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

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

(0) Obligation:

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

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

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

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

(2) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 18 less nodes.

(4) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(5) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
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)
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) ) = 2x1 + 2


POL( ISNATLIST(x1) ) = x1 + 2


POL( LENGTH(x1) ) = x1 + 2


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


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


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


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


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


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


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


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


POL( U41(x1, x2) ) = 1


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


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


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


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


POL( U91(x1, ..., x4) ) = 2x2 + x3 + 2x4 + 1


POL( U92(x1, ..., x4) ) = 2x2 + x3 + 2x4 + 1


POL( U93(x1, ..., x4) ) = 2x2 + x3 + 2x4 + 1


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


POL( isNat(x1) ) = 1


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


POL( isNatIList(x1) ) = 1


POL( U11(x1) ) = 1


POL( U21(x1) ) = 0


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


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


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


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


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


POL( length(x1) ) = 2x1


POL( s(x1) ) = x1


POL( activate(x1) ) = x1


POL( n__zeros ) = 0


POL( zeros ) = 0


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


POL( n__0 ) = 0


POL( 0 ) = 0


POL( n__length(x1) ) = 2x1


POL( n__s(x1) ) = x1


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


POL( n__nil ) = 0


POL( nil ) = 0


POL( tt ) = 0


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


POL( ACTIVATE(x1) ) = x1 + 2


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



The following usable rules [FROCOS05] were oriented:

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

(6) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 25 less nodes.

(8) Complex Obligation (AND)

(9) 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))
U511(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__length(X)) → LENGTH(X)
LENGTH(cons(N, L)) → U711(isNatList(activate(L)), activate(L), N)
U711(tt, L, N) → U721(isNat(activate(N)), activate(L))
U721(tt, L) → LENGTH(activate(L))
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__length(V1)) → ISNATLIST(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNAT(n__length(V1)) → ACTIVATE(V1)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
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)

The TRS R consists of the following rules:

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

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

(10) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U511(tt, V2) → ACTIVATE(V2)
ISNAT(n__length(V1)) → ISNATLIST(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNAT(n__length(V1)) → ACTIVATE(V1)
ISNAT(n__s(V1)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → ACTIVATE(L)
U721(tt, L) → ACTIVATE(L)
U711(tt, L, N) → ACTIVATE(N)
U711(tt, L, N) → ACTIVATE(L)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:

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


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


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


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


POL( U711(x1, ..., x3) ) = 2x2 + 2x3 + 1


POL( U721(x1, x2) ) = 2x2 + 1


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


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


POL( U61(x1, x2) ) = 2


POL( U71(x1, ..., x3) ) = 2x2 + 1


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


POL( U91(x1, ..., x4) ) = 2x2 + 2x3 + 2x4 + 1


POL( U92(x1, ..., x4) ) = 2x2 + 2x3 + 2x4 + 1


POL( U93(x1, ..., x4) ) = 2x2 + 2x3 + 2x4 + 1


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


POL( isNat(x1) ) = x1


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


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


POL( U21(x1) ) = x1


POL( U31(x1) ) = 0


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


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


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


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


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


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


POL( s(x1) ) = x1


POL( activate(x1) ) = x1


POL( n__zeros ) = 0


POL( zeros ) = 0


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


POL( n__0 ) = 0


POL( 0 ) = 0


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


POL( n__s(x1) ) = x1


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


POL( n__nil ) = 0


POL( nil ) = 0


POL( tt ) = 0


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


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



The following usable rules [FROCOS05] were oriented:

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

(11) 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))
ACTIVATE(n__length(X)) → LENGTH(X)
LENGTH(cons(N, L)) → U711(isNatList(activate(L)), activate(L), N)
U711(tt, L, N) → U721(isNat(activate(N)), activate(L))
U721(tt, L) → LENGTH(activate(L))
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
U711(tt, L, N) → ISNAT(activate(N))

The TRS R consists of the following rules:

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

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

(12) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 4 less nodes.

(13) Complex Obligation (AND)

(14) Obligation:

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

ISNAT(n__s(V1)) → ISNAT(activate(V1))

The TRS R consists of the following rules:

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

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

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ISNAT(n__s(V1)) → ISNAT(activate(V1))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(ISNAT(x1)) = -I + 2A·x1

POL(n__s(x1)) = 2A + 1A·x1

POL(activate(x1)) = 1A + 0A·x1

POL(n__zeros) = 0A

POL(zeros) = 1A

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

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

POL(n__0) = 2A

POL(0) = 2A

POL(n__length(x1)) = 2A + 0A·x1

POL(length(x1)) = 2A + 0A·x1

POL(s(x1)) = 2A + 1A·x1

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

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

POL(n__nil) = 2A

POL(nil) = 2A

POL(U81(x1)) = 2A + -I·x1

POL(isNatIList(x1)) = 2A + 0A·x1

POL(tt) = 2A

POL(U31(x1)) = -I + 0A·x1

POL(isNatList(x1)) = 1A + 0A·x1

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

POL(U51(x1, x2)) = 1A + -I·x1 + 0A·x2

POL(isNat(x1)) = -I + 0A·x1

POL(U11(x1)) = 2A + 0A·x1

POL(U61(x1, x2)) = -I + 0A·x1 + 0A·x2

POL(U21(x1)) = 2A + -I·x1

POL(U62(x1)) = 0A + 0A·x1

POL(U41(x1, x2)) = 2A + -I·x1 + 0A·x2

POL(U42(x1)) = 0A + 0A·x1

POL(U52(x1)) = 0A + 0A·x1

POL(U72(x1, x2)) = 3A + -I·x1 + 1A·x2

POL(U91(x1, x2, x3, x4)) = 0A + 0A·x1 + 1A·x2 + 1A·x3 + -I·x4

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

POL(U93(x1, x2, x3, x4)) = 2A + -I·x1 + 1A·x2 + 1A·x3 + -I·x4

The following usable rules [FROCOS05] were oriented:

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

(16) Obligation:

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

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

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

(17) PisEmptyProof (EQUIVALENT transformation)

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

(18) YES

(19) 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, V2) → ISNATLIST(activate(V2))

The TRS R consists of the following rules:

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

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

(20) Narrowing (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__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
ISNATLIST(n__cons(n__0, y1)) → U511(isNat(0), activate(y1))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(x0)), activate(y1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(x0, x1)), activate(y1))
ISNATLIST(n__cons(n__nil, y1)) → U511(isNat(nil), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))

(21) Obligation:

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

U511(tt, V2) → ISNATLIST(activate(V2))
ISNATLIST(n__cons(n__zeros, y1)) → U511(isNat(zeros), activate(y1))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
ISNATLIST(n__cons(n__0, y1)) → U511(isNat(0), activate(y1))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(x0)), activate(y1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(x0, x1)), activate(y1))
ISNATLIST(n__cons(n__nil, y1)) → U511(isNat(nil), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))

The TRS R consists of the following rules:

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

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

(22) Narrowing (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__take(x0, x1)) → ISNATLIST(take(x0, x1))
U511(tt, n__0) → ISNATLIST(0)
U511(tt, n__length(x0)) → ISNATLIST(length(x0))
U511(tt, n__s(x0)) → ISNATLIST(s(x0))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(x0, x1))
U511(tt, n__nil) → ISNATLIST(nil)
U511(tt, x0) → ISNATLIST(x0)

(23) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(24) Narrowing (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(n__zeros), activate(y0))

(25) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(26) DependencyGraphProof (EQUIVALENT transformation)

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

(27) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(28) Narrowing (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(n__zeros)

(29) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(30) DependencyGraphProof (EQUIVALENT transformation)

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

(31) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(32) Narrowing (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)

(33) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(34) DependencyGraphProof (EQUIVALENT transformation)

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

(35) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(36) Narrowing (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))

(37) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(38) Narrowing (EQUIVALENT transformation)

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

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

(39) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(40) DependencyGraphProof (EQUIVALENT transformation)

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

(41) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(42) Narrowing (EQUIVALENT transformation)

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

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

(43) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(44) Narrowing (EQUIVALENT transformation)

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

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

(45) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(46) Narrowing (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)

(47) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(48) DependencyGraphProof (EQUIVALENT transformation)

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

(49) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(50) Narrowing (EQUIVALENT transformation)

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

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

(51) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(52) DependencyGraphProof (EQUIVALENT transformation)

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

(53) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(54) Narrowing (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))

(55) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(56) DependencyGraphProof (EQUIVALENT transformation)

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

(57) Obligation:

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

ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(x0))
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__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__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(n__s(x0)), activate(y1))

The TRS R consists of the following rules:

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

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

(58) Narrowing (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(cons(n__0, n__zeros)), activate(y0))

(59) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(60) DependencyGraphProof (EQUIVALENT transformation)

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

(61) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(62) Narrowing (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(cons(n__0, n__zeros))

(63) Obligation:

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

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

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

(64) Narrowing (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))

(65) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(66) DependencyGraphProof (EQUIVALENT transformation)

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

(67) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(68) Narrowing (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))

(69) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(70) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

POL( ISNATLIST(x1) ) = x1 + 2


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


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


POL( 0 ) = 0


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


POL( isNatIList(x1) ) = 1


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


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


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


POL( activate(x1) ) = x1


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


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


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


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


POL( U61(x1, x2) ) = max{0, -1}


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


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


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


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


POL( n__0 ) = 0


POL( tt ) = 0


POL( n__length(x1) ) = 2


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


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


POL( n__s(x1) ) = 0


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


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


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


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


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


POL( length(x1) ) = 2


POL( n__zeros ) = 0


POL( zeros ) = 0


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


POL( n__nil ) = 0


POL( nil ) = 0



The following usable rules [FROCOS05] were oriented:

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

(71) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(72) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

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


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


POL( take(x1, x2) ) = x2


POL( 0 ) = 0


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


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


POL( s(x1) ) = 2


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


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


POL( activate(x1) ) = x1


POL( n__take(x1, x2) ) = x2


POL( length(x1) ) = x1 + 2


POL( nil ) = 0


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


POL( isNatList(x1) ) = 2


POL( n__length(x1) ) = x1 + 2


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


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


POL( U51(x1, x2) ) = 2


POL( U61(x1, x2) ) = 2


POL( U72(x1, x2) ) = 2


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


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


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


POL( n__0 ) = 0


POL( tt ) = 0


POL( U11(x1) ) = 1


POL( n__s(x1) ) = 2


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


POL( U42(x1) ) = 0


POL( U52(x1) ) = 2


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


POL( n__zeros ) = 0


POL( zeros ) = 0


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


POL( n__nil ) = 0



The following usable rules [FROCOS05] were oriented:

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

(73) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(74) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

POL( ISNATLIST(x1) ) = x1


POL( U511(x1, x2) ) = max{0, x1 + 2x2 - 1}


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


POL( 0 ) = 0


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


POL( isNatIList(x1) ) = 2


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


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


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


POL( activate(x1) ) = x1


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


POL( length(x1) ) = 2


POL( nil ) = 0


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


POL( isNatList(x1) ) = 1


POL( n__length(x1) ) = 2


POL( U21(x1) ) = 1


POL( U41(x1, x2) ) = x1


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


POL( U61(x1, x2) ) = 1


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


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


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


POL( isNat(x1) ) = 1


POL( n__0 ) = 0


POL( tt ) = 1


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


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


POL( U31(x1) ) = 1


POL( U42(x1) ) = 1


POL( U52(x1) ) = 1


POL( U62(x1) ) = 1


POL( n__zeros ) = 0


POL( zeros ) = 0


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


POL( n__nil ) = 0



The following usable rules [FROCOS05] were oriented:

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

(75) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(76) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

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

POL(tt) = 2A

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

POL(ISNATLIST(x1)) = 2A + 0A·x1

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

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

POL(isNat(x1)) = 2A + 1A·x1

POL(activate(x1)) = 1A + 0A·x1

POL(n__0) = 0A

POL(n__zeros) = 0A

POL(0) = 0A

POL(U81(x1)) = 2A + -I·x1

POL(isNatIList(x1)) = 1A + 1A·x1

POL(s(x1)) = 3A + 1A·x1

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

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

POL(n__length(x1)) = 2A + 0A·x1

POL(U11(x1)) = 2A + -I·x1

POL(isNatList(x1)) = -I + 0A·x1

POL(n__s(x1)) = 3A + 1A·x1

POL(U21(x1)) = 4A + 1A·x1

POL(zeros) = 1A

POL(length(x1)) = 2A + 0A·x1

POL(n__nil) = 2A

POL(nil) = 2A

POL(U31(x1)) = -I + 0A·x1

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

POL(U51(x1, x2)) = 1A + -I·x1 + 0A·x2

POL(U61(x1, x2)) = 2A + -I·x1 + 0A·x2

POL(U62(x1)) = 2A + -I·x1

POL(U41(x1, x2)) = 2A + -I·x1 + 0A·x2

POL(U42(x1)) = 2A + -I·x1

POL(U52(x1)) = -I + 0A·x1

POL(U72(x1, x2)) = 3A + 0A·x1 + 1A·x2

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

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

The following usable rules [FROCOS05] were oriented:

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

(77) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(78) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(79) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


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) ) = x1


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


POL( U721(x1, x2) ) = x2


POL( U41(x1, x2) ) = 1


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( 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) ) = 1


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


POL( U62(x1) ) = 1


POL( n__take(x1, x2) ) = x1


POL( length(x1) ) = x1


POL( s(x1) ) = 2x1


POL( activate(x1) ) = x1


POL( n__zeros ) = 0


POL( zeros ) = 0


POL( take(x1, x2) ) = x1


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] were oriented:

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

(80) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(81) DependencyGraphProof (EQUIVALENT transformation)

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

(82) TRUE

(83) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(84) Narrowing (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__take(x0, x1), y1)) → U411(isNat(take(x0, x1)), activate(y1))
ISNATILIST(n__cons(n__0, y1)) → U411(isNat(0), activate(y1))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(x0)), activate(y1))
ISNATILIST(n__cons(n__s(x0), y1)) → U411(isNat(s(x0)), activate(y1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U411(isNat(cons(x0, x1)), activate(y1))
ISNATILIST(n__cons(n__nil, y1)) → U411(isNat(nil), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U411(isNat(x0), activate(y1))

(85) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(86) Narrowing (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__take(x0, x1)) → ISNATILIST(take(x0, x1))
U411(tt, n__0) → ISNATILIST(0)
U411(tt, n__length(x0)) → ISNATILIST(length(x0))
U411(tt, n__s(x0)) → ISNATILIST(s(x0))
U411(tt, n__cons(x0, x1)) → ISNATILIST(cons(x0, x1))
U411(tt, n__nil) → ISNATILIST(nil)
U411(tt, x0) → ISNATILIST(x0)

(87) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(88) Narrowing (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(n__zeros), activate(y0))

(89) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(90) DependencyGraphProof (EQUIVALENT transformation)

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

(91) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(92) Narrowing (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(n__zeros)

(93) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(94) DependencyGraphProof (EQUIVALENT transformation)

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

(95) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(96) Narrowing (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)

(97) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(98) DependencyGraphProof (EQUIVALENT transformation)

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

(99) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(100) Narrowing (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))

(101) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(102) Narrowing (EQUIVALENT transformation)

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

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

(103) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(104) DependencyGraphProof (EQUIVALENT transformation)

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

(105) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(106) Narrowing (EQUIVALENT transformation)

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

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

(107) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(108) Narrowing (EQUIVALENT transformation)

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

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

(109) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(110) Narrowing (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)

(111) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(112) DependencyGraphProof (EQUIVALENT transformation)

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

(113) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(114) Narrowing (EQUIVALENT transformation)

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

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

(115) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(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:

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

The TRS R consists of the following rules:

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

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

(118) Narrowing (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))

(119) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(120) DependencyGraphProof (EQUIVALENT transformation)

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

(121) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(122) Narrowing (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(cons(n__0, n__zeros)), activate(y0))

(123) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(124) DependencyGraphProof (EQUIVALENT transformation)

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

(125) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(126) Narrowing (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(cons(n__0, n__zeros))

(127) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(128) Narrowing (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))

(129) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(130) DependencyGraphProof (EQUIVALENT transformation)

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

(131) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(132) Narrowing (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))

(133) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(134) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ISNATILIST(n__cons(n__take(x0, x1), y1)) → U411(isNat(take(x0, x1)), activate(y1))
ISNATILIST(n__cons(n__length(x0), y1)) → U411(isNat(length(x0)), activate(y1))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ISNATILIST(x1) ) = x1 + 2


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


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


POL( 0 ) = 0


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


POL( isNatIList(x1) ) = 1


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


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


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


POL( activate(x1) ) = x1


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


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


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


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


POL( U61(x1, x2) ) = max{0, -1}


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


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


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


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


POL( n__0 ) = 0


POL( tt ) = 0


POL( n__length(x1) ) = 2


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


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


POL( n__s(x1) ) = 0


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


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


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


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


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


POL( length(x1) ) = 2


POL( n__zeros ) = 0


POL( zeros ) = 0


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


POL( n__nil ) = 0


POL( nil ) = 0



The following usable rules [FROCOS05] were oriented:

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

(135) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(136) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

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


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


POL( take(x1, x2) ) = x2


POL( 0 ) = 0


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


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


POL( s(x1) ) = 2


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


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


POL( activate(x1) ) = x1


POL( n__take(x1, x2) ) = x2


POL( length(x1) ) = x1 + 2


POL( nil ) = 0


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


POL( isNatList(x1) ) = 2


POL( n__length(x1) ) = x1 + 2


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


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


POL( U51(x1, x2) ) = 2


POL( U61(x1, x2) ) = 2


POL( U72(x1, x2) ) = 2


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


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


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


POL( n__0 ) = 0


POL( tt ) = 0


POL( U11(x1) ) = 1


POL( n__s(x1) ) = 2


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


POL( U42(x1) ) = 0


POL( U52(x1) ) = 2


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


POL( n__zeros ) = 0


POL( zeros ) = 0


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


POL( n__nil ) = 0



The following usable rules [FROCOS05] were oriented:

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

(137) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(138) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

POL( ISNATILIST(x1) ) = x1


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


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


POL( 0 ) = 0


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


POL( isNatIList(x1) ) = 2


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


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


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


POL( activate(x1) ) = x1


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


POL( length(x1) ) = 2


POL( nil ) = 0


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


POL( isNatList(x1) ) = 1


POL( n__length(x1) ) = 2


POL( U21(x1) ) = 1


POL( U41(x1, x2) ) = x1


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


POL( U61(x1, x2) ) = 1


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


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


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


POL( isNat(x1) ) = 1


POL( n__0 ) = 0


POL( tt ) = 1


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


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


POL( U31(x1) ) = 1


POL( U42(x1) ) = 1


POL( U52(x1) ) = 1


POL( U62(x1) ) = 1


POL( n__zeros ) = 0


POL( zeros ) = 0


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


POL( n__nil ) = 0



The following usable rules [FROCOS05] were oriented:

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

(139) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(140) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U411(tt, n__take(x0, x1)) → ISNATILIST(take(x0, x1))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

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

POL(tt) = 2A

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

POL(ISNATILIST(x1)) = 2A + 0A·x1

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

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

POL(isNat(x1)) = 2A + 1A·x1

POL(activate(x1)) = 1A + 0A·x1

POL(n__0) = 0A

POL(n__zeros) = 0A

POL(0) = 0A

POL(U81(x1)) = 2A + -I·x1

POL(isNatIList(x1)) = 1A + 1A·x1

POL(s(x1)) = 3A + 1A·x1

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

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

POL(n__length(x1)) = 2A + 0A·x1

POL(U11(x1)) = 2A + -I·x1

POL(isNatList(x1)) = -I + 0A·x1

POL(n__s(x1)) = 3A + 1A·x1

POL(U21(x1)) = 4A + 1A·x1

POL(zeros) = 1A

POL(length(x1)) = 2A + 0A·x1

POL(n__nil) = 2A

POL(nil) = 2A

POL(U31(x1)) = -I + 0A·x1

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

POL(U51(x1, x2)) = 1A + -I·x1 + 0A·x2

POL(U61(x1, x2)) = 2A + -I·x1 + 0A·x2

POL(U62(x1)) = 2A + -I·x1

POL(U41(x1, x2)) = 2A + -I·x1 + 0A·x2

POL(U42(x1)) = 2A + -I·x1

POL(U52(x1)) = -I + 0A·x1

POL(U72(x1, x2)) = 3A + 0A·x1 + 1A·x2

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

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

The following usable rules [FROCOS05] were oriented:

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

(141) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(142) NonTerminationProof (EQUIVALENT transformation)

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

s = 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.



(143) NO