NO Termination w.r.t. Q proof of Transformed_CSR_04_OvConsOS_nokinds-noand_FR.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 (⇔, 272 ms)
↳6 QDP
↳7 DependencyGraphProof (⇔, 0 ms)
↳8 QDP
↳9 QDPOrderProof (⇔, 194 ms)
↳10 QDP
↳11 DependencyGraphProof (⇔, 0 ms)
↳12 AND
    ↳13 QDP
        ↳14 QDPOrderProof (⇔, 63 ms)
        ↳15 QDP
        ↳16 DependencyGraphProof (⇔, 0 ms)
        ↳17 AND
            ↳18 QDP
                ↳19 UsableRulesProof (⇔, 0 ms)
                ↳20 QDP
                ↳21 QDPSizeChangeProof (⇔, 0 ms)
                ↳22 YES
            ↳23 QDP
                ↳24 QDPOrderProof (⇔, 245 ms)
                ↳25 QDP
                ↳26 PisEmptyProof (⇔, 0 ms)
                ↳27 YES
            ↳28 QDP
                ↳29 TransformationProof (⇔, 0 ms)
                ↳30 QDP
                ↳31 TransformationProof (⇔, 0 ms)
                ↳32 QDP
                ↳33 TransformationProof (⇔, 0 ms)
                ↳34 QDP
                ↳35 DependencyGraphProof (⇔, 0 ms)
                ↳36 QDP
                ↳37 TransformationProof (⇔, 0 ms)
                ↳38 QDP
                ↳39 DependencyGraphProof (⇔, 0 ms)
                ↳40 QDP
                ↳41 TransformationProof (⇔, 0 ms)
                ↳42 QDP
                ↳43 TransformationProof (⇔, 0 ms)
                ↳44 QDP
                ↳45 DependencyGraphProof (⇔, 0 ms)
                ↳46 QDP
                ↳47 TransformationProof (⇔, 0 ms)
                ↳48 QDP
                ↳49 DependencyGraphProof (⇔, 0 ms)
                ↳50 QDP
                ↳51 TransformationProof (⇔, 0 ms)
                ↳52 QDP
                ↳53 DependencyGraphProof (⇔, 0 ms)
                ↳54 QDP
                ↳55 TransformationProof (⇔, 0 ms)
                ↳56 QDP
                ↳57 TransformationProof (⇔, 0 ms)
                ↳58 QDP
                ↳59 DependencyGraphProof (⇔, 0 ms)
                ↳60 QDP
                ↳61 TransformationProof (⇔, 0 ms)
                ↳62 QDP
                ↳63 DependencyGraphProof (⇔, 0 ms)
                ↳64 QDP
                ↳65 TransformationProof (⇔, 0 ms)
                ↳66 QDP
                ↳67 QDPOrderProof (⇔, 0 ms)
                ↳68 QDP
                ↳69 QDPOrderProof (⇔, 0 ms)
                ↳70 QDP
                ↳71 QDPOrderProof (⇔, 0 ms)
                ↳72 QDP
                ↳73 QDPOrderProof (⇔, 0 ms)
                ↳74 QDP
                ↳75 QDPOrderProof (⇔, 0 ms)
                ↳76 QDP
                ↳77 QDPOrderProof (⇔, 0 ms)
                ↳78 QDP
                ↳79 QDPOrderProof (⇔, 0 ms)
                ↳80 QDP
            ↳81 QDP
                ↳82 TransformationProof (⇔, 0 ms)
                ↳83 QDP
                ↳84 TransformationProof (⇔, 0 ms)
                ↳85 QDP
                ↳86 TransformationProof (⇔, 0 ms)
                ↳87 QDP
                ↳88 DependencyGraphProof (⇔, 0 ms)
                ↳89 QDP
                ↳90 TransformationProof (⇔, 0 ms)
                ↳91 QDP
                ↳92 DependencyGraphProof (⇔, 0 ms)
                ↳93 QDP
                ↳94 TransformationProof (⇔, 0 ms)
                ↳95 QDP
                ↳96 DependencyGraphProof (⇔, 0 ms)
                ↳97 QDP
                ↳98 TransformationProof (⇔, 0 ms)
                ↳99 QDP
                ↳100 TransformationProof (⇔, 0 ms)
                ↳101 QDP
                ↳102 DependencyGraphProof (⇔, 0 ms)
                ↳103 QDP
                ↳104 TransformationProof (⇔, 0 ms)
                ↳105 QDP
                ↳106 DependencyGraphProof (⇔, 0 ms)
                ↳107 QDP
                ↳108 TransformationProof (⇔, 0 ms)
                ↳109 QDP
                ↳110 DependencyGraphProof (⇔, 0 ms)
                ↳111 QDP
                ↳112 TransformationProof (⇔, 0 ms)
                ↳113 QDP
                ↳114 TransformationProof (⇔, 0 ms)
                ↳115 QDP
                ↳116 DependencyGraphProof (⇔, 0 ms)
                ↳117 QDP
                ↳118 TransformationProof (⇔, 0 ms)
                ↳119 QDP
                ↳120 QDPOrderProof (⇔, 0 ms)
                ↳121 QDP
                ↳122 QDPOrderProof (⇔, 0 ms)
                ↳123 QDP
                ↳124 QDPOrderProof (⇔, 0 ms)
                ↳125 QDP
                ↳126 QDPOrderProof (⇔, 0 ms)
                ↳127 QDP
                ↳128 QDPOrderProof (⇔, 0 ms)
                ↳129 QDP
                ↳130 QDPOrderProof (⇔, 0 ms)
                ↳131 QDP
                ↳132 QDPOrderProof (⇔, 0 ms)
                ↳133 QDP
                ↳134 NonTerminationLoopProof (⇒, 0 ms)
                ↳135 NO
    ↳136 QDP
        ↳137 QDPOrderProof (⇔, 171 ms)
        ↳138 QDP
        ↳139 DependencyGraphProof (⇔, 0 ms)
        ↳140 TRUE

(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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__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(activate(X1), activate(X2))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__0) → 01
ACTIVATE(n__length(X)) → LENGTH(activate(X))
ACTIVATE(n__length(X)) → ACTIVATE(X)
ACTIVATE(n__s(X)) → S(activate(X))
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → CONS(activate(X1), X2)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__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(activate(X1), activate(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__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__length(X)) → LENGTH(activate(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)
ACTIVATE(n__length(X)) → ACTIVATE(X)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__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.


TAKE(s(M), cons(N, IL)) → U911(isNatIList(activate(IL)), activate(IL), M, N)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( ISNAT(x1) ) = x1 + 2

POL( ISNATILIST(x1) ) = x1 + 2

POL( ISNATLIST(x1) ) = x1 + 2

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

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

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

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

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

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

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

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

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

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

POL( U41(x1, x2) ) = x2

POL( U51(x1, x2) ) = x2

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

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

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

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

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

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

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

POL( isNat(x1) ) = 2x1 + 2

POL( length(x1) ) = 2x1

POL( s(x1) ) = x1

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

POL( isNatList(x1) ) = x1

POL( isNatIList(x1) ) = x1

POL( U11(x1) ) = 2

POL( U21(x1) ) = 2

POL( U31(x1) ) = x1

POL( U42(x1) ) = x1

POL( U52(x1) ) = x1

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

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

POL( activate(x1) ) = x1

POL( n__zeros ) = 2

POL( zeros ) = 2

POL( n__0 ) = 0

POL( 0 ) = 0

POL( n__length(x1) ) = 2x1

POL( n__s(x1) ) = x1

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

POL( n__nil ) = 2

POL( nil ) = 2

POL( tt ) = 2

POL( U81(x1) ) = x1

POL( ACTIVATE(x1) ) = x1 + 2


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

activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
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)
length(nil) → 0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
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
U31(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)))
take(0, IL) → U81(isNatIList(IL))
U81(tt) → nil
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)))
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:

ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(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__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__length(X)) → LENGTH(activate(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)
ACTIVATE(n__length(X)) → ACTIVATE(X)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
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)
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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__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 1 SCC with 13 less nodes.

(8) 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))
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__length(V1)) → ISNATLIST(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
ISNATILIST(V) → ACTIVATE(V)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__length(X)) → LENGTH(activate(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)
ACTIVATE(n__length(X)) → ACTIVATE(X)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ISNATLIST(n__take(V1, V2)) → U611(isNat(activate(V1)), activate(V2))
U611(tt, V2) → ISNATILIST(activate(V2))
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)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
U511(tt, V2) → 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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(9) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


ISNAT(n__length(V1)) → ISNATLIST(activate(V1))
ACTIVATE(n__length(X)) → LENGTH(activate(X))
ACTIVATE(n__length(X)) → ACTIVATE(X)
ISNAT(n__length(V1)) → ACTIVATE(V1)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( ISNAT(x1) ) = 2x1

POL( ISNATILIST(x1) ) = 2x1

POL( ISNATLIST(x1) ) = 2x1

POL( LENGTH(x1) ) = 2x1

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

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

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

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

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

POL( U721(x1, x2) ) = max{0, x1 + 2x2 - 2}

POL( U41(x1, x2) ) = max{0, x1 + x2 - 1}

POL( U51(x1, x2) ) = x2

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

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

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

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

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

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

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

POL( isNat(x1) ) = 2

POL( length(x1) ) = x1 + 2

POL( s(x1) ) = x1

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

POL( isNatList(x1) ) = x1

POL( U11(x1) ) = 2

POL( U21(x1) ) = 2

POL( U31(x1) ) = x1

POL( U42(x1) ) = x1

POL( isNatIList(x1) ) = x1 + 1

POL( U52(x1) ) = x1

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

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

POL( activate(x1) ) = x1

POL( n__zeros ) = 2

POL( zeros ) = 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 ) = 2

POL( nil ) = 2

POL( tt ) = 2

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

POL( ACTIVATE(x1) ) = 2x1


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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__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))
take(X1, X2) → n__take(X1, X2)
length(nil) → 0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
U11(tt) → tt
U21(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
isNatIList(n__zeros) → tt
U62(tt) → tt
isNatIList(V) → U31(isNatList(activate(V)))
U31(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)))
take(0, IL) → U81(isNatIList(IL))
U81(tt) → nil
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)))
zeros → cons(0, n__zeros)
zeros → n__zeros
0 → n__0
nil → n__nil

(10) 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))
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
ISNATILIST(V) → ACTIVATE(V)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
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)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ISNATLIST(n__take(V1, V2)) → U611(isNat(activate(V1)), activate(V2))
U611(tt, V2) → ISNATILIST(activate(V2))
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__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)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
U511(tt, V2) → 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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(11) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 6 less nodes.

(12) Complex Obligation (AND)

(13) Obligation:

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

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__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
TAKE(0, IL) → ISNATILIST(IL)
ISNATILIST(V) → ACTIVATE(V)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
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))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U411(tt, V2) → ACTIVATE(V2)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
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))
U611(tt, V2) → ISNATILIST(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)
U511(tt, V2) → 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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(14) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


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

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

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

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

POL( U411(x1, x2) ) = 2x2 + 1

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

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

POL( U41(x1, x2) ) = 2

POL( U51(x1, x2) ) = 2

POL( U61(x1, x2) ) = 2

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

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

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

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

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

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

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

POL( length(x1) ) = 0

POL( s(x1) ) = 2x1

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

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

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

POL( U31(x1) ) = 2

POL( U42(x1) ) = 2

POL( isNatIList(x1) ) = 2

POL( U52(x1) ) = 2

POL( U62(x1) ) = 2

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

POL( activate(x1) ) = x1

POL( n__zeros ) = 2

POL( zeros ) = 2

POL( n__0 ) = 0

POL( 0 ) = 0

POL( n__length(x1) ) = 0

POL( n__s(x1) ) = 2x1

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

POL( n__nil ) = 0

POL( nil ) = 0

POL( U21(x1) ) = 0

POL( tt ) = 0

POL( U81(x1) ) = 2

POL( ACTIVATE(x1) ) = 2x1


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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
take(X1, X2) → n__take(X1, X2)
length(nil) → 0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
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(V) → U31(isNatList(activate(V)))
U31(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)))
take(0, IL) → U81(isNatIList(IL))
U81(tt) → nil
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)))
zeros → cons(0, n__zeros)
zeros → n__zeros
0 → n__0
nil → n__nil

(15) Obligation:

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

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__s(V1)) → ISNAT(activate(V1))
TAKE(0, IL) → ISNATILIST(IL)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
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))
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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(16) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 5 less nodes.

(17) Complex Obligation (AND)

(18) Obligation:

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

ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)

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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(19) UsableRulesProof (EQUIVALENT transformation)

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

(20) Obligation:

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

ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)

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

(21) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

  • ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
    The graph contains the following edges 1 > 1

  • ACTIVATE(n__s(X)) → ACTIVATE(X)
    The graph contains the following edges 1 > 1

(22) YES

(23) 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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(24) 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)) = 3A + 1A·x1

POL(n__s(x1)) = 3A + 1A·x1

POL(activate(x1)) = 1A + 0A·x1

POL(n__zeros) = 0A

POL(zeros) = 1A

POL(n__take(x1, x2)) = 3A + 1A·x1 + 0A·x2

POL(take(x1, x2)) = 3A + 1A·x1 + 0A·x2

POL(n__0) = 4A

POL(0) = 4A

POL(n__length(x1)) = 4A + 2A·x1

POL(length(x1)) = 4A + 2A·x1

POL(s(x1)) = 3A + 1A·x1

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

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

POL(n__nil) = 3A

POL(nil) = 3A

POL(U71(x1, x2, x3)) = 4A + 3A·x1 + 3A·x2 + -I·x3

POL(isNatList(x1)) = 1A + 0A·x1

POL(tt) = 2A

POL(U51(x1, x2)) = 1A + -I·x1 + 0A·x2

POL(isNat(x1)) = 2A + -I·x1

POL(U11(x1)) = 2A + -I·x1

POL(U61(x1, x2)) = 3A + -I·x1 + 0A·x2

POL(U21(x1)) = 2A + -I·x1

POL(U62(x1)) = 0A + 1A·x1

POL(isNatIList(x1)) = 2A + -I·x1

POL(U31(x1)) = 2A + -I·x1

POL(U41(x1, x2)) = 2A + 0A·x1 + -I·x2

POL(U42(x1)) = 2A + -I·x1

POL(U52(x1)) = 0A + 0A·x1

POL(U72(x1, x2)) = 5A + -I·x1 + 3A·x2

POL(U81(x1)) = 5A + -I·x1

POL(U91(x1, x2, x3, x4)) = -I + 2A·x1 + 1A·x2 + 2A·x3 + -I·x4

POL(U92(x1, x2, x3, x4)) = 3A + 2A·x1 + 1A·x2 + 2A·x3 + -I·x4

POL(U93(x1, x2, x3, x4)) = 2A + 2A·x1 + 1A·x2 + 2A·x3 + -I·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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
take(X1, X2) → n__take(X1, X2)
length(nil) → 0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
isNatList(n__nil) → tt
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)))
isNatIList(n__zeros) → tt
U62(tt) → tt
isNatIList(V) → U31(isNatList(activate(V)))
U31(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)))
take(0, IL) → U81(isNatIList(IL))
U81(tt) → nil
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)))
zeros → cons(0, n__zeros)
zeros → n__zeros
0 → n__0
nil → n__nil

(25) 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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(26) PisEmptyProof (EQUIVALENT transformation)

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

(27) YES

(28) 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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__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, 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(activate(x0), activate(x1))) → U511(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
U511(tt, n__0) → ISNATLIST(0) → U511(tt, n__0) → ISNATLIST(0)
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0))) → U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0))) → U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1)) → U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
U511(tt, n__nil) → ISNATLIST(nil) → U511(tt, n__nil) → ISNATLIST(nil)
U511(tt, x0) → ISNATLIST(x0) → U511(tt, x0) → ISNATLIST(x0)

(30) 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(activate(x0), activate(x1)))
U511(tt, n__0) → ISNATLIST(0)
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(31) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATLIST(n__cons(V1, V2)) → 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(activate(x0), activate(x1))), activate(y1)) → ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(activate(x0), activate(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(activate(x0))), activate(y1)) → ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1)) → ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1)) → ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(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))

(32) 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(activate(x0), activate(x1)))
U511(tt, n__0) → ISNATLIST(0)
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(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(activate(x0), activate(x1))), activate(y1))
ISNATLIST(n__cons(n__0, y1)) → U511(isNat(0), activate(y1))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__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__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)

(34) Obligation:

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

U511(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
U511(tt, n__0) → ISNATLIST(0)
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(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(activate(x0), activate(x1))), activate(y1))
ISNATLIST(n__cons(n__0, y1)) → U511(isNat(0), activate(y1))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__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__zeros, y1)) → U511(isNat(zeros), activate(y1))
U511(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(activate(x0), activate(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(activate(x0)))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__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__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))

(38) Obligation:

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

U511(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(activate(x0), activate(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(activate(x0)))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(39) DependencyGraphProof (EQUIVALENT transformation)

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

(40) Obligation:

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

ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(activate(x0), activate(x1))), activate(y1))
U511(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(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(activate(x0))), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__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__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))

(42) Obligation:

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

ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(activate(x0), activate(x1))), activate(y1))
U511(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
U511(tt, n__0) → ISNATLIST(0)
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(43) 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)

(44) Obligation:

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

ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(activate(x0), activate(x1))), activate(y1))
U511(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(45) DependencyGraphProof (EQUIVALENT transformation)

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

(46) Obligation:

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

U511(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(activate(x0), activate(x1))), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(47) 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)

(48) Obligation:

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

U511(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(activate(x0), activate(x1))), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), 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))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(49) DependencyGraphProof (EQUIVALENT transformation)

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

(50) Obligation:

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

ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(activate(x0), activate(x1))), activate(y1))
U511(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(51) 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))

(52) Obligation:

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

ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(activate(x0), activate(x1))), activate(y1))
U511(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
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__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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(53) DependencyGraphProof (EQUIVALENT transformation)

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

(54) Obligation:

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

U511(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(activate(x0), activate(x1))), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), 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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__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 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))

(56) Obligation:

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

U511(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(activate(x0), activate(x1))), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), 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))
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(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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(57) 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))

(58) Obligation:

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

U511(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(activate(x0), activate(x1))), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
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(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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(59) DependencyGraphProof (EQUIVALENT transformation)

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

(60) Obligation:

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

ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(activate(x0), activate(x1))), activate(y1))
U511(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
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__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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(61) 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))

(62) Obligation:

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

ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(activate(x0), activate(x1))), activate(y1))
U511(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
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__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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(63) DependencyGraphProof (EQUIVALENT transformation)

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

(64) Obligation:

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

U511(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(activate(x0), activate(x1))), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), 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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(65) 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))

(66) Obligation:

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

U511(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(activate(x0), activate(x1))), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__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(activate(x0), activate(x1))), activate(y1))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( U511(x1, x2) ) = 2x1 + 2x2

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

POL( U41(x1, x2) ) = 2

POL( U51(x1, x2) ) = 2

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

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

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

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

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

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

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

POL( length(x1) ) = 2x1

POL( s(x1) ) = 0

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

POL( U11(x1) ) = 1

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

POL( U21(x1) ) = 1

POL( isNat(x1) ) = 1

POL( U31(x1) ) = 1

POL( U42(x1) ) = 2

POL( isNatIList(x1) ) = 2

POL( U52(x1) ) = 1

POL( U62(x1) ) = 1

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

POL( activate(x1) ) = x1

POL( n__zeros ) = 1

POL( zeros ) = 1

POL( n__0 ) = 0

POL( 0 ) = 0

POL( n__length(x1) ) = 2x1

POL( n__s(x1) ) = 0

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

POL( n__nil ) = 0

POL( nil ) = 0

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

POL( tt ) = 1


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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
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)))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
0 → n__0
isNatList(n__nil) → 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)))
isNatIList(n__zeros) → tt
U62(tt) → tt
isNatIList(V) → U31(isNatList(activate(V)))
U31(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)))
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)))
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(activate(x0), activate(x1)))
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__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.


ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(activate(x0))), activate(y1))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( U511(x1, x2) ) = 2x2 + 1

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

POL( U41(x1, x2) ) = 1

POL( U51(x1, x2) ) = max{0, 2x1 - 1}

POL( U61(x1, x2) ) = 2

POL( U71(x1, ..., x3) ) = max{0, x1 + 2x3 - 1}

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

POL( U91(x1, ..., x4) ) = max{0, x1 + x2 + 2x4 - 1}

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

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

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

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

POL( s(x1) ) = 0

POL( take(x1, x2) ) = x2

POL( U11(x1) ) = 1

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

POL( U21(x1) ) = 1

POL( isNat(x1) ) = x1 + 1

POL( U31(x1) ) = 1

POL( U42(x1) ) = 1

POL( isNatIList(x1) ) = 1

POL( U52(x1) ) = 1

POL( U62(x1) ) = max{0, 2x1 - 1}

POL( n__take(x1, x2) ) = x2

POL( activate(x1) ) = x1

POL( n__zeros ) = 0

POL( zeros ) = 0

POL( n__0 ) = 0

POL( 0 ) = 0

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

POL( n__s(x1) ) = 0

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

POL( n__nil ) = 0

POL( nil ) = 0

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

POL( tt ) = 1


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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
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)))
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
0 → n__0
isNatList(n__nil) → 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)))
isNatIList(n__zeros) → tt
U62(tt) → tt
isNatIList(V) → U31(isNatList(activate(V)))
U31(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)))
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)))
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(activate(x0), activate(x1)))
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(71) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(activate(x0))), activate(y1))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( U511(x1, x2) ) = x1 + x2 + 2

POL( ISNATLIST(x1) ) = x1 + 2

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

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

POL( U61(x1, x2) ) = 0

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

POL( U72(x1, x2) ) = 2

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

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

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

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

POL( length(x1) ) = 2

POL( s(x1) ) = 1

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

POL( U11(x1) ) = 0

POL( isNatList(x1) ) = 0

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

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

POL( U31(x1) ) = max{0, -1}

POL( U42(x1) ) = 0

POL( isNatIList(x1) ) = 0

POL( U52(x1) ) = 0

POL( U62(x1) ) = x1

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

POL( activate(x1) ) = x1

POL( n__zeros ) = 1

POL( zeros ) = 1

POL( n__0 ) = 0

POL( 0 ) = 0

POL( n__length(x1) ) = 2

POL( n__s(x1) ) = 1

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

POL( n__nil ) = 0

POL( nil ) = 0

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

POL( tt ) = 0


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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
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)
s(X) → n__s(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
cons(X1, X2) → n__cons(X1, X2)
0 → n__0
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)))
isNatIList(n__zeros) → tt
U62(tt) → tt
isNatIList(V) → U31(isNatList(activate(V)))
U31(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)))
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)))
zeros → cons(0, n__zeros)
zeros → n__zeros
nil → n__nil

(72) Obligation:

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

U511(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
U511(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(73) 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(activate(x0)))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( U511(x1, x2) ) = max{0, 2x1 + 2x2 - 2}

POL( ISNATLIST(x1) ) = x1 + 2

POL( U41(x1, x2) ) = 2

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

POL( U61(x1, x2) ) = 2

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

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

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

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

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

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

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

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

POL( take(x1, x2) ) = x2

POL( U11(x1) ) = 2

POL( isNatList(x1) ) = x1 + 2

POL( U21(x1) ) = 2

POL( isNat(x1) ) = x1 + 2

POL( U31(x1) ) = 2

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

POL( isNatIList(x1) ) = 2

POL( U52(x1) ) = 2

POL( U62(x1) ) = 2

POL( n__take(x1, x2) ) = x2

POL( activate(x1) ) = x1

POL( n__zeros ) = 0

POL( zeros ) = 0

POL( n__0 ) = 0

POL( 0 ) = 0

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

POL( n__s(x1) ) = 0

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

POL( n__nil ) = 0

POL( nil ) = 0

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

POL( tt ) = 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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
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)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
0 → n__0
isNatList(n__nil) → 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)))
isNatIList(n__zeros) → tt
U62(tt) → tt
isNatIList(V) → U31(isNatList(activate(V)))
U31(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)))
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)))
zeros → cons(0, n__zeros)
zeros → n__zeros
nil → n__nil

(74) Obligation:

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

U511(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(75) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


U511(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( U511(x1, x2) ) = x1 + 2x2

POL( ISNATLIST(x1) ) = x1 + 2

POL( U41(x1, x2) ) = x1

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

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

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

POL( U72(x1, x2) ) = 2

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

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

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

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

POL( length(x1) ) = 2

POL( s(x1) ) = 2

POL( take(x1, x2) ) = x2

POL( U11(x1) ) = 2

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

POL( U21(x1) ) = 2

POL( isNat(x1) ) = 2

POL( U31(x1) ) = 2

POL( U42(x1) ) = 2

POL( isNatIList(x1) ) = 2

POL( U52(x1) ) = x1

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

POL( n__take(x1, x2) ) = x2

POL( activate(x1) ) = x1

POL( n__zeros ) = 0

POL( zeros ) = 0

POL( n__0 ) = 0

POL( 0 ) = 0

POL( n__length(x1) ) = 2

POL( n__s(x1) ) = 2

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

POL( n__nil ) = 0

POL( nil ) = 0

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

POL( tt ) = 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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
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)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
0 → n__0
length(nil) → 0
length(X) → n__length(X)
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
isNatList(n__nil) → 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)))
isNatIList(n__zeros) → tt
U62(tt) → tt
isNatIList(V) → U31(isNatList(activate(V)))
U31(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)))
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)))
zeros → cons(0, n__zeros)
zeros → n__zeros
nil → n__nil

(76) Obligation:

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

U511(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(77) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


U511(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(U511(x1, x2)) = -I + 0A·x1 + 1A·x2

POL(tt) = 5A

POL(n__take(x1, x2)) = 5A + 2A·x1 + 2A·x2

POL(ISNATLIST(x1)) = 5A + 0A·x1

POL(take(x1, x2)) = 5A + 2A·x1 + 2A·x2

POL(activate(x1)) = 2A + 0A·x1

POL(n__cons(x1, x2)) = 2A + 0A·x1 + 1A·x2

POL(cons(x1, x2)) = 2A + 0A·x1 + 1A·x2

POL(isNat(x1)) = 5A + -I·x1

POL(n__zeros) = 0A

POL(0) = 2A

POL(n__0) = 2A

POL(zeros) = 2A

POL(n__length(x1)) = 4A + 0A·x1

POL(length(x1)) = 4A + 0A·x1

POL(n__s(x1)) = 4A + 1A·x1

POL(s(x1)) = 4A + 1A·x1

POL(n__nil) = 5A

POL(nil) = 5A

POL(U81(x1)) = 5A + -I·x1

POL(isNatIList(x1)) = 5A + 0A·x1

POL(U91(x1, x2, x3, x4)) = 5A + 1A·x1 + 3A·x2 + 3A·x3 + 0A·x4

POL(U11(x1)) = 5A + -I·x1

POL(isNatList(x1)) = 3A + 0A·x1

POL(U21(x1)) = 5A + -I·x1

POL(U71(x1, x2, x3)) = 3A + 0A·x1 + 1A·x2 + -I·x3

POL(U51(x1, x2)) = 3A + -I·x1 + 0A·x2

POL(U61(x1, x2)) = 1A + 0A·x1 + -I·x2

POL(U62(x1)) = 5A + -I·x1

POL(U31(x1)) = 5A + -I·x1

POL(U41(x1, x2)) = 5A + -I·x1 + -I·x2

POL(U42(x1)) = 5A + -I·x1

POL(U52(x1)) = 3A + 0A·x1

POL(U72(x1, x2)) = 5A + -I·x1 + 1A·x2

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

POL(U93(x1, x2, x3, x4)) = 4A + 1A·x1 + 3A·x2 + 3A·x3 + 0A·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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
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)
cons(X1, X2) → n__cons(X1, X2)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
0 → n__0
length(nil) → 0
length(X) → n__length(X)
s(X) → n__s(X)
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
isNatList(n__nil) → 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)))
isNatIList(n__zeros) → tt
U62(tt) → tt
isNatIList(V) → U31(isNatList(activate(V)))
U31(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)))
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)))
zeros → cons(0, n__zeros)
zeros → n__zeros
nil → n__nil

(78) Obligation:

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

U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(79) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(activate(x0), x1)), activate(y1))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

POL(U511(x1, x2)) = 0 +
[0,1]
·x1 +
[1,0]
·x2

POL(tt) =
/0\
\0/

POL(n__cons(x1, x2)) =
/0\
\1/
 +
/01\
\00/
·x1 +
/10\
\00/
·x2

POL(ISNATLIST(x1)) = 0 +
[1,0]
·x1

POL(cons(x1, x2)) =
/0\
\1/
 +
/01\
\00/
·x1 +
/10\
\00/
·x2

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

POL(isNat(x1)) =
/1\
\0/
 +
/11\
\00/
·x1

POL(n__zeros) =
/1\
\1/

POL(0) =
/0\
\0/

POL(n__0) =
/0\
\0/

POL(zeros) =
/1\
\1/

POL(n__take(x1, x2)) =
/0\
\1/
 +
/00\
\11/
·x1 +
/10\
\00/
·x2

POL(take(x1, x2)) =
/0\
\1/
 +
/00\
\11/
·x1 +
/10\
\00/
·x2

POL(n__length(x1)) =
/1\
\1/
 +
/00\
\00/
·x1

POL(length(x1)) =
/1\
\1/
 +
/00\
\00/
·x1

POL(n__s(x1)) =
/1\
\1/
 +
/00\
\00/
·x1

POL(s(x1)) =
/1\
\1/
 +
/00\
\00/
·x1

POL(n__nil) =
/0\
\0/

POL(nil) =
/0\
\0/

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

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

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

POL(U71(x1, x2, x3)) =
/1\
\1/
 +
/00\
\00/
·x1 +
/00\
\00/
·x2 +
/00\
\00/
·x3

POL(U51(x1, x2)) =
/0\
\1/
 +
/00\
\01/
·x1 +
/00\
\00/
·x2

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

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

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

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

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

POL(U42(x1)) =
/0\
\0/
 +
/00\
\00/
·x1

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

POL(U72(x1, x2)) =
/1\
\1/
 +
/00\
\00/
·x1 +
/00\
\00/
·x2

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

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

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

POL(U93(x1, x2, x3, x4)) =
/0\
\1/
 +
/00\
\00/
·x1 +
/10\
\00/
·x2 +
/00\
\00/
·x3 +
/01\
\00/
·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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
cons(X1, X2) → n__cons(X1, X2)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
0 → n__0
take(X1, X2) → n__take(X1, X2)
length(nil) → 0
length(X) → n__length(X)
s(X) → n__s(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)))
isNatIList(n__zeros) → tt
U62(tt) → tt
isNatIList(V) → U31(isNatList(activate(V)))
U31(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)))
take(0, IL) → U81(isNatIList(IL))
U81(tt) → nil
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)))
zeros → cons(0, n__zeros)
zeros → n__zeros
nil → n__nil

(80) Obligation:

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

U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(81) 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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(82) 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(activate(x0), activate(x1))), activate(y1)) → ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(activate(x0), activate(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(activate(x0))), activate(y1)) → ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(activate(x0))), activate(y1))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1)) → ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1)) → ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(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))

(83) 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(activate(x0), activate(x1))), activate(y1))
ISNATILIST(n__cons(n__0, y1)) → U411(isNat(0), activate(y1))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(activate(x0))), activate(y1))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

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

(85) 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(activate(x0), activate(x1))), activate(y1))
ISNATILIST(n__cons(n__0, y1)) → U411(isNat(0), activate(y1))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(activate(x0))), activate(y1))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(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(activate(x0), activate(x1)))
U411(tt, n__0) → ISNATILIST(0)
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(86) 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))

(87) Obligation:

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

ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(activate(x0), activate(x1))), activate(y1))
ISNATILIST(n__cons(n__0, y1)) → U411(isNat(0), activate(y1))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(activate(x0))), activate(y1))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(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(activate(x0), activate(x1)))
U411(tt, n__0) → ISNATILIST(0)
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(88) DependencyGraphProof (EQUIVALENT transformation)

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

(89) 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(activate(x0), activate(x1))), activate(y1))
U411(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(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(activate(x0))), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(90) 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)

(91) Obligation:

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

ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(activate(x0), activate(x1))), activate(y1))
U411(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(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(activate(x0))), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(92) DependencyGraphProof (EQUIVALENT transformation)

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

(93) Obligation:

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

U411(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(activate(x0), activate(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(activate(x0)))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(activate(x0))), activate(y1))
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(94) 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)

(95) Obligation:

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

U411(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(activate(x0), activate(x1))), activate(y1))
ISNATILIST(n__cons(n__0, y1)) → U411(isNat(0), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(activate(x0))), activate(y1))
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(96) DependencyGraphProof (EQUIVALENT transformation)

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

(97) Obligation:

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

ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(activate(x0), activate(x1))), activate(y1))
U411(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
ISNATILIST(n__cons(n__0, y1)) → U411(isNat(0), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(activate(x0))), activate(y1))
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(98) 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))

(99) Obligation:

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

ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(activate(x0), activate(x1))), activate(y1))
U411(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(activate(x0))), activate(y1))
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(100) 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)

(101) Obligation:

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

ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(activate(x0), activate(x1))), activate(y1))
U411(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(activate(x0))), activate(y1))
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
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__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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(102) DependencyGraphProof (EQUIVALENT transformation)

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

(103) Obligation:

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

U411(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(activate(x0), activate(x1))), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(activate(x0))), activate(y1))
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(104) 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))

(105) Obligation:

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

U411(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(activate(x0), activate(x1))), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(activate(x0))), activate(y1))
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), 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))
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__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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(106) DependencyGraphProof (EQUIVALENT transformation)

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

(107) Obligation:

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

ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(activate(x0), activate(x1))), activate(y1))
U411(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(activate(x0))), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
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))

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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(108) 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))

(109) Obligation:

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

ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(activate(x0), activate(x1))), activate(y1))
U411(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(activate(x0))), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
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))
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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(110) DependencyGraphProof (EQUIVALENT transformation)

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

(111) Obligation:

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

U411(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(activate(x0), activate(x1))), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(activate(x0))), activate(y1))
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), 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))
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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(112) 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))

(113) Obligation:

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

U411(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(activate(x0), activate(x1))), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(activate(x0))), activate(y1))
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), 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))
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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(114) 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))

(115) Obligation:

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

U411(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(activate(x0), activate(x1))), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(activate(x0))), activate(y1))
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), 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__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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(116) DependencyGraphProof (EQUIVALENT transformation)

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

(117) Obligation:

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

ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(activate(x0), activate(x1))), activate(y1))
U411(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(activate(x0))), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(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__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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(118) 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))

(119) Obligation:

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

ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(activate(x0), activate(x1))), activate(y1))
U411(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(activate(x0))), activate(y1))
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(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__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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(120) 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(activate(x0))), activate(y1))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( U411(x1, x2) ) = x2 + 1

POL( ISNATILIST(x1) ) = x1 + 1

POL( U41(x1, x2) ) = 2

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

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

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

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

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

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

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

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

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

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

POL( take(x1, x2) ) = x2

POL( U11(x1) ) = 2

POL( isNatList(x1) ) = x1 + 2

POL( U21(x1) ) = 2

POL( isNat(x1) ) = 2

POL( U31(x1) ) = 2

POL( U42(x1) ) = 2

POL( isNatIList(x1) ) = x1 + 2

POL( U52(x1) ) = 2

POL( U62(x1) ) = x1

POL( n__take(x1, x2) ) = x2

POL( activate(x1) ) = x1

POL( n__zeros ) = 0

POL( zeros ) = 0

POL( n__0 ) = 0

POL( 0 ) = 0

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

POL( n__s(x1) ) = 0

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

POL( n__nil ) = 0

POL( nil ) = 0

POL( U81(x1) ) = 0

POL( tt ) = 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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
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)))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
0 → n__0
isNatList(n__nil) → 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)))
isNatIList(n__zeros) → tt
U62(tt) → tt
isNatIList(V) → U31(isNatList(activate(V)))
U31(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)))
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)))
zeros → cons(0, n__zeros)
zeros → n__zeros
nil → n__nil

(121) Obligation:

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

ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(activate(x0), activate(x1))), activate(y1))
U411(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(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__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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(122) 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(activate(x0), activate(x1))), activate(y1))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( U411(x1, x2) ) = x1 + 2x2

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

POL( U41(x1, x2) ) = 2

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

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

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

POL( U72(x1, x2) ) = x1 + 2x2 + 1

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

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

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

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

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

POL( s(x1) ) = x1

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

POL( U11(x1) ) = x1 + 1

POL( isNatList(x1) ) = 2x1

POL( U21(x1) ) = 1

POL( isNat(x1) ) = x1 + 1

POL( U31(x1) ) = 2

POL( U42(x1) ) = 2

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

POL( U52(x1) ) = x1

POL( U62(x1) ) = x1 + 2

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

POL( activate(x1) ) = x1

POL( n__zeros ) = 0

POL( zeros ) = 0

POL( n__0 ) = 0

POL( 0 ) = 0

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

POL( n__s(x1) ) = x1

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

POL( n__nil ) = 1

POL( nil ) = 1

POL( U81(x1) ) = x1

POL( tt ) = 1


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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
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)))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
0 → n__0
isNatList(n__nil) → 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)))
isNatIList(n__zeros) → tt
U62(tt) → tt
isNatIList(V) → U31(isNatList(activate(V)))
U31(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)))
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)))
zeros → cons(0, n__zeros)
zeros → n__zeros
nil → n__nil

(123) Obligation:

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

U411(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(activate(x0))), activate(y1))
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(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__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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(124) 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(s(activate(x0))), activate(y1))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( U411(x1, x2) ) = 2x1 + 2x2 + 2

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

POL( U41(x1, x2) ) = 0

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

POL( U61(x1, x2) ) = 0

POL( U71(x1, ..., x3) ) = 2x1 + 2x3 + 1

POL( U72(x1, x2) ) = 1

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

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

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

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

POL( length(x1) ) = x1 + 2

POL( s(x1) ) = 1

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

POL( U11(x1) ) = 0

POL( isNatList(x1) ) = 0

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

POL( isNat(x1) ) = 0

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

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

POL( isNatIList(x1) ) = 0

POL( U52(x1) ) = 0

POL( U62(x1) ) = 0

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

POL( activate(x1) ) = x1

POL( n__zeros ) = 2

POL( zeros ) = 2

POL( n__0 ) = 0

POL( 0 ) = 0

POL( n__length(x1) ) = x1 + 2

POL( n__s(x1) ) = 1

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

POL( n__nil ) = 0

POL( nil ) = 0

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

POL( tt ) = 0


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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
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)
s(X) → n__s(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
cons(X1, X2) → n__cons(X1, X2)
0 → n__0
isNatList(n__nil) → 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)))
isNatIList(n__zeros) → tt
U62(tt) → tt
isNatIList(V) → U31(isNatList(activate(V)))
U31(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)))
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)))
zeros → cons(0, n__zeros)
zeros → n__zeros
nil → n__nil

(125) Obligation:

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

U411(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
U411(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(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__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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(126) 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(activate(x0)))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( U411(x1, x2) ) = 2x2 + 2

POL( ISNATILIST(x1) ) = x1 + 2

POL( U41(x1, x2) ) = max{0, 2x1 - 1}

POL( U51(x1, x2) ) = max{0, 2x1 + 2x2 - 1}

POL( U61(x1, x2) ) = x1

POL( U71(x1, ..., x3) ) = x1 + x2 + 1

POL( U72(x1, x2) ) = x1

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

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

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

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

POL( length(x1) ) = x1 + 2

POL( s(x1) ) = 0

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

POL( U11(x1) ) = 1

POL( isNatList(x1) ) = x1 + 1

POL( U21(x1) ) = max{0, 2x1 - 1}

POL( isNat(x1) ) = 1

POL( U31(x1) ) = 1

POL( U42(x1) ) = 1

POL( isNatIList(x1) ) = 1

POL( U52(x1) ) = max{0, 2x1 - 1}

POL( U62(x1) ) = max{0, 2x1 - 1}

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

POL( activate(x1) ) = x1

POL( n__zeros ) = 0

POL( zeros ) = 0

POL( n__0 ) = 0

POL( 0 ) = 0

POL( n__length(x1) ) = x1 + 2

POL( n__s(x1) ) = 0

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

POL( n__nil ) = 0

POL( nil ) = 0

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

POL( tt ) = 1


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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
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)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
0 → n__0
isNatList(n__nil) → 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)))
isNatIList(n__zeros) → tt
U62(tt) → tt
isNatIList(V) → U31(isNatList(activate(V)))
U31(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)))
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)))
zeros → cons(0, n__zeros)
zeros → n__zeros
nil → n__nil

(127) Obligation:

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

U411(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(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__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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(128) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


U411(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( U411(x1, x2) ) = max{0, 2x1 + 2x2 - 1}

POL( ISNATILIST(x1) ) = x1 + 1

POL( U41(x1, x2) ) = 1

POL( U51(x1, x2) ) = x1 + 2x2 + 1

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

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

POL( U72(x1, x2) ) = 2

POL( U91(x1, ..., x4) ) = max{0, 2x1 + 2x2 + 2x4 - 2}

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

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

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

POL( length(x1) ) = x1 + 2

POL( s(x1) ) = 2

POL( take(x1, x2) ) = x2

POL( U11(x1) ) = 2

POL( isNatList(x1) ) = x1 + 2

POL( U21(x1) ) = 2

POL( isNat(x1) ) = x1 + 1

POL( U31(x1) ) = 1

POL( U42(x1) ) = 1

POL( isNatIList(x1) ) = 1

POL( U52(x1) ) = x1

POL( U62(x1) ) = x1 + 1

POL( n__take(x1, x2) ) = x2

POL( activate(x1) ) = x1

POL( n__zeros ) = 0

POL( zeros ) = 0

POL( n__0 ) = 0

POL( 0 ) = 0

POL( n__length(x1) ) = x1 + 2

POL( n__s(x1) ) = 2

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

POL( n__nil ) = 0

POL( nil ) = 0

POL( U81(x1) ) = 0

POL( tt ) = 1


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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
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)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
0 → n__0
length(nil) → 0
length(X) → n__length(X)
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
isNatList(n__nil) → 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)))
isNatIList(n__zeros) → tt
U62(tt) → tt
isNatIList(V) → U31(isNatList(activate(V)))
U31(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)))
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)))
zeros → cons(0, n__zeros)
zeros → n__zeros
nil → n__nil

(129) Obligation:

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

U411(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(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__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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(130) 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(activate(x0), activate(x1)))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(U411(x1, x2)) = 3A + -I·x1 + 1A·x2

POL(tt) = 4A

POL(n__take(x1, x2)) = 3A + 1A·x1 + 1A·x2

POL(ISNATILIST(x1)) = 3A + 0A·x1

POL(take(x1, x2)) = 3A + 1A·x1 + 1A·x2

POL(activate(x1)) = 2A + 0A·x1

POL(n__cons(x1, x2)) = 2A + -I·x1 + 1A·x2

POL(isNat(x1)) = 4A + -I·x1

POL(cons(x1, x2)) = 2A + -I·x1 + 1A·x2

POL(n__0) = 2A

POL(n__zeros) = 1A

POL(0) = 2A

POL(zeros) = 2A

POL(n__length(x1)) = 3A + 0A·x1

POL(length(x1)) = 3A + 0A·x1

POL(n__s(x1)) = 3A + 1A·x1

POL(s(x1)) = 3A + 1A·x1

POL(n__nil) = 3A

POL(nil) = 3A

POL(U81(x1)) = 3A + -I·x1

POL(isNatIList(x1)) = 4A + -I·x1

POL(U91(x1, x2, x3, x4)) = 3A + 0A·x1 + 2A·x2 + 2A·x3 + -I·x4

POL(U11(x1)) = 4A + -I·x1

POL(isNatList(x1)) = 2A + 1A·x1

POL(U21(x1)) = 4A + -I·x1

POL(U71(x1, x2, x3)) = 3A + 0A·x1 + 1A·x2 + -I·x3

POL(U51(x1, x2)) = 3A + -I·x1 + 1A·x2

POL(U61(x1, x2)) = 4A + -I·x1 + -I·x2

POL(U62(x1)) = 4A + 0A·x1

POL(U31(x1)) = 4A + -I·x1

POL(U41(x1, x2)) = 4A + -I·x1 + -I·x2

POL(U42(x1)) = 4A + -I·x1

POL(U52(x1)) = 2A + 0A·x1

POL(U72(x1, x2)) = 4A + -I·x1 + 1A·x2

POL(U92(x1, x2, x3, x4)) = 4A + 0A·x1 + 2A·x2 + 2A·x3 + -I·x4

POL(U93(x1, x2, x3, x4)) = 4A + -I·x1 + 2A·x2 + 2A·x3 + -I·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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
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)
cons(X1, X2) → n__cons(X1, X2)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
0 → n__0
length(nil) → 0
length(X) → n__length(X)
s(X) → n__s(X)
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
isNatList(n__nil) → 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)))
isNatIList(n__zeros) → tt
U62(tt) → tt
isNatIList(V) → U31(isNatList(activate(V)))
U31(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)))
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)))
zeros → cons(0, n__zeros)
zeros → n__zeros
nil → n__nil

(131) Obligation:

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

ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(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__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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(132) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(activate(x0), x1)), activate(y1))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

POL(ISNATILIST(x1)) = 1 +
[0,1]
·x1

POL(n__cons(x1, x2)) =
/1\
\0/
 +
/10\
\10/
·x1 +
/00\
\01/
·x2

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

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

POL(cons(x1, x2)) =
/1\
\0/
 +
/10\
\10/
·x1 +
/00\
\01/
·x2

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

POL(tt) =
/1\
\0/

POL(n__0) =
/0\
\1/

POL(n__zeros) =
/1\
\1/

POL(0) =
/0\
\1/

POL(zeros) =
/1\
\1/

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

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

POL(n__length(x1)) =
/1\
\1/
 +
/11\
\01/
·x1

POL(length(x1)) =
/1\
\1/
 +
/11\
\01/
·x1

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

POL(s(x1)) =
/0\
\1/
 +
/00\
\00/
·x1

POL(n__nil) =
/1\
\0/

POL(nil) =
/1\
\0/

POL(U11(x1)) =
/1\
\0/
 +
/00\
\00/
·x1

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

POL(U21(x1)) =
/1\
\0/
 +
/00\
\00/
·x1

POL(U71(x1, x2, x3)) =
/0\
\1/
 +
/00\
\00/
·x1 +
/01\
\01/
·x2 +
/00\
\00/
·x3

POL(U51(x1, x2)) =
/1\
\1/
 +
/01\
\01/
·x1 +
/00\
\00/
·x2

POL(U61(x1, x2)) =
/1\
\0/
 +
/10\
\00/
·x1 +
/10\
\00/
·x2

POL(U62(x1)) =
/0\
\0/
 +
/10\
\00/
·x1

POL(isNatIList(x1)) =
/1\
\0/
 +
/10\
\00/
·x1

POL(U31(x1)) =
/1\
\0/
 +
/01\
\00/
·x1

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

POL(U42(x1)) =
/1\
\0/
 +
/00\
\00/
·x1

POL(U52(x1)) =
/1\
\0/
 +
/00\
\00/
·x1

POL(U72(x1, x2)) =
/0\
\1/
 +
/00\
\00/
·x1 +
/01\
\01/
·x2

POL(U81(x1)) =
/1\
\0/
 +
/00\
\01/
·x1

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

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

POL(U93(x1, x2, x3, x4)) =
/1\
\0/
 +
/01\
\00/
·x1 +
/00\
\01/
·x2 +
/00\
\00/
·x3 +
/10\
\10/
·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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
cons(X1, X2) → n__cons(X1, X2)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
0 → n__0
take(X1, X2) → n__take(X1, X2)
length(nil) → 0
length(X) → n__length(X)
s(X) → n__s(X)
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
isNatList(n__nil) → 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)))
isNatIList(n__zeros) → tt
U62(tt) → tt
isNatIList(V) → U31(isNatList(activate(V)))
U31(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)))
take(0, IL) → U81(isNatIList(IL))
U81(tt) → nil
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)))
zeros → cons(0, n__zeros)
zeros → n__zeros
nil → n__nil

(133) Obligation:

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

U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(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__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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(134) 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.



(135) NO

(136) 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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(137) 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 + 2

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

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

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) ) = 2x3

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

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

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

POL( isNatList(x1) ) = x1

POL( length(x1) ) = x1

POL( s(x1) ) = 2x1

POL( take(x1, x2) ) = x1

POL( isNat(x1) ) = x1

POL( U11(x1) ) = x1

POL( U21(x1) ) = max{0, 2x1 - 1}

POL( U31(x1) ) = 1

POL( U42(x1) ) = 1

POL( isNatIList(x1) ) = 2

POL( U52(x1) ) = 2x1

POL( U62(x1) ) = 1

POL( n__take(x1, x2) ) = x1

POL( activate(x1) ) = x1

POL( n__zeros ) = 0

POL( zeros ) = 0

POL( n__0 ) = 1

POL( 0 ) = 1

POL( n__length(x1) ) = x1

POL( n__s(x1) ) = 2x1

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

POL( n__nil ) = 1

POL( nil ) = 1

POL( tt ) = 1

POL( U81(x1) ) = 1


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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__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)
length(nil) → 0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
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(V) → U31(isNatList(activate(V)))
U31(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)))
take(0, IL) → U81(isNatIList(IL))
U81(tt) → nil
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)))
zeros → cons(0, n__zeros)
zeros → n__zeros
0 → n__0
nil → n__nil

(138) 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(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(139) DependencyGraphProof (EQUIVALENT transformation)

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

(140) TRUE