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

0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 DependencyGraphProof (⇔)
↳4 QDP
↳5 QDPOrderProof (⇔)
↳6 QDP
↳7 DependencyGraphProof (⇔)
↳8 AND
  ↳9 QDP
    ↳10 Narrowing (⇔)
    ↳11 QDP
    ↳12 Narrowing (⇔)
    ↳13 QDP
    ↳14 DependencyGraphProof (⇔)
    ↳15 QDP
    ↳16 Narrowing (⇔)
    ↳17 QDP
    ↳18 DependencyGraphProof (⇔)
    ↳19 QDP
    ↳20 Narrowing (⇔)
    ↳21 QDP
    ↳22 DependencyGraphProof (⇔)
    ↳23 QDP
    ↳24 Narrowing (⇔)
    ↳25 QDP
    ↳26 Narrowing (⇔)
    ↳27 QDP
    ↳28 DependencyGraphProof (⇔)
    ↳29 QDP
    ↳30 Narrowing (⇔)
    ↳31 QDP
    ↳32 Narrowing (⇔)
    ↳33 QDP
    ↳34 QDPOrderProof (⇔)
    ↳35 QDP
    ↳36 DependencyGraphProof (⇔)
    ↳37 AND
      ↳38 QDP
        ↳39 UsableRulesProof (⇔)
        ↳40 QDP
        ↳41 MNOCProof (⇔)
        ↳42 QDP
        ↳43 Rewriting (⇔)
        ↳44 QDP
        ↳45 UsableRulesProof (⇔)
        ↳46 QDP
        ↳47 QReductionProof (⇔)
        ↳48 QDP
        ↳49 Instantiation (⇔)
        ↳50 QDP
        ↳51 NonTerminationProof (⇔)
        ↳52 FALSE
      ↳53 QDP
  ↳54 QDP

(0) Obligation:

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

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

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

(2) Obligation:

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

ISNATILIST(IL) → ISNATLIST(activate(IL))
ISNATILIST(IL) → ACTIVATE(IL)
ISNAT(n__s(N)) → ISNAT(activate(N))
ISNAT(n__s(N)) → ACTIVATE(N)
ISNAT(n__length(L)) → ISNATLIST(activate(L))
ISNAT(n__length(L)) → ACTIVATE(L)
ISNATILIST(n__cons(N, IL)) → AND(isNat(activate(N)), isNatIList(activate(IL)))
ISNATILIST(n__cons(N, IL)) → ISNAT(activate(N))
ISNATILIST(n__cons(N, IL)) → ACTIVATE(N)
ISNATILIST(n__cons(N, IL)) → ISNATILIST(activate(IL))
ISNATILIST(n__cons(N, IL)) → ACTIVATE(IL)
ISNATLIST(n__cons(N, L)) → AND(isNat(activate(N)), isNatList(activate(L)))
ISNATLIST(n__cons(N, L)) → ISNAT(activate(N))
ISNATLIST(n__cons(N, L)) → ACTIVATE(N)
ISNATLIST(n__cons(N, L)) → ISNATLIST(activate(L))
ISNATLIST(n__cons(N, L)) → ACTIVATE(L)
ISNATLIST(n__take(N, IL)) → AND(isNat(activate(N)), isNatIList(activate(IL)))
ISNATLIST(n__take(N, IL)) → ISNAT(activate(N))
ISNATLIST(n__take(N, IL)) → ACTIVATE(N)
ISNATLIST(n__take(N, IL)) → ISNATILIST(activate(IL))
ISNATLIST(n__take(N, IL)) → ACTIVATE(IL)
ZEROS → CONS(0, n__zeros)
ZEROS → 01
TAKE(0, IL) → UTAKE1(isNatIList(IL))
TAKE(0, IL) → ISNATILIST(IL)
UTAKE1(tt) → NIL
TAKE(s(M), cons(N, IL)) → UTAKE2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
TAKE(s(M), cons(N, IL)) → AND(isNat(M), and(isNat(N), isNatIList(activate(IL))))
TAKE(s(M), cons(N, IL)) → ISNAT(M)
TAKE(s(M), cons(N, IL)) → AND(isNat(N), isNatIList(activate(IL)))
TAKE(s(M), cons(N, IL)) → ISNAT(N)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
UTAKE2(tt, M, N, IL) → CONS(activate(N), n__take(activate(M), activate(IL)))
UTAKE2(tt, M, N, IL) → ACTIVATE(N)
UTAKE2(tt, M, N, IL) → ACTIVATE(M)
UTAKE2(tt, M, N, IL) → ACTIVATE(IL)
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(activate(L))), activate(L))
LENGTH(cons(N, L)) → AND(isNat(N), isNatList(activate(L)))
LENGTH(cons(N, L)) → ISNAT(N)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
LENGTH(cons(N, L)) → ACTIVATE(L)
ULENGTH(tt, L) → S(length(activate(L)))
ULENGTH(tt, L) → LENGTH(activate(L))
ULENGTH(tt, L) → ACTIVATE(L)
ACTIVATE(n__0) → 01
ACTIVATE(n__s(X)) → S(X)
ACTIVATE(n__length(X)) → LENGTH(X)
ACTIVATE(n__zeros) → ZEROS
ACTIVATE(n__cons(X1, X2)) → CONS(X1, X2)
ACTIVATE(n__nil) → NIL
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 17 less nodes.

(4) Obligation:

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

ISNATLIST(n__cons(N, L)) → ISNAT(activate(N))
ISNAT(n__s(N)) → ISNAT(activate(N))
ISNAT(n__s(N)) → ACTIVATE(N)
ACTIVATE(n__length(X)) → LENGTH(X)
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(activate(L))), activate(L))
ULENGTH(tt, L) → LENGTH(activate(L))
LENGTH(cons(N, L)) → ISNAT(N)
ISNAT(n__length(L)) → ISNATLIST(activate(L))
ISNATLIST(n__cons(N, L)) → ACTIVATE(N)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
TAKE(0, IL) → ISNATILIST(IL)
ISNATILIST(IL) → ISNATLIST(activate(IL))
ISNATLIST(n__cons(N, L)) → ISNATLIST(activate(L))
ISNATLIST(n__cons(N, L)) → ACTIVATE(L)
ISNATLIST(n__take(N, IL)) → ISNAT(activate(N))
ISNAT(n__length(L)) → ACTIVATE(L)
ISNATLIST(n__take(N, IL)) → ACTIVATE(N)
ISNATLIST(n__take(N, IL)) → ISNATILIST(activate(IL))
ISNATILIST(IL) → ACTIVATE(IL)
ISNATILIST(n__cons(N, IL)) → ISNAT(activate(N))
ISNATILIST(n__cons(N, IL)) → ACTIVATE(N)
ISNATILIST(n__cons(N, IL)) → ISNATILIST(activate(IL))
ISNATILIST(n__cons(N, IL)) → ACTIVATE(IL)
ISNATLIST(n__take(N, IL)) → ACTIVATE(IL)
TAKE(s(M), cons(N, IL)) → UTAKE2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
UTAKE2(tt, M, N, IL) → ACTIVATE(N)
UTAKE2(tt, M, N, IL) → ACTIVATE(M)
UTAKE2(tt, M, N, IL) → ACTIVATE(IL)
TAKE(s(M), cons(N, IL)) → ISNAT(M)
TAKE(s(M), cons(N, IL)) → ISNAT(N)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
LENGTH(cons(N, L)) → ACTIVATE(L)
ULENGTH(tt, L) → ACTIVATE(L)

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(5) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ISNATILIST(IL) → ISNATLIST(activate(IL))
ISNATLIST(n__take(N, IL)) → ISNAT(activate(N))
ISNATLIST(n__take(N, IL)) → ACTIVATE(N)
ISNATILIST(IL) → ACTIVATE(IL)
ISNATILIST(n__cons(N, IL)) → ISNAT(activate(N))
ISNATILIST(n__cons(N, IL)) → ACTIVATE(N)
ISNATILIST(n__cons(N, IL)) → ACTIVATE(IL)
ISNATLIST(n__take(N, IL)) → ACTIVATE(IL)
TAKE(s(M), cons(N, IL)) → UTAKE2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
TAKE(s(M), cons(N, IL)) → ISNAT(M)
TAKE(s(M), cons(N, IL)) → ISNAT(N)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(ACTIVATE(x1)) = x1   
POL(ISNAT(x1)) = x1   
POL(ISNATILIST(x1)) = 1 + x1   
POL(ISNATLIST(x1)) = x1   
POL(LENGTH(x1)) = x1   
POL(TAKE(x1, x2)) = 1 + x1 + x2   
POL(ULENGTH(x1, x2)) = x2   
POL(UTAKE2(x1, x2, x3, x4)) = x2 + x3 + x4   
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)) = 0   
POL(length(x1)) = x1   
POL(n__0) = 0   
POL(n__cons(x1, x2)) = x1 + x2   
POL(n__length(x1)) = x1   
POL(n__nil) = 0   
POL(n__s(x1)) = x1   
POL(n__take(x1, x2)) = 1 + x1 + x2   
POL(n__zeros) = 1   
POL(nil) = 0   
POL(s(x1)) = x1   
POL(take(x1, x2)) = 1 + x1 + x2   
POL(tt) = 0   
POL(uLength(x1, x2)) = x2   
POL(uTake1(x1)) = 1   
POL(uTake2(x1, x2, x3, x4)) = 1 + x2 + x3 + x4   
POL(zeros) = 1   

The following usable rules [FROCOS05] were oriented:

activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X
activate(n__nil) → nil
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__zeros) → zeros
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__0) → 0
take(X1, X2) → n__take(X1, X2)
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
zeros → n__zeros
length(X) → n__length(X)
s(X) → n__s(X)
0 → n__0
uLength(tt, L) → s(length(activate(L)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))

(6) Obligation:

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

ISNATLIST(n__cons(N, L)) → ISNAT(activate(N))
ISNAT(n__s(N)) → ISNAT(activate(N))
ISNAT(n__s(N)) → ACTIVATE(N)
ACTIVATE(n__length(X)) → LENGTH(X)
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(activate(L))), activate(L))
ULENGTH(tt, L) → LENGTH(activate(L))
LENGTH(cons(N, L)) → ISNAT(N)
ISNAT(n__length(L)) → ISNATLIST(activate(L))
ISNATLIST(n__cons(N, L)) → ACTIVATE(N)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
TAKE(0, IL) → ISNATILIST(IL)
ISNATLIST(n__cons(N, L)) → ISNATLIST(activate(L))
ISNATLIST(n__cons(N, L)) → ACTIVATE(L)
ISNAT(n__length(L)) → ACTIVATE(L)
ISNATLIST(n__take(N, IL)) → ISNATILIST(activate(IL))
ISNATILIST(n__cons(N, IL)) → ISNATILIST(activate(IL))
UTAKE2(tt, M, N, IL) → ACTIVATE(N)
UTAKE2(tt, M, N, IL) → ACTIVATE(M)
UTAKE2(tt, M, N, IL) → ACTIVATE(IL)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
LENGTH(cons(N, L)) → ACTIVATE(L)
ULENGTH(tt, L) → ACTIVATE(L)

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 7 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

ISNATILIST(n__cons(N, IL)) → ISNATILIST(activate(IL))

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(10) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATILIST(n__cons(N, IL)) → ISNATILIST(activate(IL)) at position [0] we obtained the following new rules [LPAR04]:

ISNATILIST(n__cons(y0, n__0)) → ISNATILIST(0)
ISNATILIST(n__cons(y0, n__s(x0))) → ISNATILIST(s(x0))
ISNATILIST(n__cons(y0, n__length(x0))) → ISNATILIST(length(x0))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(zeros)
ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(cons(x0, x1))
ISNATILIST(n__cons(y0, n__nil)) → ISNATILIST(nil)
ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(y0, x0)) → ISNATILIST(x0)

(11) Obligation:

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

ISNATILIST(n__cons(y0, n__0)) → ISNATILIST(0)
ISNATILIST(n__cons(y0, n__s(x0))) → ISNATILIST(s(x0))
ISNATILIST(n__cons(y0, n__length(x0))) → ISNATILIST(length(x0))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(zeros)
ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(cons(x0, x1))
ISNATILIST(n__cons(y0, n__nil)) → ISNATILIST(nil)
ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(y0, x0)) → ISNATILIST(x0)

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(12) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATILIST(n__cons(y0, n__0)) → ISNATILIST(0) at position [0] we obtained the following new rules [LPAR04]:

ISNATILIST(n__cons(y0, n__0)) → ISNATILIST(n__0)

(13) Obligation:

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

ISNATILIST(n__cons(y0, n__s(x0))) → ISNATILIST(s(x0))
ISNATILIST(n__cons(y0, n__length(x0))) → ISNATILIST(length(x0))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(zeros)
ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(cons(x0, x1))
ISNATILIST(n__cons(y0, n__nil)) → ISNATILIST(nil)
ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(y0, x0)) → ISNATILIST(x0)
ISNATILIST(n__cons(y0, n__0)) → ISNATILIST(n__0)

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(14) DependencyGraphProof (EQUIVALENT transformation)

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

(15) Obligation:

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

ISNATILIST(n__cons(y0, n__s(x0))) → ISNATILIST(s(x0))
ISNATILIST(n__cons(y0, n__length(x0))) → ISNATILIST(length(x0))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(zeros)
ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(cons(x0, x1))
ISNATILIST(n__cons(y0, n__nil)) → ISNATILIST(nil)
ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(y0, x0)) → ISNATILIST(x0)

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(16) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATILIST(n__cons(y0, n__s(x0))) → ISNATILIST(s(x0)) at position [0] we obtained the following new rules [LPAR04]:

ISNATILIST(n__cons(y0, n__s(x0))) → ISNATILIST(n__s(x0))

(17) Obligation:

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

ISNATILIST(n__cons(y0, n__length(x0))) → ISNATILIST(length(x0))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(zeros)
ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(cons(x0, x1))
ISNATILIST(n__cons(y0, n__nil)) → ISNATILIST(nil)
ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(y0, x0)) → ISNATILIST(x0)
ISNATILIST(n__cons(y0, n__s(x0))) → ISNATILIST(n__s(x0))

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(18) DependencyGraphProof (EQUIVALENT transformation)

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

(19) Obligation:

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

ISNATILIST(n__cons(y0, n__length(x0))) → ISNATILIST(length(x0))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(zeros)
ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(cons(x0, x1))
ISNATILIST(n__cons(y0, n__nil)) → ISNATILIST(nil)
ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(y0, x0)) → ISNATILIST(x0)

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(20) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(zeros) at position [0] we obtained the following new rules [LPAR04]:

ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__zeros)

(21) Obligation:

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

ISNATILIST(n__cons(y0, n__length(x0))) → ISNATILIST(length(x0))
ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(cons(x0, x1))
ISNATILIST(n__cons(y0, n__nil)) → ISNATILIST(nil)
ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(y0, x0)) → ISNATILIST(x0)
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__zeros)

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
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 1 SCC with 1 less node.

(23) Obligation:

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

ISNATILIST(n__cons(y0, n__length(x0))) → ISNATILIST(length(x0))
ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(cons(x0, x1))
ISNATILIST(n__cons(y0, n__nil)) → ISNATILIST(nil)
ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(y0, x0)) → ISNATILIST(x0)
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(cons(0, n__zeros))

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(24) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(cons(x0, x1)) at position [0] we obtained the following new rules [LPAR04]:

ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(n__cons(x0, x1))

(25) Obligation:

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

ISNATILIST(n__cons(y0, n__length(x0))) → ISNATILIST(length(x0))
ISNATILIST(n__cons(y0, n__nil)) → ISNATILIST(nil)
ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(y0, x0)) → ISNATILIST(x0)
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(n__cons(x0, x1))

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(26) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATILIST(n__cons(y0, n__nil)) → ISNATILIST(nil) at position [0] we obtained the following new rules [LPAR04]:

ISNATILIST(n__cons(y0, n__nil)) → ISNATILIST(n__nil)

(27) Obligation:

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

ISNATILIST(n__cons(y0, n__length(x0))) → ISNATILIST(length(x0))
ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(y0, x0)) → ISNATILIST(x0)
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(n__cons(x0, x1))
ISNATILIST(n__cons(y0, n__nil)) → ISNATILIST(n__nil)

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(28) DependencyGraphProof (EQUIVALENT transformation)

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

(29) Obligation:

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

ISNATILIST(n__cons(y0, n__length(x0))) → ISNATILIST(length(x0))
ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(y0, x0)) → ISNATILIST(x0)
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(n__cons(x0, x1))

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(30) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(cons(0, n__zeros)) at position [0] we obtained the following new rules [LPAR04]:

ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(0, n__zeros))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(cons(n__0, n__zeros))

(31) Obligation:

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

ISNATILIST(n__cons(y0, n__length(x0))) → ISNATILIST(length(x0))
ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(y0, x0)) → ISNATILIST(x0)
ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(n__cons(x0, x1))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(0, n__zeros))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(cons(n__0, n__zeros))

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
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(y0, n__zeros)) → ISNATILIST(cons(n__0, n__zeros)) at position [0] we obtained the following new rules [LPAR04]:

ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(n__0, n__zeros))

(33) Obligation:

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

ISNATILIST(n__cons(y0, n__length(x0))) → ISNATILIST(length(x0))
ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(y0, x0)) → ISNATILIST(x0)
ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(n__cons(x0, x1))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(0, n__zeros))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(34) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ISNATILIST(n__cons(y0, n__length(x0))) → ISNATILIST(length(x0))
ISNATILIST(n__cons(y0, x0)) → ISNATILIST(x0)
ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(n__cons(x0, x1))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 1   
POL(ISNATILIST(x1)) = x1   
POL(activate(x1)) = 1   
POL(and(x1, x2)) = x2   
POL(cons(x1, x2)) = 1 + x2   
POL(isNat(x1)) = 0   
POL(isNatIList(x1)) = 0   
POL(isNatList(x1)) = 0   
POL(length(x1)) = 1   
POL(n__0) = 0   
POL(n__cons(x1, x2)) = 1 + x2   
POL(n__length(x1)) = 1   
POL(n__nil) = 0   
POL(n__s(x1)) = 0   
POL(n__take(x1, x2)) = 0   
POL(n__zeros) = 0   
POL(nil) = 0   
POL(s(x1)) = 0   
POL(take(x1, x2)) = 1   
POL(tt) = 0   
POL(uLength(x1, x2)) = 1   
POL(uTake1(x1)) = 0   
POL(uTake2(x1, x2, x3, x4)) = 1   
POL(zeros) = 0   

The following usable rules [FROCOS05] were oriented:

activate(n__take(X1, X2)) → take(X1, X2)
activate(n__length(X)) → length(X)
take(X1, X2) → n__take(X1, X2)
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
length(X) → n__length(X)
s(X) → n__s(X)
uLength(tt, L) → s(length(activate(L)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))

(35) Obligation:

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

ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(0, n__zeros))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(36) DependencyGraphProof (EQUIVALENT transformation)

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

(37) Complex Obligation (AND)

(38) Obligation:

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

ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(0, n__zeros))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(39) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(40) Obligation:

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

ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(0, n__zeros))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

0 → n__0

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

(41) MNOCProof (EQUIVALENT transformation)

We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.

(42) Obligation:

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

ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(0, n__zeros))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

0 → n__0

The set Q consists of the following terms:

0

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

(43) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(0, n__zeros)) at position [0,0] we obtained the following new rules [LPAR04]:

ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(n__0, n__zeros))

(44) Obligation:

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

ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

0 → n__0

The set Q consists of the following terms:

0

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

(45) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(46) Obligation:

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

ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(n__0, n__zeros))

R is empty.
The set Q consists of the following terms:

0

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

(47) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

0

(48) Obligation:

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

ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(n__0, n__zeros))

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

(49) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(n__0, n__zeros)) we obtained the following new rules [LPAR04]:

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

(50) Obligation:

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

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

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

(51) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = ISNATILIST(n__cons(n__0, n__zeros)) evaluates to t =ISNATILIST(n__cons(n__0, n__zeros))

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



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from ISNATILIST(n__cons(n__0, n__zeros)) to ISNATILIST(n__cons(n__0, n__zeros)).



(52) FALSE

(53) Obligation:

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

ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(take(x0, x1))

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(54) Obligation:

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

ISNAT(n__s(N)) → ISNAT(activate(N))
ISNAT(n__s(N)) → ACTIVATE(N)
ACTIVATE(n__length(X)) → LENGTH(X)
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(activate(L))), activate(L))
ULENGTH(tt, L) → LENGTH(activate(L))
LENGTH(cons(N, L)) → ISNAT(N)
ISNAT(n__length(L)) → ISNATLIST(activate(L))
ISNATLIST(n__cons(N, L)) → ISNAT(activate(N))
ISNAT(n__length(L)) → ACTIVATE(L)
ISNATLIST(n__cons(N, L)) → ACTIVATE(N)
ISNATLIST(n__cons(N, L)) → ISNATLIST(activate(L))
ISNATLIST(n__cons(N, L)) → ACTIVATE(L)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
LENGTH(cons(N, L)) → ACTIVATE(L)
ULENGTH(tt, L) → ACTIVATE(L)

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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