NO Termination w.r.t. Q proof of Transformed_CSR_04_OvConsOS_nokinds-noand_Z.ari

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

0 QTRS
↳1 DependencyPairsProof (⇔, 0 ms)
↳2 QDP
↳3 DependencyGraphProof (⇔, 0 ms)
↳4 QDP
↳5 QDPOrderProof (⇔, 273 ms)
↳6 QDP
↳7 DependencyGraphProof (⇔, 0 ms)
↳8 AND
    ↳9 QDP
        ↳10 QDPOrderProof (⇔, 154 ms)
        ↳11 QDP
        ↳12 PisEmptyProof (⇔, 0 ms)
        ↳13 YES
    ↳14 QDP
        ↳15 TransformationProof (⇔, 0 ms)
        ↳16 QDP
        ↳17 TransformationProof (⇔, 0 ms)
        ↳18 QDP
        ↳19 TransformationProof (⇔, 0 ms)
        ↳20 QDP
        ↳21 DependencyGraphProof (⇔, 0 ms)
        ↳22 QDP
        ↳23 TransformationProof (⇔, 0 ms)
        ↳24 QDP
        ↳25 DependencyGraphProof (⇔, 0 ms)
        ↳26 QDP
        ↳27 TransformationProof (⇔, 0 ms)
        ↳28 QDP
        ↳29 TransformationProof (⇔, 0 ms)
        ↳30 QDP
        ↳31 DependencyGraphProof (⇔, 0 ms)
        ↳32 QDP
        ↳33 TransformationProof (⇔, 0 ms)
        ↳34 QDP
        ↳35 DependencyGraphProof (⇔, 0 ms)
        ↳36 QDP
        ↳37 TransformationProof (⇔, 0 ms)
        ↳38 QDP
        ↳39 TransformationProof (⇔, 0 ms)
        ↳40 QDP
        ↳41 TransformationProof (⇔, 0 ms)
        ↳42 QDP
        ↳43 DependencyGraphProof (⇔, 0 ms)
        ↳44 QDP
        ↳45 TransformationProof (⇔, 0 ms)
        ↳46 QDP
        ↳47 DependencyGraphProof (⇔, 0 ms)
        ↳48 QDP
        ↳49 TransformationProof (⇔, 0 ms)
        ↳50 QDP
        ↳51 DependencyGraphProof (⇔, 0 ms)
        ↳52 QDP
        ↳53 TransformationProof (⇔, 0 ms)
        ↳54 QDP
        ↳55 TransformationProof (⇔, 0 ms)
        ↳56 QDP
        ↳57 DependencyGraphProof (⇔, 0 ms)
        ↳58 QDP
        ↳59 TransformationProof (⇔, 0 ms)
        ↳60 QDP
        ↳61 DependencyGraphProof (⇔, 0 ms)
        ↳62 QDP
        ↳63 TransformationProof (⇔, 0 ms)
        ↳64 QDP
        ↳65 QDPOrderProof (⇔, 0 ms)
        ↳66 QDP
        ↳67 QDPOrderProof (⇔, 0 ms)
        ↳68 QDP
        ↳69 QDPOrderProof (⇔, 0 ms)
        ↳70 QDP
    ↳71 QDP
        ↳72 QDPOrderProof (⇔, 65 ms)
        ↳73 QDP
        ↳74 DependencyGraphProof (⇔, 0 ms)
        ↳75 TRUE
    ↳76 QDP
        ↳77 TransformationProof (⇔, 0 ms)
        ↳78 QDP
        ↳79 TransformationProof (⇔, 0 ms)
        ↳80 QDP
        ↳81 TransformationProof (⇔, 0 ms)
        ↳82 QDP
        ↳83 DependencyGraphProof (⇔, 0 ms)
        ↳84 QDP
        ↳85 TransformationProof (⇔, 0 ms)
        ↳86 QDP
        ↳87 DependencyGraphProof (⇔, 0 ms)
        ↳88 QDP
        ↳89 TransformationProof (⇔, 0 ms)
        ↳90 QDP
        ↳91 DependencyGraphProof (⇔, 0 ms)
        ↳92 QDP
        ↳93 TransformationProof (⇔, 0 ms)
        ↳94 QDP
        ↳95 TransformationProof (⇔, 0 ms)
        ↳96 QDP
        ↳97 DependencyGraphProof (⇔, 0 ms)
        ↳98 QDP
        ↳99 TransformationProof (⇔, 0 ms)
        ↳100 QDP
        ↳101 TransformationProof (⇔, 0 ms)
        ↳102 QDP
        ↳103 TransformationProof (⇔, 0 ms)
        ↳104 QDP
        ↳105 DependencyGraphProof (⇔, 0 ms)
        ↳106 QDP
        ↳107 TransformationProof (⇔, 0 ms)
        ↳108 QDP
        ↳109 DependencyGraphProof (⇔, 0 ms)
        ↳110 QDP
        ↳111 TransformationProof (⇔, 0 ms)
        ↳112 QDP
        ↳113 DependencyGraphProof (⇔, 0 ms)
        ↳114 QDP
        ↳115 TransformationProof (⇔, 0 ms)
        ↳116 QDP
        ↳117 DependencyGraphProof (⇔, 0 ms)
        ↳118 QDP
        ↳119 TransformationProof (⇔, 0 ms)
        ↳120 QDP
        ↳121 TransformationProof (⇔, 0 ms)
        ↳122 QDP
        ↳123 DependencyGraphProof (⇔, 0 ms)
        ↳124 QDP
        ↳125 TransformationProof (⇔, 0 ms)
        ↳126 QDP
        ↳127 QDPOrderProof (⇔, 0 ms)
        ↳128 QDP
        ↳129 QDPOrderProof (⇔, 0 ms)
        ↳130 QDP
        ↳131 QDPOrderProof (⇔, 0 ms)
        ↳132 QDP
        ↳133 QDPOrderProof (⇔, 0 ms)
        ↳134 QDP
        ↳135 QDPOrderProof (⇔, 0 ms)
        ↳136 QDP
        ↳137 NonTerminationLoopProof (⇒, 0 ms)
        ↳138 NO

(0) Obligation:

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

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
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:

ZEROS → CONS(0, n__zeros)
ZEROS → 01
U411(tt, V2) → U421(isNatIList(activate(V2)))
U411(tt, V2) → ISNATILIST(activate(V2))
U411(tt, V2) → ACTIVATE(V2)
U511(tt, V2) → U521(isNatList(activate(V2)))
U511(tt, V2) → ISNATLIST(activate(V2))
U511(tt, V2) → ACTIVATE(V2)
U611(tt, V2) → U621(isNatIList(activate(V2)))
U611(tt, V2) → ISNATILIST(activate(V2))
U611(tt, V2) → ACTIVATE(V2)
U711(tt, L, N) → U721(isNat(activate(N)), activate(L))
U711(tt, L, N) → ISNAT(activate(N))
U711(tt, L, N) → ACTIVATE(N)
U711(tt, L, N) → ACTIVATE(L)
U721(tt, L) → S(length(activate(L)))
U721(tt, L) → LENGTH(activate(L))
U721(tt, L) → ACTIVATE(L)
U811(tt) → NIL
U911(tt, IL, M, N) → U921(isNat(activate(M)), activate(IL), activate(M), activate(N))
U911(tt, IL, M, N) → ISNAT(activate(M))
U911(tt, IL, M, N) → ACTIVATE(M)
U911(tt, IL, M, N) → ACTIVATE(IL)
U911(tt, IL, M, N) → ACTIVATE(N)
U921(tt, IL, M, N) → U931(isNat(activate(N)), activate(IL), activate(M), activate(N))
U921(tt, IL, M, N) → ISNAT(activate(N))
U921(tt, IL, M, N) → ACTIVATE(N)
U921(tt, IL, M, N) → ACTIVATE(IL)
U921(tt, IL, M, N) → ACTIVATE(M)
U931(tt, IL, M, N) → CONS(activate(N), n__take(activate(M), activate(IL)))
U931(tt, IL, M, N) → ACTIVATE(N)
U931(tt, IL, M, N) → ACTIVATE(M)
U931(tt, IL, M, N) → ACTIVATE(IL)
ISNAT(n__length(V1)) → U111(isNatList(activate(V1)))
ISNAT(n__length(V1)) → ISNATLIST(activate(V1))
ISNAT(n__length(V1)) → ACTIVATE(V1)
ISNAT(n__s(V1)) → U211(isNat(activate(V1)))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNATILIST(V) → U311(isNatList(activate(V)))
ISNATILIST(V) → ISNATLIST(activate(V))
ISNATILIST(V) → ACTIVATE(V)
ISNATILIST(n__cons(V1, V2)) → U411(isNat(activate(V1)), 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)) → U511(isNat(activate(V1)), activate(V2))
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__take(V1, V2)) → U611(isNat(activate(V1)), activate(V2))
ISNATLIST(n__take(V1, V2)) → ISNAT(activate(V1))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
LENGTH(nil) → 01
LENGTH(cons(N, L)) → U711(isNatList(activate(L)), activate(L), N)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
LENGTH(cons(N, L)) → ACTIVATE(L)
TAKE(0, IL) → U811(isNatIList(IL))
TAKE(0, IL) → ISNATILIST(IL)
TAKE(s(M), cons(N, IL)) → U911(isNatIList(activate(IL)), activate(IL), M, N)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
ACTIVATE(n__zeros) → ZEROS
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
ACTIVATE(n__0) → 01
ACTIVATE(n__length(X)) → LENGTH(X)
ACTIVATE(n__s(X)) → S(X)
ACTIVATE(n__cons(X1, X2)) → CONS(X1, X2)
ACTIVATE(n__nil) → NIL

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
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 18 less nodes.

(4) Obligation:

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

ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
TAKE(0, IL) → ISNATILIST(IL)
ISNATILIST(V) → ISNATLIST(activate(V))
ISNATLIST(n__cons(V1, V2)) → U511(isNat(activate(V1)), activate(V2))
U511(tt, V2) → ISNATLIST(activate(V2))
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__length(V1)) → ISNATLIST(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → LENGTH(X)
LENGTH(cons(N, L)) → U711(isNatList(activate(L)), activate(L), N)
U711(tt, L, N) → U721(isNat(activate(N)), activate(L))
U721(tt, L) → LENGTH(activate(L))
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__take(V1, V2)) → U611(isNat(activate(V1)), activate(V2))
U611(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(V) → ACTIVATE(V)
ISNATILIST(n__cons(V1, V2)) → U411(isNat(activate(V1)), activate(V2))
U411(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__length(V1)) → ACTIVATE(V1)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U411(tt, V2) → ACTIVATE(V2)
U611(tt, V2) → ACTIVATE(V2)
ISNATLIST(n__take(V1, V2)) → ISNAT(activate(V1))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
LENGTH(cons(N, L)) → ACTIVATE(L)
U721(tt, L) → ACTIVATE(L)
U711(tt, L, N) → ISNAT(activate(N))
U711(tt, L, N) → ACTIVATE(N)
U711(tt, L, N) → ACTIVATE(L)
U511(tt, V2) → ACTIVATE(V2)
TAKE(s(M), cons(N, IL)) → U911(isNatIList(activate(IL)), activate(IL), M, N)
U911(tt, IL, M, N) → U921(isNat(activate(M)), activate(IL), activate(M), activate(N))
U921(tt, IL, M, N) → U931(isNat(activate(N)), activate(IL), activate(M), activate(N))
U931(tt, IL, M, N) → ACTIVATE(N)
U931(tt, IL, M, N) → ACTIVATE(M)
U931(tt, IL, M, N) → ACTIVATE(IL)
U921(tt, IL, M, N) → ISNAT(activate(N))
U921(tt, IL, M, N) → ACTIVATE(N)
U921(tt, IL, M, N) → ACTIVATE(IL)
U921(tt, IL, M, N) → ACTIVATE(M)
U911(tt, IL, M, N) → ISNAT(activate(M))
U911(tt, IL, M, N) → ACTIVATE(M)
U911(tt, IL, M, N) → ACTIVATE(IL)
U911(tt, IL, M, N) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
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,JAR06].


The following pairs can be oriented strictly and are deleted.


ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__length(V1)) → ISNATLIST(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → LENGTH(X)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
U611(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(V) → ACTIVATE(V)
ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__length(V1)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U411(tt, V2) → ACTIVATE(V2)
U611(tt, V2) → ACTIVATE(V2)
ISNATLIST(n__take(V1, V2)) → ISNAT(activate(V1))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
LENGTH(cons(N, L)) → ACTIVATE(L)
U721(tt, L) → ACTIVATE(L)
U711(tt, L, N) → ISNAT(activate(N))
U711(tt, L, N) → ACTIVATE(N)
U711(tt, L, N) → ACTIVATE(L)
U511(tt, V2) → ACTIVATE(V2)
TAKE(s(M), cons(N, IL)) → U911(isNatIList(activate(IL)), activate(IL), M, N)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( ISNAT(x1) ) = 2x1 + 1

POL( ISNATILIST(x1) ) = 2x1 + 2

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

POL( LENGTH(x1) ) = 2x1 + 2

POL( U411(x1, x2) ) = 2x2 + 2

POL( U511(x1, x2) ) = 2x2 + 2

POL( U611(x1, x2) ) = 2x1 + 2x2 + 2

POL( U711(x1, ..., x3) ) = 2x2 + 2x3 + 2

POL( U721(x1, x2) ) = 2x2 + 2

POL( U911(x1, ..., x4) ) = 2x2 + 2x3 + 2x4 + 1

POL( U921(x1, ..., x4) ) = 2x2 + 2x3 + 2x4 + 1

POL( U931(x1, ..., x4) ) = 2x2 + 2x3 + 2x4 + 1

POL( U41(x1, x2) ) = 2

POL( U51(x1, x2) ) = 2

POL( U61(x1, x2) ) = 2

POL( U71(x1, ..., x3) ) = max{0, 2x1 + x2 - 2}

POL( U72(x1, x2) ) = max{0, 2x1 + x2 - 2}

POL( U91(x1, ..., x4) ) = 2x2 + x3 + x4 + 2

POL( U92(x1, ..., x4) ) = 2x2 + x3 + x4 + 2

POL( U93(x1, ..., x4) ) = max{0, 2x1 + 2x2 + x3 + x4 - 2}

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

POL( isNat(x1) ) = 2

POL( isNatList(x1) ) = 2

POL( isNatIList(x1) ) = 2

POL( U11(x1) ) = 2

POL( U21(x1) ) = 2

POL( U31(x1) ) = x1

POL( U42(x1) ) = 2

POL( U52(x1) ) = 2

POL( U62(x1) ) = 2

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

POL( length(x1) ) = x1 + 2

POL( s(x1) ) = x1

POL( activate(x1) ) = x1

POL( n__zeros ) = 2

POL( zeros ) = 2

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

POL( n__0 ) = 0

POL( 0 ) = 0

POL( n__length(x1) ) = x1 + 2

POL( n__s(x1) ) = x1

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

POL( n__nil ) = 1

POL( nil ) = 1

POL( tt ) = 2

POL( U81(x1) ) = max{0, x1 - 1}

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

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


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

activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
take(X1, X2) → n__take(X1, X2)
take(0, IL) → U81(isNatIList(IL))
U81(tt) → nil
U31(tt) → tt
length(nil) → 0
length(X) → n__length(X)
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
U11(tt) → tt
U21(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
s(X) → n__s(X)
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
cons(X1, X2) → n__cons(X1, X2)
zeros → cons(0, n__zeros)
zeros → n__zeros
0 → n__0
nil → n__nil

(6) Obligation:

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

TAKE(0, IL) → ISNATILIST(IL)
ISNATILIST(V) → ISNATLIST(activate(V))
ISNATLIST(n__cons(V1, V2)) → U511(isNat(activate(V1)), activate(V2))
U511(tt, V2) → ISNATLIST(activate(V2))
LENGTH(cons(N, L)) → U711(isNatList(activate(L)), activate(L), N)
U711(tt, L, N) → U721(isNat(activate(N)), activate(L))
U721(tt, L) → LENGTH(activate(L))
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ISNATLIST(n__take(V1, V2)) → U611(isNat(activate(V1)), activate(V2))
ISNATILIST(n__cons(V1, V2)) → U411(isNat(activate(V1)), activate(V2))
U411(tt, V2) → ISNATILIST(activate(V2))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U911(tt, IL, M, N) → U921(isNat(activate(M)), activate(IL), activate(M), activate(N))
U921(tt, IL, M, N) → U931(isNat(activate(N)), activate(IL), activate(M), activate(N))
U931(tt, IL, M, N) → ACTIVATE(N)
U931(tt, IL, M, N) → ACTIVATE(M)
U931(tt, IL, M, N) → ACTIVATE(IL)
U921(tt, IL, M, N) → ISNAT(activate(N))
U921(tt, IL, M, N) → ACTIVATE(N)
U921(tt, IL, M, N) → ACTIVATE(IL)
U921(tt, IL, M, N) → ACTIVATE(M)
U911(tt, IL, M, N) → ISNAT(activate(M))
U911(tt, IL, M, N) → ACTIVATE(M)
U911(tt, IL, M, N) → ACTIVATE(IL)
U911(tt, IL, M, N) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
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 4 SCCs with 19 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

ISNAT(n__s(V1)) → ISNAT(activate(V1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(10) 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: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(ISNAT(x1)) = 4A + 5A·x1

POL(n__s(x1)) = 2A + 1A·x1

POL(activate(x1)) = 1A + 0A·x1

POL(n__zeros) = 0A

POL(zeros) = 1A

POL(n__take(x1, x2)) = 3A + 2A·x1 + 0A·x2

POL(take(x1, x2)) = 3A + 2A·x1 + 0A·x2

POL(n__0) = 0A

POL(0) = 0A

POL(n__length(x1)) = 5A + 3A·x1

POL(length(x1)) = 5A + 3A·x1

POL(s(x1)) = 2A + 1A·x1

POL(n__cons(x1, x2)) = -I + 1A·x1 + 1A·x2

POL(cons(x1, x2)) = 0A + 1A·x1 + 1A·x2

POL(n__nil) = 2A

POL(nil) = 2A

POL(U81(x1)) = 3A + -I·x1

POL(isNatIList(x1)) = 3A + 0A·x1

POL(tt) = 2A

POL(U31(x1)) = 0A + 0A·x1

POL(isNatList(x1)) = 1A + 0A·x1

POL(U71(x1, x2, x3)) = -I + 4A·x1 + 4A·x2 + 4A·x3

POL(U51(x1, x2)) = 1A + -I·x1 + 0A·x2

POL(isNat(x1)) = 2A + 0A·x1

POL(U11(x1)) = 3A + -I·x1

POL(U61(x1, x2)) = 0A + 1A·x1 + 0A·x2

POL(U21(x1)) = 2A + -I·x1

POL(U62(x1)) = -I + 0A·x1

POL(U41(x1, x2)) = 3A + -I·x1 + 1A·x2

POL(U42(x1)) = 2A + -I·x1

POL(U52(x1)) = 0A + 0A·x1

POL(U72(x1, x2)) = 0A + 4A·x1 + 4A·x2

POL(U91(x1, x2, x3, x4)) = 4A + -I·x1 + 1A·x2 + 3A·x3 + 1A·x4

POL(U92(x1, x2, x3, x4)) = 4A + 0A·x1 + 1A·x2 + 3A·x3 + 1A·x4

POL(U93(x1, x2, x3, x4)) = 4A + 0A·x1 + 1A·x2 + 3A·x3 + 1A·x4

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

activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
take(X1, X2) → n__take(X1, X2)
take(0, IL) → U81(isNatIList(IL))
isNatIList(n__zeros) → tt
U81(tt) → nil
isNatIList(V) → U31(isNatList(activate(V)))
isNatList(n__nil) → tt
U31(tt) → tt
length(nil) → 0
length(X) → n__length(X)
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
U11(tt) → tt
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
s(X) → n__s(X)
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
cons(X1, X2) → n__cons(X1, X2)
zeros → cons(0, n__zeros)
zeros → n__zeros
0 → n__0
nil → n__nil

(11) Obligation:

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

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(12) PisEmptyProof (EQUIVALENT transformation)

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

(13) YES

(14) Obligation:

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

U511(tt, V2) → ISNATLIST(activate(V2))
ISNATLIST(n__cons(V1, V2)) → U511(isNat(activate(V1)), activate(V2))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(15) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U511(tt, V2) → ISNATLIST(activate(V2)) at position [0] we obtained the following new rules [LPAR04]:

U511(tt, n__zeros) → ISNATLIST(zeros) → U511(tt, n__zeros) → ISNATLIST(zeros)
U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1)) → U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U511(tt, n__0) → ISNATLIST(0) → U511(tt, n__0) → ISNATLIST(0)
U511(tt, n__length(x0)) → ISNATLIST(length(x0)) → U511(tt, n__length(x0)) → ISNATLIST(length(x0))
U511(tt, n__s(x0)) → ISNATLIST(s(x0)) → U511(tt, n__s(x0)) → ISNATLIST(s(x0))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(x0, x1)) → U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(x0, x1))
U511(tt, n__nil) → ISNATLIST(nil) → U511(tt, n__nil) → ISNATLIST(nil)
U511(tt, x0) → ISNATLIST(x0) → U511(tt, x0) → ISNATLIST(x0)

(16) Obligation:

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

ISNATLIST(n__cons(V1, V2)) → U511(isNat(activate(V1)), activate(V2))
U511(tt, n__zeros) → ISNATLIST(zeros)
U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U511(tt, n__0) → ISNATLIST(0)
U511(tt, n__length(x0)) → ISNATLIST(length(x0))
U511(tt, n__s(x0)) → ISNATLIST(s(x0))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(x0, x1))
U511(tt, n__nil) → ISNATLIST(nil)
U511(tt, x0) → ISNATLIST(x0)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(17) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATLIST(n__cons(V1, V2)) → U511(isNat(activate(V1)), activate(V2)) at position [0] we obtained the following new rules [LPAR04]:

ISNATLIST(n__cons(n__zeros, y1)) → U511(isNat(zeros), activate(y1)) → ISNATLIST(n__cons(n__zeros, y1)) → U511(isNat(zeros), activate(y1))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1)) → ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
ISNATLIST(n__cons(n__0, y1)) → U511(isNat(0), activate(y1)) → ISNATLIST(n__cons(n__0, y1)) → U511(isNat(0), activate(y1))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), activate(y1)) → ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(x0)), activate(y1)) → ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(x0)), activate(y1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(x0, x1)), activate(y1)) → ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(x0, x1)), activate(y1))
ISNATLIST(n__cons(n__nil, y1)) → U511(isNat(nil), activate(y1)) → ISNATLIST(n__cons(n__nil, y1)) → U511(isNat(nil), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1)) → ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))

(18) Obligation:

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

U511(tt, n__zeros) → ISNATLIST(zeros)
U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U511(tt, n__0) → ISNATLIST(0)
U511(tt, n__length(x0)) → ISNATLIST(length(x0))
U511(tt, n__s(x0)) → ISNATLIST(s(x0))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(x0, x1))
U511(tt, n__nil) → ISNATLIST(nil)
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__zeros, y1)) → U511(isNat(zeros), activate(y1))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
ISNATLIST(n__cons(n__0, y1)) → U511(isNat(0), activate(y1))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(x0)), activate(y1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(x0, x1)), activate(y1))
ISNATLIST(n__cons(n__nil, y1)) → U511(isNat(nil), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(19) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U511(tt, n__zeros) → ISNATLIST(zeros) at position [0] we obtained the following new rules [LPAR04]:

U511(tt, n__zeros) → ISNATLIST(cons(0, n__zeros)) → U511(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
U511(tt, n__zeros) → ISNATLIST(n__zeros) → U511(tt, n__zeros) → ISNATLIST(n__zeros)

(20) Obligation:

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

U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U511(tt, n__0) → ISNATLIST(0)
U511(tt, n__length(x0)) → ISNATLIST(length(x0))
U511(tt, n__s(x0)) → ISNATLIST(s(x0))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(x0, x1))
U511(tt, n__nil) → ISNATLIST(nil)
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__zeros, y1)) → U511(isNat(zeros), activate(y1))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
ISNATLIST(n__cons(n__0, y1)) → U511(isNat(0), activate(y1))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(x0)), activate(y1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(x0, x1)), activate(y1))
ISNATLIST(n__cons(n__nil, y1)) → U511(isNat(nil), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
U511(tt, n__zeros) → ISNATLIST(n__zeros)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(21) DependencyGraphProof (EQUIVALENT transformation)

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

(22) Obligation:

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

ISNATLIST(n__cons(n__zeros, y1)) → U511(isNat(zeros), activate(y1))
U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
U511(tt, n__0) → ISNATLIST(0)
ISNATLIST(n__cons(n__0, y1)) → U511(isNat(0), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(x0))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(x0))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(x0)), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(x0, x1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(x0, x1)), activate(y1))
U511(tt, n__nil) → ISNATLIST(nil)
ISNATLIST(n__cons(n__nil, y1)) → U511(isNat(nil), activate(y1))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(23) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATLIST(n__cons(n__zeros, y1)) → U511(isNat(zeros), activate(y1)) at position [0] we obtained the following new rules [LPAR04]:

ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(0, n__zeros)), activate(y0)) → ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(0, n__zeros)), activate(y0))
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(n__zeros), activate(y0)) → ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(n__zeros), activate(y0))

(24) Obligation:

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

U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
U511(tt, n__0) → ISNATLIST(0)
ISNATLIST(n__cons(n__0, y1)) → U511(isNat(0), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(x0))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(x0))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(x0)), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(x0, x1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(x0, x1)), activate(y1))
U511(tt, n__nil) → ISNATLIST(nil)
ISNATLIST(n__cons(n__nil, y1)) → U511(isNat(nil), activate(y1))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(0, n__zeros)), activate(y0))
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(n__zeros), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(25) DependencyGraphProof (EQUIVALENT transformation)

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

(26) Obligation:

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

ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(n__0, y1)) → U511(isNat(0), activate(y1))
U511(tt, n__0) → ISNATLIST(0)
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(x0))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(x0)), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(x0))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(x0, x1)), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(x0, x1))
ISNATLIST(n__cons(n__nil, y1)) → U511(isNat(nil), activate(y1))
U511(tt, n__nil) → ISNATLIST(nil)
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(0, n__zeros)), activate(y0))
U511(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(27) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATLIST(n__cons(n__0, y1)) → U511(isNat(0), activate(y1)) at position [0] we obtained the following new rules [LPAR04]:

ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0)) → ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))

(28) Obligation:

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

ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U511(tt, n__0) → ISNATLIST(0)
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(x0))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(x0)), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(x0))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(x0, x1)), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(x0, x1))
ISNATLIST(n__cons(n__nil, y1)) → U511(isNat(nil), activate(y1))
U511(tt, n__nil) → ISNATLIST(nil)
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(0, n__zeros)), activate(y0))
U511(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(29) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U511(tt, n__0) → ISNATLIST(0) at position [0] we obtained the following new rules [LPAR04]:

U511(tt, n__0) → ISNATLIST(n__0) → U511(tt, n__0) → ISNATLIST(n__0)

(30) Obligation:

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

ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(x0))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(x0)), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(x0))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(x0, x1)), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(x0, x1))
ISNATLIST(n__cons(n__nil, y1)) → U511(isNat(nil), activate(y1))
U511(tt, n__nil) → ISNATLIST(nil)
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(0, n__zeros)), activate(y0))
U511(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
U511(tt, n__0) → ISNATLIST(n__0)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(31) DependencyGraphProof (EQUIVALENT transformation)

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

(32) Obligation:

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

U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(x0))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(x0))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(x0)), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(x0, x1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(x0, x1)), activate(y1))
U511(tt, n__nil) → ISNATLIST(nil)
ISNATLIST(n__cons(n__nil, y1)) → U511(isNat(nil), activate(y1))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(0, n__zeros)), activate(y0))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(33) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U511(tt, n__s(x0)) → ISNATLIST(s(x0)) at position [0] we obtained the following new rules [LPAR04]:

U511(tt, n__s(x0)) → ISNATLIST(n__s(x0)) → U511(tt, n__s(x0)) → ISNATLIST(n__s(x0))

(34) Obligation:

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

U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(x0))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(x0)), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(x0, x1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(x0, x1)), activate(y1))
U511(tt, n__nil) → ISNATLIST(nil)
ISNATLIST(n__cons(n__nil, y1)) → U511(isNat(nil), activate(y1))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(0, n__zeros)), activate(y0))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
U511(tt, n__s(x0)) → ISNATLIST(n__s(x0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(35) DependencyGraphProof (EQUIVALENT transformation)

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

(36) Obligation:

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

ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(x0))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(x0)), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(x0, x1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(x0, x1)), activate(y1))
U511(tt, n__nil) → ISNATLIST(nil)
ISNATLIST(n__cons(n__nil, y1)) → U511(isNat(nil), activate(y1))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(0, n__zeros)), activate(y0))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(37) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(x0)), activate(y1)) at position [0] we obtained the following new rules [LPAR04]:

ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(n__s(x0)), activate(y1)) → ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(n__s(x0)), activate(y1))

(38) Obligation:

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

ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(x0))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(x0, x1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(x0, x1)), activate(y1))
U511(tt, n__nil) → ISNATLIST(nil)
ISNATLIST(n__cons(n__nil, y1)) → U511(isNat(nil), activate(y1))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(0, n__zeros)), activate(y0))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(n__s(x0)), activate(y1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(39) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(x0, x1)) at position [0] we obtained the following new rules [LPAR04]:

U511(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1)) → U511(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))

(40) Obligation:

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

ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(x0))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(x0, x1)), activate(y1))
U511(tt, n__nil) → ISNATLIST(nil)
ISNATLIST(n__cons(n__nil, y1)) → U511(isNat(nil), activate(y1))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(0, n__zeros)), activate(y0))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(n__s(x0)), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(41) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(x0, x1)), activate(y1)) at position [0] we obtained the following new rules [LPAR04]:

ISNATLIST(n__cons(n__cons(x0, x1), y2)) → U511(isNat(n__cons(x0, x1)), activate(y2)) → ISNATLIST(n__cons(n__cons(x0, x1), y2)) → U511(isNat(n__cons(x0, x1)), activate(y2))

(42) Obligation:

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

ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(x0))
U511(tt, n__nil) → ISNATLIST(nil)
ISNATLIST(n__cons(n__nil, y1)) → U511(isNat(nil), activate(y1))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(0, n__zeros)), activate(y0))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(n__s(x0)), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
ISNATLIST(n__cons(n__cons(x0, x1), y2)) → U511(isNat(n__cons(x0, x1)), activate(y2))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(43) DependencyGraphProof (EQUIVALENT transformation)

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

(44) Obligation:

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

U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(x0))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
U511(tt, n__nil) → ISNATLIST(nil)
ISNATLIST(n__cons(n__nil, y1)) → U511(isNat(nil), activate(y1))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(0, n__zeros)), activate(y0))
U511(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(n__s(x0)), activate(y1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(45) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U511(tt, n__nil) → ISNATLIST(nil) at position [0] we obtained the following new rules [LPAR04]:

U511(tt, n__nil) → ISNATLIST(n__nil) → U511(tt, n__nil) → ISNATLIST(n__nil)

(46) Obligation:

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

U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(x0))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
ISNATLIST(n__cons(n__nil, y1)) → U511(isNat(nil), activate(y1))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(0, n__zeros)), activate(y0))
U511(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(n__s(x0)), activate(y1))
U511(tt, n__nil) → ISNATLIST(n__nil)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(47) DependencyGraphProof (EQUIVALENT transformation)

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

(48) Obligation:

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

ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(x0))
ISNATLIST(n__cons(n__nil, y1)) → U511(isNat(nil), activate(y1))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(0, n__zeros)), activate(y0))
U511(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(n__s(x0)), activate(y1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(49) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATLIST(n__cons(n__nil, y1)) → U511(isNat(nil), activate(y1)) at position [0] we obtained the following new rules [LPAR04]:

ISNATLIST(n__cons(n__nil, y0)) → U511(isNat(n__nil), activate(y0)) → ISNATLIST(n__cons(n__nil, y0)) → U511(isNat(n__nil), activate(y0))

(50) Obligation:

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

ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(x0))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(0, n__zeros)), activate(y0))
U511(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(n__s(x0)), activate(y1))
ISNATLIST(n__cons(n__nil, y0)) → U511(isNat(n__nil), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(51) DependencyGraphProof (EQUIVALENT transformation)

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

(52) Obligation:

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

U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(x0))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(0, n__zeros)), activate(y0))
U511(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(n__s(x0)), activate(y1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(53) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U511(tt, n__zeros) → ISNATLIST(cons(0, n__zeros)) at position [0] we obtained the following new rules [LPAR04]:

U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros)) → U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U511(tt, n__zeros) → ISNATLIST(cons(n__0, n__zeros)) → U511(tt, n__zeros) → ISNATLIST(cons(n__0, n__zeros))

(54) Obligation:

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

U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(x0))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(0, n__zeros)), activate(y0))
U511(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(n__s(x0)), activate(y1))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U511(tt, n__zeros) → ISNATLIST(cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(55) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(0, n__zeros)), activate(y0)) at position [0] we obtained the following new rules [LPAR04]:

ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(n__cons(0, n__zeros)), activate(y0)) → ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(n__cons(0, n__zeros)), activate(y0))
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(n__0, n__zeros)), activate(y0)) → ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(n__0, n__zeros)), activate(y0))

(56) Obligation:

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

U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(x0))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(n__s(x0)), activate(y1))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U511(tt, n__zeros) → ISNATLIST(cons(n__0, n__zeros))
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(n__cons(0, n__zeros)), activate(y0))
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(n__0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(57) DependencyGraphProof (EQUIVALENT transformation)

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

(58) Obligation:

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

ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(x0))
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
U511(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(n__s(x0)), activate(y1))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(n__0, n__zeros)), activate(y0))
U511(tt, n__zeros) → ISNATLIST(cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(59) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(n__0, n__zeros)), activate(y0)) at position [0] we obtained the following new rules [LPAR04]:

ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(n__cons(n__0, n__zeros)), activate(y0)) → ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(n__cons(n__0, n__zeros)), activate(y0))

(60) Obligation:

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

ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(x0))
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
U511(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(n__s(x0)), activate(y1))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U511(tt, n__zeros) → ISNATLIST(cons(n__0, n__zeros))
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(n__cons(n__0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(61) DependencyGraphProof (EQUIVALENT transformation)

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

(62) Obligation:

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

U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(x0))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(n__s(x0)), activate(y1))
U511(tt, n__zeros) → ISNATLIST(cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(63) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U511(tt, n__zeros) → ISNATLIST(cons(n__0, n__zeros)) at position [0] we obtained the following new rules [LPAR04]:

U511(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros)) → U511(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))

(64) Obligation:

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

U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(x0))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(n__s(x0)), activate(y1))
U511(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(65) 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__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(n__s(x0)), activate(y1))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( ISNATLIST(x1) ) = 2x1 + 1

POL( U511(x1, x2) ) = x1 + 2x2 + 1

POL( take(x1, x2) ) = x2

POL( 0 ) = 0

POL( U81(x1) ) = max{0, -2}

POL( isNatIList(x1) ) = 0

POL( s(x1) ) = 1

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

POL( U91(x1, ..., x4) ) = x1 + 2x2 + x4

POL( activate(x1) ) = x1

POL( n__take(x1, x2) ) = x2

POL( U21(x1) ) = 0

POL( U41(x1, x2) ) = 0

POL( U51(x1, x2) ) = max{0, -2}

POL( U61(x1, x2) ) = 1

POL( U72(x1, x2) ) = x1 + 1

POL( U92(x1, ..., x4) ) = 2x2 + x4

POL( U93(x1, ..., x4) ) = 2x2 + x4

POL( isNat(x1) ) = x1

POL( n__0 ) = 0

POL( tt ) = 0

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

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

POL( isNatList(x1) ) = 1

POL( n__s(x1) ) = 1

POL( U71(x1, ..., x3) ) = x3 + 1

POL( U31(x1) ) = max{0, -2}

POL( U42(x1) ) = 0

POL( U52(x1) ) = max{0, x1 - 2}

POL( U62(x1) ) = max{0, -2}

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

POL( n__zeros ) = 0

POL( zeros ) = 0

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

POL( n__nil ) = 0

POL( nil ) = 0


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

take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
take(X1, X2) → n__take(X1, X2)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
0 → n__0
isNatIList(n__zeros) → tt
U81(tt) → nil
isNatIList(V) → U31(isNatList(activate(V)))
isNatList(n__nil) → tt
U31(tt) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U11(tt) → tt
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
U21(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
s(X) → n__s(X)
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
cons(X1, X2) → n__cons(X1, X2)
zeros → cons(0, n__zeros)
zeros → n__zeros
nil → n__nil

(66) Obligation:

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

U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(x0))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U511(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(67) 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__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( ISNATLIST(x1) ) = 2x1

POL( U511(x1, x2) ) = 2x1 + 2x2

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

POL( 0 ) = 0

POL( U81(x1) ) = max{0, -2}

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

POL( s(x1) ) = 0

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

POL( U91(x1, ..., x4) ) = x2 + 2x4 + 2

POL( activate(x1) ) = x1

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

POL( U21(x1) ) = max{0, -2}

POL( U41(x1, x2) ) = x1 + 1

POL( U51(x1, x2) ) = 1

POL( U61(x1, x2) ) = max{0, -2}

POL( U72(x1, x2) ) = 0

POL( U92(x1, ..., x4) ) = x2 + 2x4 + 2

POL( U93(x1, ..., x4) ) = x2 + 2x4 + 2

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

POL( n__0 ) = 0

POL( tt ) = 0

POL( n__length(x1) ) = 0

POL( U11(x1) ) = 0

POL( isNatList(x1) ) = 1

POL( n__s(x1) ) = 0

POL( U71(x1, ..., x3) ) = max{0, x1 - 2}

POL( U31(x1) ) = 0

POL( U42(x1) ) = max{0, -1}

POL( U52(x1) ) = 1

POL( U62(x1) ) = max{0, -2}

POL( length(x1) ) = 0

POL( n__zeros ) = 0

POL( zeros ) = 0

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

POL( n__nil ) = 0

POL( nil ) = 0


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

take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
take(X1, X2) → n__take(X1, X2)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
0 → n__0
U81(tt) → nil
isNatIList(V) → U31(isNatList(activate(V)))
isNatList(n__nil) → tt
U31(tt) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U11(tt) → tt
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
U21(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
s(X) → n__s(X)
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
cons(X1, X2) → n__cons(X1, X2)
zeros → cons(0, n__zeros)
zeros → n__zeros
nil → n__nil

(68) Obligation:

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

U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U511(tt, n__length(x0)) → ISNATLIST(length(x0))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U511(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(69) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


U511(tt, n__length(x0)) → ISNATLIST(length(x0))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( ISNATLIST(x1) ) = x1 + 1

POL( U511(x1, x2) ) = 2x2 + 1

POL( take(x1, x2) ) = 0

POL( 0 ) = 0

POL( U81(x1) ) = max{0, -2}

POL( isNatIList(x1) ) = 2

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

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

POL( U91(x1, ..., x4) ) = 0

POL( activate(x1) ) = x1

POL( n__take(x1, x2) ) = 0

POL( length(x1) ) = 2

POL( nil ) = 0

POL( U71(x1, ..., x3) ) = 2

POL( isNatList(x1) ) = x1 + 2

POL( n__length(x1) ) = 2

POL( U21(x1) ) = 0

POL( U41(x1, x2) ) = 2

POL( U51(x1, x2) ) = 2x2 + 2

POL( U61(x1, x2) ) = 1

POL( U72(x1, x2) ) = 0

POL( U92(x1, ..., x4) ) = 0

POL( U93(x1, ..., x4) ) = max{0, x1 - 1}

POL( isNat(x1) ) = 1

POL( n__0 ) = 0

POL( tt ) = 0

POL( U11(x1) ) = 1

POL( n__s(x1) ) = 0

POL( U31(x1) ) = max{0, -2}

POL( U42(x1) ) = 2

POL( U52(x1) ) = 2

POL( U62(x1) ) = max{0, x1 - 2}

POL( n__zeros ) = 0

POL( zeros ) = 0

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

POL( n__nil ) = 0


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

take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
take(X1, X2) → n__take(X1, X2)
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
0 → n__0
isNatIList(n__zeros) → tt
U81(tt) → nil
isNatIList(V) → U31(isNatList(activate(V)))
U31(tt) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U11(tt) → tt
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
U21(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
s(X) → n__s(X)
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
cons(X1, X2) → n__cons(X1, X2)
zeros → cons(0, n__zeros)
zeros → n__zeros
nil → n__nil

(70) Obligation:

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

U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U511(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(71) Obligation:

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

LENGTH(cons(N, L)) → U711(isNatList(activate(L)), activate(L), N)
U711(tt, L, N) → U721(isNat(activate(N)), activate(L))
U721(tt, L) → LENGTH(activate(L))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
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.


U711(tt, L, N) → U721(isNat(activate(N)), activate(L))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( LENGTH(x1) ) = 2x1

POL( U711(x1, ..., x3) ) = max{0, x1 + 2x2 - 1}

POL( U721(x1, x2) ) = 2x2

POL( U41(x1, x2) ) = 2

POL( U51(x1, x2) ) = 2x2

POL( U61(x1, x2) ) = x1

POL( U71(x1, ..., x3) ) = 2x2

POL( U72(x1, x2) ) = 2x2

POL( U91(x1, ..., x4) ) = 2x2 + 2x3

POL( U92(x1, ..., x4) ) = 2x2 + 2x3

POL( U93(x1, ..., x4) ) = 2x2 + 2x3

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

POL( isNatList(x1) ) = 2x1

POL( isNat(x1) ) = 2x1

POL( U11(x1) ) = x1

POL( U21(x1) ) = x1

POL( U31(x1) ) = 2

POL( U42(x1) ) = x1

POL( isNatIList(x1) ) = 2

POL( U52(x1) ) = x1

POL( U62(x1) ) = 2

POL( n__take(x1, x2) ) = x1 + x2

POL( length(x1) ) = x1

POL( s(x1) ) = 2x1

POL( activate(x1) ) = x1

POL( n__zeros ) = 0

POL( zeros ) = 0

POL( take(x1, x2) ) = x1 + x2

POL( n__0 ) = 2

POL( 0 ) = 2

POL( n__length(x1) ) = x1

POL( n__s(x1) ) = 2x1

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

POL( n__nil ) = 2

POL( nil ) = 2

POL( tt ) = 2

POL( U81(x1) ) = x1


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

activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
take(X1, X2) → n__take(X1, X2)
take(0, IL) → U81(isNatIList(IL))
isNatIList(n__zeros) → tt
U81(tt) → nil
isNatIList(V) → U31(isNatList(activate(V)))
U31(tt) → tt
length(nil) → 0
length(X) → n__length(X)
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
U11(tt) → tt
U21(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
s(X) → n__s(X)
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
cons(X1, X2) → n__cons(X1, X2)
zeros → cons(0, n__zeros)
zeros → n__zeros
0 → n__0
nil → n__nil

(73) Obligation:

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

LENGTH(cons(N, L)) → U711(isNatList(activate(L)), activate(L), N)
U721(tt, L) → LENGTH(activate(L))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(74) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(75) TRUE

(76) Obligation:

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

ISNATILIST(n__cons(V1, V2)) → U411(isNat(activate(V1)), activate(V2))
U411(tt, V2) → ISNATILIST(activate(V2))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(77) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATILIST(n__cons(V1, V2)) → U411(isNat(activate(V1)), activate(V2)) at position [0] we obtained the following new rules [LPAR04]:

ISNATILIST(n__cons(n__zeros, y1)) → U411(isNat(zeros), activate(y1)) → ISNATILIST(n__cons(n__zeros, y1)) → U411(isNat(zeros), activate(y1))
ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(x0, x1)), activate(y1)) → ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(x0, x1)), activate(y1))
ISNATILIST(n__cons(n__0, y1)) → U411(isNat(0), activate(y1)) → ISNATILIST(n__cons(n__0, y1)) → U411(isNat(0), activate(y1))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(x0)), activate(y1)) → ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(x0)), activate(y1))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(x0)), activate(y1)) → ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(x0)), activate(y1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(x0, x1)), activate(y1)) → ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(x0, x1)), activate(y1))
ISNATILIST(n__cons(n__nil, y1)) → U411(isNat(nil), activate(y1)) → ISNATILIST(n__cons(n__nil, y1)) → U411(isNat(nil), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1)) → ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))

(78) Obligation:

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

U411(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(n__cons(n__zeros, y1)) → U411(isNat(zeros), activate(y1))
ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(x0, x1)), activate(y1))
ISNATILIST(n__cons(n__0, y1)) → U411(isNat(0), activate(y1))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(x0)), activate(y1))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(x0)), activate(y1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(x0, x1)), activate(y1))
ISNATILIST(n__cons(n__nil, y1)) → U411(isNat(nil), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(79) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U411(tt, V2) → ISNATILIST(activate(V2)) at position [0] we obtained the following new rules [LPAR04]:

U411(tt, n__zeros) → ISNATILIST(zeros) → U411(tt, n__zeros) → ISNATILIST(zeros)
U411(tt, n__take(x0, x1)) → ISNATILIST(take(x0, x1)) → U411(tt, n__take(x0, x1)) → ISNATILIST(take(x0, x1))
U411(tt, n__0) → ISNATILIST(0) → U411(tt, n__0) → ISNATILIST(0)
U411(tt, n__length(x0)) → ISNATILIST(length(x0)) → U411(tt, n__length(x0)) → ISNATILIST(length(x0))
U411(tt, n__s(x0)) → ISNATILIST(s(x0)) → U411(tt, n__s(x0)) → ISNATILIST(s(x0))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(x0, x1)) → U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(x0, x1))
U411(tt, n__nil) → ISNATILIST(nil) → U411(tt, n__nil) → ISNATILIST(nil)
U411(tt, x0) → ISNATILIST(x0) → U411(tt, x0) → ISNATILIST(x0)

(80) Obligation:

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

ISNATILIST(n__cons(n__zeros, y1)) → U411(isNat(zeros), activate(y1))
ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(x0, x1)), activate(y1))
ISNATILIST(n__cons(n__0, y1)) → U411(isNat(0), activate(y1))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(x0)), activate(y1))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(x0)), activate(y1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(x0, x1)), activate(y1))
ISNATILIST(n__cons(n__nil, y1)) → U411(isNat(nil), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
U411(tt, n__zeros) → ISNATILIST(zeros)
U411(tt, n__take(x0, x1)) → ISNATILIST(take(x0, x1))
U411(tt, n__0) → ISNATILIST(0)
U411(tt, n__length(x0)) → ISNATILIST(length(x0))
U411(tt, n__s(x0)) → ISNATILIST(s(x0))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(x0, x1))
U411(tt, n__nil) → ISNATILIST(nil)
U411(tt, x0) → ISNATILIST(x0)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(81) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATILIST(n__cons(n__zeros, y1)) → U411(isNat(zeros), activate(y1)) at position [0] we obtained the following new rules [LPAR04]:

ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(cons(0, n__zeros)), activate(y0)) → ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(cons(0, n__zeros)), activate(y0))
ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(n__zeros), activate(y0)) → ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(n__zeros), activate(y0))

(82) Obligation:

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

ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(x0, x1)), activate(y1))
ISNATILIST(n__cons(n__0, y1)) → U411(isNat(0), activate(y1))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(x0)), activate(y1))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(x0)), activate(y1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(x0, x1)), activate(y1))
ISNATILIST(n__cons(n__nil, y1)) → U411(isNat(nil), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
U411(tt, n__zeros) → ISNATILIST(zeros)
U411(tt, n__take(x0, x1)) → ISNATILIST(take(x0, x1))
U411(tt, n__0) → ISNATILIST(0)
U411(tt, n__length(x0)) → ISNATILIST(length(x0))
U411(tt, n__s(x0)) → ISNATILIST(s(x0))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(x0, x1))
U411(tt, n__nil) → ISNATILIST(nil)
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(cons(0, n__zeros)), activate(y0))
ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(n__zeros), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(83) DependencyGraphProof (EQUIVALENT transformation)

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

(84) Obligation:

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

U411(tt, n__zeros) → ISNATILIST(zeros)
ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(x0, x1)), activate(y1))
U411(tt, n__take(x0, x1)) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(n__0, y1)) → U411(isNat(0), activate(y1))
U411(tt, n__0) → ISNATILIST(0)
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(x0)), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(x0))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(x0)), activate(y1))
U411(tt, n__s(x0)) → ISNATILIST(s(x0))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(x0, x1)), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(x0, x1))
ISNATILIST(n__cons(n__nil, y1)) → U411(isNat(nil), activate(y1))
U411(tt, n__nil) → ISNATILIST(nil)
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(cons(0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(85) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U411(tt, n__zeros) → ISNATILIST(zeros) at position [0] we obtained the following new rules [LPAR04]:

U411(tt, n__zeros) → ISNATILIST(cons(0, n__zeros)) → U411(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
U411(tt, n__zeros) → ISNATILIST(n__zeros) → U411(tt, n__zeros) → ISNATILIST(n__zeros)

(86) Obligation:

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

ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(x0, x1)), activate(y1))
U411(tt, n__take(x0, x1)) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(n__0, y1)) → U411(isNat(0), activate(y1))
U411(tt, n__0) → ISNATILIST(0)
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(x0)), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(x0))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(x0)), activate(y1))
U411(tt, n__s(x0)) → ISNATILIST(s(x0))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(x0, x1)), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(x0, x1))
ISNATILIST(n__cons(n__nil, y1)) → U411(isNat(nil), activate(y1))
U411(tt, n__nil) → ISNATILIST(nil)
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(cons(0, n__zeros)), activate(y0))
U411(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
U411(tt, n__zeros) → ISNATILIST(n__zeros)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(87) DependencyGraphProof (EQUIVALENT transformation)

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

(88) Obligation:

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

U411(tt, n__take(x0, x1)) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(x0, x1)), activate(y1))
U411(tt, n__0) → ISNATILIST(0)
ISNATILIST(n__cons(n__0, y1)) → U411(isNat(0), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(x0))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(x0)), activate(y1))
U411(tt, n__s(x0)) → ISNATILIST(s(x0))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(x0)), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(x0, x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(x0, x1)), activate(y1))
U411(tt, n__nil) → ISNATILIST(nil)
ISNATILIST(n__cons(n__nil, y1)) → U411(isNat(nil), activate(y1))
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
U411(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(cons(0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(89) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U411(tt, n__0) → ISNATILIST(0) at position [0] we obtained the following new rules [LPAR04]:

U411(tt, n__0) → ISNATILIST(n__0) → U411(tt, n__0) → ISNATILIST(n__0)

(90) Obligation:

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

U411(tt, n__take(x0, x1)) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(x0, x1)), activate(y1))
ISNATILIST(n__cons(n__0, y1)) → U411(isNat(0), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(x0))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(x0)), activate(y1))
U411(tt, n__s(x0)) → ISNATILIST(s(x0))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(x0)), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(x0, x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(x0, x1)), activate(y1))
U411(tt, n__nil) → ISNATILIST(nil)
ISNATILIST(n__cons(n__nil, y1)) → U411(isNat(nil), activate(y1))
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
U411(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(cons(0, n__zeros)), activate(y0))
U411(tt, n__0) → ISNATILIST(n__0)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(91) DependencyGraphProof (EQUIVALENT transformation)

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

(92) Obligation:

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

ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(x0, x1)), activate(y1))
U411(tt, n__take(x0, x1)) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(n__0, y1)) → U411(isNat(0), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(x0))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(x0)), activate(y1))
U411(tt, n__s(x0)) → ISNATILIST(s(x0))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(x0)), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(x0, x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(x0, x1)), activate(y1))
U411(tt, n__nil) → ISNATILIST(nil)
ISNATILIST(n__cons(n__nil, y1)) → U411(isNat(nil), activate(y1))
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
U411(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(cons(0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(93) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATILIST(n__cons(n__0, y1)) → U411(isNat(0), activate(y1)) at position [0] we obtained the following new rules [LPAR04]:

ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0)) → ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))

(94) Obligation:

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

ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(x0, x1)), activate(y1))
U411(tt, n__take(x0, x1)) → ISNATILIST(take(x0, x1))
U411(tt, n__length(x0)) → ISNATILIST(length(x0))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(x0)), activate(y1))
U411(tt, n__s(x0)) → ISNATILIST(s(x0))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(x0)), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(x0, x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(x0, x1)), activate(y1))
U411(tt, n__nil) → ISNATILIST(nil)
ISNATILIST(n__cons(n__nil, y1)) → U411(isNat(nil), activate(y1))
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
U411(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(cons(0, n__zeros)), activate(y0))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(95) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U411(tt, n__s(x0)) → ISNATILIST(s(x0)) at position [0] we obtained the following new rules [LPAR04]:

U411(tt, n__s(x0)) → ISNATILIST(n__s(x0)) → U411(tt, n__s(x0)) → ISNATILIST(n__s(x0))

(96) Obligation:

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

ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(x0, x1)), activate(y1))
U411(tt, n__take(x0, x1)) → ISNATILIST(take(x0, x1))
U411(tt, n__length(x0)) → ISNATILIST(length(x0))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(x0)), activate(y1))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(x0)), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(x0, x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(x0, x1)), activate(y1))
U411(tt, n__nil) → ISNATILIST(nil)
ISNATILIST(n__cons(n__nil, y1)) → U411(isNat(nil), activate(y1))
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
U411(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(cons(0, n__zeros)), activate(y0))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
U411(tt, n__s(x0)) → ISNATILIST(n__s(x0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(97) DependencyGraphProof (EQUIVALENT transformation)

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

(98) Obligation:

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

U411(tt, n__take(x0, x1)) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(x0, x1)), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(x0))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(x0)), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(x0, x1))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(x0)), activate(y1))
U411(tt, n__nil) → ISNATILIST(nil)
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(x0, x1)), activate(y1))
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__nil, y1)) → U411(isNat(nil), activate(y1))
U411(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(cons(0, n__zeros)), activate(y0))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(99) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(x0, x1)) at position [0] we obtained the following new rules [LPAR04]:

U411(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1)) → U411(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))

(100) Obligation:

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

U411(tt, n__take(x0, x1)) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(x0, x1)), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(x0))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(x0)), activate(y1))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(x0)), activate(y1))
U411(tt, n__nil) → ISNATILIST(nil)
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(x0, x1)), activate(y1))
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__nil, y1)) → U411(isNat(nil), activate(y1))
U411(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(cons(0, n__zeros)), activate(y0))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
U411(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(101) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(x0)), activate(y1)) at position [0] we obtained the following new rules [LPAR04]:

ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(n__s(x0)), activate(y1)) → ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(n__s(x0)), activate(y1))

(102) Obligation:

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

U411(tt, n__take(x0, x1)) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(x0, x1)), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(x0))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(x0)), activate(y1))
U411(tt, n__nil) → ISNATILIST(nil)
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(x0, x1)), activate(y1))
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__nil, y1)) → U411(isNat(nil), activate(y1))
U411(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(cons(0, n__zeros)), activate(y0))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
U411(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(n__s(x0)), activate(y1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(103) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U411(tt, n__nil) → ISNATILIST(nil) at position [0] we obtained the following new rules [LPAR04]:

U411(tt, n__nil) → ISNATILIST(n__nil) → U411(tt, n__nil) → ISNATILIST(n__nil)

(104) Obligation:

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

U411(tt, n__take(x0, x1)) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(x0, x1)), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(x0))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(x0)), activate(y1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(x0, x1)), activate(y1))
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__nil, y1)) → U411(isNat(nil), activate(y1))
U411(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(cons(0, n__zeros)), activate(y0))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
U411(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(n__s(x0)), activate(y1))
U411(tt, n__nil) → ISNATILIST(n__nil)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(105) DependencyGraphProof (EQUIVALENT transformation)

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

(106) Obligation:

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

ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(x0, x1)), activate(y1))
U411(tt, n__take(x0, x1)) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(x0)), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(x0))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(x0, x1)), activate(y1))
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__nil, y1)) → U411(isNat(nil), activate(y1))
U411(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))
ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(cons(0, n__zeros)), activate(y0))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(n__s(x0)), activate(y1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(107) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(x0, x1)), activate(y1)) at position [0] we obtained the following new rules [LPAR04]:

ISNATILIST(n__cons(n__cons(x0, x1), y2)) → U411(isNat(n__cons(x0, x1)), activate(y2)) → ISNATILIST(n__cons(n__cons(x0, x1), y2)) → U411(isNat(n__cons(x0, x1)), activate(y2))

(108) Obligation:

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

ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(x0, x1)), activate(y1))
U411(tt, n__take(x0, x1)) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(x0)), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(x0))
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__nil, y1)) → U411(isNat(nil), activate(y1))
U411(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))
ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(cons(0, n__zeros)), activate(y0))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(n__s(x0)), activate(y1))
ISNATILIST(n__cons(n__cons(x0, x1), y2)) → U411(isNat(n__cons(x0, x1)), activate(y2))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(109) DependencyGraphProof (EQUIVALENT transformation)

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

(110) Obligation:

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

U411(tt, n__take(x0, x1)) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(x0, x1)), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(x0))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(x0)), activate(y1))
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__nil, y1)) → U411(isNat(nil), activate(y1))
U411(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))
ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(cons(0, n__zeros)), activate(y0))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(n__s(x0)), activate(y1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(111) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATILIST(n__cons(n__nil, y1)) → U411(isNat(nil), activate(y1)) at position [0] we obtained the following new rules [LPAR04]:

ISNATILIST(n__cons(n__nil, y0)) → U411(isNat(n__nil), activate(y0)) → ISNATILIST(n__cons(n__nil, y0)) → U411(isNat(n__nil), activate(y0))

(112) Obligation:

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

U411(tt, n__take(x0, x1)) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(x0, x1)), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(x0))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(x0)), activate(y1))
U411(tt, x0) → ISNATILIST(x0)
U411(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))
ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(cons(0, n__zeros)), activate(y0))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(n__s(x0)), activate(y1))
ISNATILIST(n__cons(n__nil, y0)) → U411(isNat(n__nil), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(113) DependencyGraphProof (EQUIVALENT transformation)

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

(114) Obligation:

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

ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(x0, x1)), activate(y1))
U411(tt, n__take(x0, x1)) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(x0)), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(x0))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(cons(0, n__zeros)), activate(y0))
U411(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
U411(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(n__s(x0)), activate(y1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(115) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(cons(0, n__zeros)), activate(y0)) at position [0] we obtained the following new rules [LPAR04]:

ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(n__cons(0, n__zeros)), activate(y0)) → ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(n__cons(0, n__zeros)), activate(y0))
ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(cons(n__0, n__zeros)), activate(y0)) → ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(cons(n__0, n__zeros)), activate(y0))

(116) Obligation:

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

ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(x0, x1)), activate(y1))
U411(tt, n__take(x0, x1)) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(x0)), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(x0))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
U411(tt, x0) → ISNATILIST(x0)
U411(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
U411(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(n__s(x0)), activate(y1))
ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(n__cons(0, n__zeros)), activate(y0))
ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(cons(n__0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(117) DependencyGraphProof (EQUIVALENT transformation)

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

(118) Obligation:

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

U411(tt, n__take(x0, x1)) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(x0, x1)), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(x0))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(x0)), activate(y1))
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
U411(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
U411(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(n__s(x0)), activate(y1))
ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(cons(n__0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(119) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U411(tt, n__zeros) → ISNATILIST(cons(0, n__zeros)) at position [0] we obtained the following new rules [LPAR04]:

U411(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros)) → U411(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U411(tt, n__zeros) → ISNATILIST(cons(n__0, n__zeros)) → U411(tt, n__zeros) → ISNATILIST(cons(n__0, n__zeros))

(120) Obligation:

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

U411(tt, n__take(x0, x1)) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(x0, x1)), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(x0))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(x0)), activate(y1))
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
U411(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(n__s(x0)), activate(y1))
ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(cons(n__0, n__zeros)), activate(y0))
U411(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U411(tt, n__zeros) → ISNATILIST(cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(121) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(cons(n__0, n__zeros)), activate(y0)) at position [0] we obtained the following new rules [LPAR04]:

ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(n__cons(n__0, n__zeros)), activate(y0)) → ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(n__cons(n__0, n__zeros)), activate(y0))

(122) Obligation:

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

U411(tt, n__take(x0, x1)) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(x0, x1)), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(x0))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(x0)), activate(y1))
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
U411(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(n__s(x0)), activate(y1))
U411(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U411(tt, n__zeros) → ISNATILIST(cons(n__0, n__zeros))
ISNATILIST(n__cons(n__zeros, y0)) → U411(isNat(n__cons(n__0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(123) DependencyGraphProof (EQUIVALENT transformation)

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

(124) Obligation:

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

ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(x0, x1)), activate(y1))
U411(tt, n__take(x0, x1)) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(x0)), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(x0))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
U411(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(n__s(x0)), activate(y1))
U411(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U411(tt, n__zeros) → ISNATILIST(cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(125) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U411(tt, n__zeros) → ISNATILIST(cons(n__0, n__zeros)) at position [0] we obtained the following new rules [LPAR04]:

U411(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros)) → U411(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))

(126) Obligation:

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

ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(x0, x1)), activate(y1))
U411(tt, n__take(x0, x1)) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(x0)), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(x0))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
U411(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(n__s(x0)), activate(y1))
U411(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U411(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(127) 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__take(x0, x1), y1)) → U411(isNat(take(x0, x1)), activate(y1))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( ISNATILIST(x1) ) = 2x1 + 1

POL( U411(x1, x2) ) = max{0, x1 + 2x2 - 1}

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

POL( 0 ) = 0

POL( U81(x1) ) = x1

POL( isNatIList(x1) ) = x1 + 2

POL( s(x1) ) = 2x1

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

POL( U91(x1, ..., x4) ) = 2x2 + 2x3 + 2x4 + 2

POL( activate(x1) ) = x1

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

POL( U21(x1) ) = 2

POL( U41(x1, x2) ) = 2

POL( U51(x1, x2) ) = 2

POL( U61(x1, x2) ) = 2

POL( U72(x1, x2) ) = max{0, -1}

POL( U92(x1, ..., x4) ) = 2x2 + 2x3 + 2x4 + 2

POL( U93(x1, ..., x4) ) = x1 + 2x2 + 2x3 + x4

POL( isNat(x1) ) = x1 + 2

POL( n__0 ) = 0

POL( tt ) = 2

POL( n__length(x1) ) = 0

POL( U11(x1) ) = x1

POL( isNatList(x1) ) = 2

POL( n__s(x1) ) = 2x1

POL( U71(x1, ..., x3) ) = max{0, -1}

POL( U31(x1) ) = max{0, 2x1 - 2}

POL( U42(x1) ) = 2

POL( U52(x1) ) = x1

POL( U62(x1) ) = 2

POL( length(x1) ) = 0

POL( n__zeros ) = 1

POL( zeros ) = 1

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

POL( n__nil ) = 2

POL( nil ) = 2


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

take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
take(X1, X2) → n__take(X1, X2)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
0 → n__0
isNatIList(n__zeros) → tt
U81(tt) → nil
isNatIList(V) → U31(isNatList(activate(V)))
isNatList(n__nil) → tt
U31(tt) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U11(tt) → tt
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
U21(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
s(X) → n__s(X)
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
cons(X1, X2) → n__cons(X1, X2)
zeros → cons(0, n__zeros)
zeros → n__zeros
nil → n__nil

(128) Obligation:

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

U411(tt, n__take(x0, x1)) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(x0)), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(x0))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
U411(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(n__s(x0)), activate(y1))
U411(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U411(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(129) 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__length(x0), y1)) → U411(isNat(length(x0)), activate(y1))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( ISNATILIST(x1) ) = 2x1

POL( U411(x1, x2) ) = 2x2

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

POL( 0 ) = 0

POL( U81(x1) ) = max{0, -2}

POL( isNatIList(x1) ) = 2x1 + 2

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

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

POL( U91(x1, ..., x4) ) = x2 + 2x4 + 1

POL( activate(x1) ) = x1

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

POL( length(x1) ) = x1 + 2

POL( nil ) = 0

POL( U71(x1, ..., x3) ) = max{0, x1 - 2}

POL( isNatList(x1) ) = x1 + 2

POL( n__length(x1) ) = x1 + 2

POL( U21(x1) ) = x1

POL( U41(x1, x2) ) = x1 + 2x2 + 1

POL( U51(x1, x2) ) = x2 + 1

POL( U61(x1, x2) ) = x1 + x2 + 2

POL( U72(x1, x2) ) = max{0, x1 - 2}

POL( U92(x1, ..., x4) ) = max{0, 2x1 + x2 + 2x4 - 1}

POL( U93(x1, ..., x4) ) = max{0, 2x1 + x2 + 2x4 - 1}

POL( isNat(x1) ) = 1

POL( n__0 ) = 0

POL( tt ) = 1

POL( U11(x1) ) = 1

POL( n__s(x1) ) = 0

POL( U31(x1) ) = x1

POL( U42(x1) ) = x1

POL( U52(x1) ) = 1

POL( U62(x1) ) = 2

POL( n__zeros ) = 0

POL( zeros ) = 0

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

POL( n__nil ) = 0


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

take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
take(X1, X2) → n__take(X1, X2)
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
0 → n__0
isNatIList(n__zeros) → tt
U81(tt) → nil
isNatIList(V) → U31(isNatList(activate(V)))
isNatList(n__nil) → tt
U31(tt) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U11(tt) → tt
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
U21(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
s(X) → n__s(X)
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
cons(X1, X2) → n__cons(X1, X2)
zeros → cons(0, n__zeros)
zeros → n__zeros
nil → n__nil

(130) Obligation:

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

U411(tt, n__take(x0, x1)) → ISNATILIST(take(x0, x1))
U411(tt, n__length(x0)) → ISNATILIST(length(x0))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
U411(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(n__s(x0)), activate(y1))
U411(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U411(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(131) 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)) → U411(isNat(n__s(x0)), activate(y1))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( ISNATILIST(x1) ) = 2x1

POL( U411(x1, x2) ) = x1 + 2x2

POL( take(x1, x2) ) = x2

POL( 0 ) = 0

POL( U81(x1) ) = 0

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

POL( s(x1) ) = 1

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

POL( U91(x1, ..., x4) ) = 2x2 + 2x4

POL( activate(x1) ) = x1

POL( n__take(x1, x2) ) = x2

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

POL( nil ) = 0

POL( U71(x1, ..., x3) ) = x1 + 1

POL( isNatList(x1) ) = 0

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

POL( U21(x1) ) = 1

POL( U41(x1, x2) ) = 2x2 + 1

POL( U51(x1, x2) ) = 0

POL( U61(x1, x2) ) = 0

POL( U72(x1, x2) ) = 1

POL( U92(x1, ..., x4) ) = 2x2 + 2x4

POL( U93(x1, ..., x4) ) = 2x2 + 2x4

POL( isNat(x1) ) = 2x1

POL( n__0 ) = 0

POL( tt ) = 0

POL( U11(x1) ) = x1 + 1

POL( n__s(x1) ) = 1

POL( U31(x1) ) = x1 + 1

POL( U42(x1) ) = x1

POL( U52(x1) ) = 0

POL( U62(x1) ) = 0

POL( n__zeros ) = 0

POL( zeros ) = 0

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

POL( n__nil ) = 0


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

take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
take(X1, X2) → n__take(X1, X2)
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
0 → n__0
isNatIList(n__zeros) → tt
U81(tt) → nil
isNatIList(V) → U31(isNatList(activate(V)))
isNatList(n__nil) → tt
U31(tt) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U11(tt) → tt
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
U21(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
s(X) → n__s(X)
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
cons(X1, X2) → n__cons(X1, X2)
zeros → cons(0, n__zeros)
zeros → n__zeros
nil → n__nil

(132) Obligation:

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

U411(tt, n__take(x0, x1)) → ISNATILIST(take(x0, x1))
U411(tt, n__length(x0)) → ISNATILIST(length(x0))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
U411(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))
U411(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U411(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(133) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


U411(tt, n__length(x0)) → ISNATILIST(length(x0))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( ISNATILIST(x1) ) = x1

POL( U411(x1, x2) ) = x1 + 2x2

POL( take(x1, x2) ) = 0

POL( 0 ) = 1

POL( U81(x1) ) = 0

POL( isNatIList(x1) ) = 1

POL( s(x1) ) = 1

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

POL( U91(x1, ..., x4) ) = max{0, -1}

POL( activate(x1) ) = x1

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

POL( length(x1) ) = 2x1 + 2

POL( nil ) = 0

POL( U71(x1, ..., x3) ) = 2x2 + 2

POL( isNatList(x1) ) = x1 + 2

POL( n__length(x1) ) = 2x1 + 2

POL( U21(x1) ) = max{0, -2}

POL( U41(x1, x2) ) = max{0, -2}

POL( U51(x1, x2) ) = 2x2 + 2

POL( U61(x1, x2) ) = 2

POL( U72(x1, x2) ) = 2

POL( U92(x1, ..., x4) ) = 0

POL( U93(x1, ..., x4) ) = max{0, -2}

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

POL( n__0 ) = 1

POL( tt ) = 0

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

POL( n__s(x1) ) = 1

POL( U31(x1) ) = max{0, -2}

POL( U42(x1) ) = max{0, 2x1 - 2}

POL( U52(x1) ) = 2

POL( U62(x1) ) = 2

POL( n__zeros ) = 0

POL( zeros ) = 0

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

POL( n__nil ) = 0


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

take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
take(X1, X2) → n__take(X1, X2)
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
0 → n__0
isNatIList(n__zeros) → tt
U81(tt) → nil
isNatIList(V) → U31(isNatList(activate(V)))
U31(tt) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U11(tt) → tt
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
U21(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
s(X) → n__s(X)
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
cons(X1, X2) → n__cons(X1, X2)
zeros → cons(0, n__zeros)
zeros → n__zeros
nil → n__nil

(134) Obligation:

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

U411(tt, n__take(x0, x1)) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
U411(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))
U411(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U411(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(135) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


U411(tt, n__take(x0, x1)) → ISNATILIST(take(x0, x1))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

POL(U411(x1, x2)) = 1 +
[0,1]
·x1 +
[1,1]
·x2

POL(tt) =
/1\
\0/

POL(n__take(x1, x2)) =
/1\
\0/
 +
/00\
\10/
·x1 +
/00\
\10/
·x2

POL(ISNATILIST(x1)) = 1 +
[0,1]
·x1

POL(take(x1, x2)) =
/1\
\0/
 +
/00\
\10/
·x1 +
/00\
\10/
·x2

POL(n__cons(x1, x2)) =
/0\
\0/
 +
/00\
\00/
·x1 +
/10\
\11/
·x2

POL(isNat(x1)) =
/0\
\0/
 +
/01\
\00/
·x1

POL(activate(x1)) =
/0\
\0/
 +
/10\
\01/
·x1

POL(n__0) =
/0\
\1/

POL(n__zeros) =
/0\
\1/

POL(0) =
/0\
\1/

POL(U81(x1)) =
/1\
\0/
 +
/00\
\00/
·x1

POL(isNatIList(x1)) =
/1\
\0/
 +
/01\
\00/
·x1

POL(s(x1)) =
/0\
\0/
 +
/11\
\01/
·x1

POL(cons(x1, x2)) =
/0\
\0/
 +
/00\
\00/
·x1 +
/10\
\11/
·x2

POL(U91(x1, x2, x3, x4)) =
/1\
\0/
 +
/01\
\00/
·x1 +
/00\
\10/
·x2 +
/00\
\11/
·x3 +
/00\
\00/
·x4

POL(n__length(x1)) =
/0\
\0/
 +
/01\
\10/
·x1

POL(U11(x1)) =
/0\
\0/
 +
/10\
\00/
·x1

POL(isNatList(x1)) =
/0\
\0/
 +
/10\
\00/
·x1

POL(n__s(x1)) =
/0\
\0/
 +
/11\
\01/
·x1

POL(U21(x1)) =
/0\
\0/
 +
/10\
\00/
·x1

POL(zeros) =
/0\
\1/

POL(length(x1)) =
/0\
\0/
 +
/01\
\10/
·x1

POL(n__nil) =
/1\
\0/

POL(nil) =
/1\
\0/

POL(U31(x1)) =
/1\
\0/
 +
/00\
\00/
·x1

POL(U71(x1, x2, x3)) =
/0\
\0/
 +
/01\
\00/
·x1 +
/11\
\10/
·x2 +
/00\
\00/
·x3

POL(U51(x1, x2)) =
/0\
\0/
 +
/00\
\00/
·x1 +
/10\
\00/
·x2

POL(U61(x1, x2)) =
/1\
\0/
 +
/00\
\01/
·x1 +
/00\
\00/
·x2

POL(U62(x1)) =
/1\
\0/
 +
/00\
\01/
·x1

POL(U41(x1, x2)) =
/1\
\0/
 +
/00\
\01/
·x1 +
/01\
\00/
·x2

POL(U42(x1)) =
/0\
\0/
 +
/11\
\01/
·x1

POL(U52(x1)) =
/0\
\0/
 +
/10\
\00/
·x1

POL(U72(x1, x2)) =
/0\
\0/
 +
/00\
\00/
·x1 +
/11\
\10/
·x2

POL(U92(x1, x2, x3, x4)) =
/1\
\0/
 +
/01\
\11/
·x1 +
/00\
\10/
·x2 +
/00\
\10/
·x3 +
/00\
\00/
·x4

POL(U93(x1, x2, x3, x4)) =
/1\
\1/
 +
/01\
\00/
·x1 +
/00\
\10/
·x2 +
/00\
\10/
·x3 +
/00\
\00/
·x4

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

take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
take(X1, X2) → n__take(X1, X2)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
0 → n__0
isNatIList(n__zeros) → tt
U81(tt) → nil
isNatIList(V) → U31(isNatList(activate(V)))
isNatList(n__nil) → tt
U31(tt) → tt
length(nil) → 0
length(X) → n__length(X)
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U11(tt) → tt
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
U21(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
s(X) → n__s(X)
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
cons(X1, X2) → n__cons(X1, X2)
zeros → cons(0, n__zeros)
zeros → n__zeros
nil → n__nil

(136) Obligation:

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

ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))
U411(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__0, y0)) → U411(isNat(n__0), activate(y0))
U411(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))
U411(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U411(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(137) NonTerminationLoopProof (COMPLETE transformation)

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

s = U411(isNat(n__0), activate(n__zeros)) evaluates to t =U411(isNat(n__0), activate(n__zeros))

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



Rewriting sequence

U411(isNat(n__0), activate(n__zeros)) → U411(isNat(n__0), n__zeros)
with rule activate(X) → X at position [1] and matcher [X / n__zeros]

U411(isNat(n__0), n__zeros) → U411(tt, n__zeros)
with rule isNat(n__0) → tt at position [0] and matcher [ ]

U411(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
with rule U411(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros)) at position [] and matcher [ ]

ISNATILIST(n__cons(n__0, n__zeros)) → U411(isNat(n__0), activate(n__zeros))
with rule ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.



(138) NO