nil) → nil
activate(n__isNat(X)) → isNat(X)
activate(X) → X
0n__0
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatList(X) → n__isNatList(X)
length(X) → n__length(X)
isNatIList(X) → n__isNatIList(X)
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U11(tt, L) → s(length(activate(L)))
niln__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__s(x0), y1)) → AND(isNat(s(activate(x0))), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(activate(x0), x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))

The TRS R consists of the following rules:

zeroscons(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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(53) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(activate(x0))), n__isNatList(activate(y1)))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

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

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

zeroscons(0, n__zeros)
zerosn__zeros
isNat(n__0) → tt
isNat(n__s(V1)) → isNat(activate(V1))
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
and(tt, X) → activate(X)
activate(n__isNatList(X)) → isNatList(X)
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
activate(n__nil) → nil
activate(n__isNat(X)) → isNat(X)
activate(X) → X
0n__0
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatList(X) → n__isNatList(X)
length(X) → n__length(X)
isNatIList(X) → n__isNatIList(X)
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U11(tt, L) → s(length(activate(L)))
niln__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(n__cons(x0, x1), y1)) → AND(isNat(cons(activate(x0), x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))

The TRS R consists of the following rules:

zeroscons(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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(55) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


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 interpretation [POLO]:

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

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

zeroscons(0, n__zeros)
zerosn__zeros
isNat(n__0) → tt
isNat(n__s(V1)) → isNat(activate(V1))
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
and(tt, X) → activate(X)
activate(n__isNatList(X)) → isNatList(X)
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
activate(n__nil) → nil
activate(n__isNat(X)) → isNat(X)
activate(X) → X
0n__0
cons(X1, X2) → n__cons(X1, X2)
isNatList(X) → n__isNatList(X)
length(X) → n__length(X)
s(X) → n__s(X)
isNatIList(X) → n__isNatIList(X)
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U11(tt, L) → s(length(activate(L)))
niln__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(n__cons(x0, x1), y1)) → AND(isNat(cons(activate(x0), x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))

The TRS R consists of the following rules:

zeroscons(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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(57) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(activate(x0), x1)), n__isNatList(activate(y1)))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

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

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

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

(58) Obligation:

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

AND(tt, n__isNatList(y_3)) → ACTIVATE(n__isNatList(y_3))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(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:

zeroscons(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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(59) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

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

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

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

(60) 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:

zeroscons(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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(61) Obligation:

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

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

The TRS R consists of the following rules:

zeroscons(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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(62) 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:

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


Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(ACTIVATE(x1)) = x1   
POL(AND(x1, x2)) = 2·x1 + 2·x2   
POL(ISNATILIST(x1)) = 2·x1   
POL(U11(x1, x2)) = 2·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)) = 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) = 2   
POL(n__s(x1)) = x1   
POL(n__zeros) = 0   
POL(nil) = 2   
POL(s(x1)) = x1   
POL(tt) = 0   
POL(zeros) = 0   

(63) Obligation:

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

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

zeroscons(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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(64) 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:

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


Used ordering: Polynomial interpretation [POLO]:

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

(65) Obligation:

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

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

zeroscons(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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(66) 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:

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)) = 2·x1 + x2   
POL(ISNATILIST(x1)) = 2 + 2·x1   
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 + 2·x1   
POL(isNatList(x1)) = x1   
POL(length(x1)) = 2·x1   
POL(n__0) = 0   
POL(n__cons(x1, x2)) = 2·x1 + 2·x2   
POL(n__isNat(x1)) = x1   
POL(n__isNatIList(x1)) = 2 + 2·x1   
POL(n__isNatList(x1)) = x1   
POL(n__length(x1)) = 2·x1   
POL(n__nil) = 0   
POL(n__s(x1)) = x1   
POL(n__zeros) = 0   
POL(nil) = 0   
POL(s(x1)) = x1   
POL(tt) = 0   
POL(zeros) = 0   

(67) Obligation:

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

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

The TRS R consists of the following rules:

zeroscons(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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(68) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


ISNATILIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatIList(activate(y1)))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

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

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

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

(69) Obligation:

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

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

The TRS R consists of the following rules:

zeroscons(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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(70) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


ISNATILIST(n__cons(n__s(x0), y1)) → AND(isNat(s(activate(x0))), n__isNatIList(activate(y1)))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

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

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

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

(71) Obligation:

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

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

The TRS R consists of the following rules:

zeroscons(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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(72) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


ISNATILIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatIList(activate(y1)))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

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

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

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

(73) Obligation:

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

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

The TRS R consists of the following rules:

zeroscons(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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(74) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


ISNATILIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatIList(activate(y1)))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

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

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

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

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

The TRS R consists of the following rules:

zeroscons(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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(76) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


ISNATILIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(activate(x0), x1)), n__isNatIList(activate(y1)))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

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

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

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

(77) 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:

zeroscons(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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(78) NonTerminationLoopProof (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__isNatIList(activate(n__zeros))) evaluates to t =ACTIVATE(n__isNatIList(activate(n__zeros)))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Matcher: [ ]
  • Semiunifier: [ ]




Rewriting sequence

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

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

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

ACTIVATE(n__isNatIList(cons(n__0, n__zeros)))ACTIVATE(n__isNatIList(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__isNatIList(n__cons(n__0, n__zeros)))ISNATILIST(n__cons(n__0, n__zeros))
with rule ACTIVATE(n__isNatIList(X)) → ISNATILIST(X) at position [] and matcher [X / n__cons(n__0, n__zeros)]

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

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

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



(79) NO

(80) 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:

zeroscons(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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(81) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


U111(tt, L) → LENGTH(activate(L))
LENGTH(cons(N, L)) → U111(and(isNatList(activate(L)), n__isNat(N)), activate(L))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( LENGTH(x1) ) = 2


POL( U111(x1, x2) ) = x1 + 1


POL( activate(x1) ) = 2x1


POL( n__zeros ) = 2


POL( zeros ) = 2


POL( n__0 ) = 0


POL( 0 ) = 0


POL( n__length(x1) ) = 1


POL( length(x1) ) = 2


POL( n__s(x1) ) = 2


POL( s(x1) ) = 2


POL( n__cons(x1, x2) ) = 0


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


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


POL( isNatIList(x1) ) = 0


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 ) = 0


POL( nil ) = 0


POL( n__isNat(x1) ) = 1


POL( U11(x1, x2) ) = 2



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

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

(82) Obligation:

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

zeroscons(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))
zerosn__zeros
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(83) PisEmptyProof (EQUIVALENT transformation)

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

(84) YES