Termination w.r.t. Q proof of /tmp/tmp32fHT4/LengthOfFiniteLists_nokinds-noand

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

0 QTRS

↳1 QTRSRRRProof (⇔)

↳2 QTRS

↳3 QTRSRRRProof (⇔)

↳4 QTRS

↳5 QTRSRRRProof (⇔)

↳6 QTRS

↳7 QTRSRRRProof (⇔)

↳8 QTRS

↳9 DependencyPairsProof (⇔)

↳10 QDP

↳11 DependencyGraphProof (⇔)

↳12 AND

  ↳13 QDP

    ↳14 UsableRulesReductionPairsProof (⇔)

    ↳15 QDP

    ↳16 DependencyGraphProof (⇔)

    ↳17 AND

      ↳18 QDP

        ↳19 QDPOrderProof (⇔)

        ↳20 QDP

        ↳21 PisEmptyProof (⇔)

        ↳22 TRUE

      ↳23 QDP

        ↳24 Narrowing (⇔)

        ↳25 QDP

        ↳26 Narrowing (⇔)

        ↳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 DependencyGraphProof (⇔)

        ↳45 QDP

        ↳46 Narrowing (⇔)

        ↳47 QDP

        ↳48 Narrowing (⇔)

        ↳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 DependencyGraphProof (⇔)

        ↳71 QDP

        ↳72 Narrowing (⇔)

        ↳73 QDP

        ↳74 MRRProof (⇔)

        ↳75 QDP

        ↳76 MRRProof (⇔)

        ↳77 QDP

        ↳78 QDPOrderProof (⇔)

        ↳79 QDP

        ↳80 NonTerminationProof (⇔)

        ↳81 FALSE

      ↳82 QDP

  ↳83 QDP

    ↳84 UsableRulesReductionPairsProof (⇔)

    ↳85 QDP

    ↳86 Narrowing (⇔)

    ↳87 QDP

    ↳88 Narrowing (⇔)

    ↳89 QDP

    ↳90 Narrowing (⇔)

    ↳91 QDP

    ↳92 DependencyGraphProof (⇔)

    ↳93 QDP

    ↳94 Narrowing (⇔)

    ↳95 QDP

    ↳96 DependencyGraphProof (⇔)

    ↳97 QDP

    ↳98 Narrowing (⇔)

    ↳99 QDP

    ↳100 Narrowing (⇔)

    ↳101 QDP

    ↳102 DependencyGraphProof (⇔)

    ↳103 QDP

    ↳104 Narrowing (⇔)

    ↳105 QDP

    ↳106 DependencyGraphProof (⇔)

    ↳107 QDP

    ↳108 Narrowing (⇔)

    ↳109 QDP

    ↳110 Narrowing (⇔)

    ↳111 QDP

    ↳112 Narrowing (⇔)

    ↳113 QDP

    ↳114 DependencyGraphProof (⇔)

    ↳115 QDP

    ↳116 Narrowing (⇔)

    ↳117 QDP

    ↳118 DependencyGraphProof (⇔)

    ↳119 QDP

    ↳120 Narrowing (⇔)

    ↳121 QDP

    ↳122 DependencyGraphProof (⇔)

    ↳123 QDP

    ↳124 Narrowing (⇔)

    ↳125 QDP

    ↳126 Narrowing (⇔)

    ↳127 QDP

    ↳128 DependencyGraphProof (⇔)

    ↳129 QDP

    ↳130 Narrowing (⇔)

    ↳131 QDP

    ↳132 DependencyGraphProof (⇔)

    ↳133 QDP

    ↳134 Narrowing (⇔)

    ↳135 QDP

    ↳136 MRRProof (⇔)

    ↳137 QDP

    ↳138 MRRProof (⇔)

    ↳139 QDP

    ↳140 QDPOrderProof (⇔)

    ↳141 QDP

    ↳142 NonTerminationProof (⇔)

    ↳143 FALSE

(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, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
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))
length(nil) → 0
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
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__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) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(U11(x₁)) = x₁
POL(U21(x₁)) = x₁
POL(U31(x₁)) = x₁
POL(U41(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U42(x₁)) = x₁
POL(U51(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U52(x₁)) = x₁
POL(U61(x₁, x₂, x₃)) = x₁ + 2·x₂ + 2·x₃
POL(U62(x₁, x₂)) = 2·x₁ + 2·x₂
POL(activate(x₁)) = x₁
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = 2·x₁
POL(isNatList(x₁)) = 2·x₁
POL(length(x₁)) = 2·x₁
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(n__length(x₁)) = 2·x₁
POL(n__nil) = 1
POL(n__s(x₁)) = x₁
POL(n__zeros) = 0
POL(nil) = 1
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

isNatList(n__nil) → tt
length(nil) → 0

(2) 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, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
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__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
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__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.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(U11(x₁)) = 1 + x₁
POL(U21(x₁)) = x₁
POL(U31(x₁)) = x₁
POL(U41(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U42(x₁)) = x₁
POL(U51(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U52(x₁)) = x₁
POL(U61(x₁, x₂, x₃)) = 1 + x₁ + 2·x₂ + 2·x₃
POL(U62(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(activate(x₁)) = x₁
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = 2·x₁
POL(isNatIList(x₁)) = 2·x₁
POL(isNatList(x₁)) = 2·x₁
POL(length(x₁)) = 1 + 2·x₁
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(n__length(x₁)) = 1 + 2·x₁
POL(n__nil) = 0
POL(n__s(x₁)) = x₁
POL(n__zeros) = 0
POL(nil) = 0
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

U11(tt) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))

(4) Obligation:

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

zeros → cons(0, n__zeros)
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, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
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__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
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__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.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(U21(x₁)) = x₁
POL(U31(x₁)) = x₁
POL(U41(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(U42(x₁)) = x₁
POL(U51(x₁, x₂)) = x₁ + 2·x₂
POL(U52(x₁)) = x₁
POL(U61(x₁, x₂, x₃)) = x₁ + 2·x₂ + x₃
POL(U62(x₁, x₂)) = x₁ + 2·x₂
POL(activate(x₁)) = x₁
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = 1 + 2·x₁
POL(isNatList(x₁)) = 2·x₁
POL(length(x₁)) = 2·x₁
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(n__length(x₁)) = 2·x₁
POL(n__nil) = 0
POL(n__s(x₁)) = x₁
POL(n__zeros) = 0
POL(nil) = 0
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt

(6) Obligation:

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

zeros → cons(0, n__zeros)
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, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
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__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.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(U21(x₁)) = x₁
POL(U31(x₁)) = 1 + 2·x₁
POL(U41(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U42(x₁)) = 2·x₁
POL(U51(x₁, x₂)) = x₁ + 2·x₂
POL(U52(x₁)) = x₁
POL(U61(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + 2·x₃
POL(U62(x₁, x₂)) = 2·x₁ + 2·x₂
POL(activate(x₁)) = x₁
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 2·x₁
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(n__length(x₁)) = 2·x₁
POL(n__nil) = 0
POL(n__s(x₁)) = x₁
POL(n__zeros) = 0
POL(nil) = 0
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

U31(tt) → tt

(8) Obligation:

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

zeros → cons(0, n__zeros)
U21(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
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__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.

(9) DependencyPairsProof (EQUIVALENT transformation)

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

(10) 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, L, N) → U62¹(isNat(activate(N)), activate(L))
U61¹(tt, L, N) → ISNAT(activate(N))
U61¹(tt, L, N) → ACTIVATE(N)
U61¹(tt, L, N) → ACTIVATE(L)
U62¹(tt, L) → S(length(activate(L)))
U62¹(tt, L) → LENGTH(activate(L))
U62¹(tt, L) → ACTIVATE(L)
ISNAT(n__s(V1)) → U21¹(isNat(activate(V1)))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
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)
LENGTH(cons(N, L)) → U61¹(isNatList(activate(L)), activate(L), N)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
LENGTH(cons(N, L)) → ACTIVATE(L)
ACTIVATE(n__zeros) → ZEROS
ACTIVATE(n__0) → 0¹
ACTIVATE(n__length(X)) → LENGTH(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)
U21(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
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__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.

(11) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 15 less nodes.

(12) Complex Obligation (AND)

(13) Obligation:

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

ACTIVATE(n__length(X)) → LENGTH(X)
LENGTH(cons(N, L)) → U61¹(isNatList(activate(L)), activate(L), N)
U61¹(tt, L, N) → U62¹(isNat(activate(N)), activate(L))
U62¹(tt, L) → LENGTH(activate(L))
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ISNATLIST(n__cons(V1, V2)) → U51¹(isNat(activate(V1)), activate(V2))
U51¹(tt, V2) → ISNATLIST(activate(V2))
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
U51¹(tt, V2) → ACTIVATE(V2)
LENGTH(cons(N, L)) → ACTIVATE(L)
U62¹(tt, L) → ACTIVATE(L)
U61¹(tt, L, N) → ISNAT(activate(N))
U61¹(tt, L, N) → ACTIVATE(N)
U61¹(tt, L, N) → ACTIVATE(L)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U21(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
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__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.

(14) UsableRulesReductionPairsProof (EQUIVALENT transformation)

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

ACTIVATE(n__length(X)) → LENGTH(X)

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(0) = 0
POL(ACTIVATE(x₁)) = 2·x₁
POL(ISNAT(x₁)) = 2·x₁
POL(ISNATLIST(x₁)) = 2·x₁
POL(LENGTH(x₁)) = 2·x₁
POL(U21(x₁)) = x₁
POL(U51(x₁, x₂)) = x₁ + 2·x₂
POL(U51¹(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U52(x₁)) = 2·x₁
POL(U61(x₁, x₂, x₃)) = 2 + 2·x₁ + 2·x₂ + 2·x₃
POL(U61¹(x₁, x₂, x₃)) = x₁ + 2·x₂ + 2·x₃
POL(U62(x₁, x₂)) = 2 + x₁ + 2·x₂
POL(U62¹(x₁, x₂)) = x₁ + 2·x₂
POL(activate(x₁)) = x₁
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = 2·x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 2 + 2·x₁
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(n__length(x₁)) = 2 + 2·x₁
POL(n__nil) = 0
POL(n__s(x₁)) = x₁
POL(n__zeros) = 0
POL(nil) = 0
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

(15) Obligation:

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

LENGTH(cons(N, L)) → U61¹(isNatList(activate(L)), activate(L), N)
U61¹(tt, L, N) → U62¹(isNat(activate(N)), activate(L))
U62¹(tt, L) → LENGTH(activate(L))
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ISNATLIST(n__cons(V1, V2)) → U51¹(isNat(activate(V1)), activate(V2))
U51¹(tt, V2) → ISNATLIST(activate(V2))
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
U51¹(tt, V2) → ACTIVATE(V2)
LENGTH(cons(N, L)) → ACTIVATE(L)
U62¹(tt, L) → ACTIVATE(L)
U61¹(tt, L, N) → ISNAT(activate(N))
U61¹(tt, L, N) → ACTIVATE(N)
U61¹(tt, L, N) → ACTIVATE(L)

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(16) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 11 less nodes.

(17) Complex Obligation (AND)

(18) 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:

activate(n__zeros) → zeros
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
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(19) 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: Polynomial interpretation [POLO]:

POL(0) = 0
POL(ISNAT(x₁)) = x₁
POL(U21(x₁)) = 1
POL(U51(x₁, x₂)) = 0
POL(U52(x₁)) = x₁
POL(U61(x₁, x₂, x₃)) = x₁
POL(U62(x₁, x₂)) = 1
POL(activate(x₁)) = x₁
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = x₁
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = 0
POL(n__length(x₁)) = 0
POL(n__nil) = 0
POL(n__s(x₁)) = 1 + x₁
POL(n__zeros) = 1
POL(nil) = 0
POL(s(x₁)) = 1 + x₁
POL(tt) = 1
POL(zeros) = 1

The following usable rules [FROCOS05] were oriented:

isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(X) → n__length(X)
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
s(X) → n__s(X)
0 → n__0
U52(tt) → tt
zeros → n__zeros
zeros → cons(0, n__zeros)
U62(tt, L) → s(length(activate(L)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U21(tt) → tt
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)

(20) Obligation:

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

activate(n__zeros) → zeros
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
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(21) PisEmptyProof (EQUIVALENT transformation)

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

(22) TRUE

(23) Obligation:

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(24) 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__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)

(25) Obligation:

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

ISNATLIST(n__cons(V1, V2)) → U51¹(isNat(activate(V1)), activate(V2))
U51¹(tt, n__zeros) → ISNATLIST(zeros)
U51¹(tt, n__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:

activate(n__zeros) → zeros
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
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(26) 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__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))

(27) Obligation:

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

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

activate(n__zeros) → zeros
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
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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:

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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:

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(32) 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))

(33) Obligation:

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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__0, y1)) → U51¹(isNat(0), activate(y1))
U51¹(tt, n__0) → ISNATLIST(0)
ISNATLIST(n__cons(n__length(x0), y1)) → U51¹(isNat(length(x0)), activate(y1))
U51¹(tt, n__length(x0)) → ISNATLIST(length(x0))
ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(s(x0)), activate(y1))
U51¹(tt, n__s(x0)) → ISNATLIST(s(x0))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U51¹(isNat(cons(x0, x1)), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(x0, x1))
ISNATLIST(n__cons(n__nil, y1)) → U51¹(isNat(nil), activate(y1))
U51¹(tt, n__nil) → ISNATLIST(nil)
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__zeros, y0)) → U51¹(isNat(cons(0, n__zeros)), activate(y0))
U51¹(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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:

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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__0) → ISNATLIST(0) at position [0] we obtained the following new rules [LPAR04]:

U51¹(tt, n__0) → ISNATLIST(n__0)

(39) Obligation:

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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__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:

activate(n__zeros) → zeros
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
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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__s(x0)) → ISNATLIST(s(x0)) at position [0] we obtained the following new rules [LPAR04]:

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

(43) Obligation:

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

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:

activate(n__zeros) → zeros
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
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(44) DependencyGraphProof (EQUIVALENT transformation)

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

(45) Obligation:

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(46) 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))

(47) Obligation:

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(48) 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))

(49) Obligation:

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(54) 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)

(55) Obligation:

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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__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))

(59) Obligation:

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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(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))

(65) Obligation:

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(68) 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))

(69) Obligation:

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(70) DependencyGraphProof (EQUIVALENT transformation)

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

(71) Obligation:

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(72) 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))

(73) Obligation:

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(74) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

ISNATLIST(n__cons(n__length(x0), y1)) → U51¹(isNat(length(x0)), activate(y1))

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(ISNATLIST(x₁)) = 2·x₁
POL(U21(x₁)) = x₁
POL(U51(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U51¹(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U52(x₁)) = 2·x₁
POL(U61(x₁, x₂, x₃)) = 2 + x₁ + 2·x₂ + 2·x₃
POL(U62(x₁, x₂)) = 2 + x₁ + 2·x₂
POL(activate(x₁)) = x₁
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 2 + 2·x₁
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(n__length(x₁)) = 2 + 2·x₁
POL(n__nil) = 0
POL(n__s(x₁)) = x₁
POL(n__zeros) = 0
POL(nil) = 0
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

(75) Obligation:

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(76) MRRProof (EQUIVALENT transformation)

U51¹(tt, n__length(x0)) → ISNATLIST(length(x0))

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(ISNATLIST(x₁)) = x₁
POL(U21(x₁)) = x₁
POL(U51(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U51¹(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U52(x₁)) = 2·x₁
POL(U61(x₁, x₂, x₃)) = 1 + 2·x₁ + 2·x₂ + 2·x₃
POL(U62(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(activate(x₁)) = x₁
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 1 + 2·x₁
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(n__length(x₁)) = 1 + 2·x₁
POL(n__nil) = 0
POL(n__s(x₁)) = x₁
POL(n__zeros) = 0
POL(nil) = 0
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

(77) Obligation:

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(78) 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 interpretation [POLO]:

POL(0) = 0
POL(ISNATLIST(x₁)) = x₁
POL(U21(x₁)) = 0
POL(U51(x₁, x₂)) = x₂
POL(U51¹(x₁, x₂)) = x₂
POL(U52(x₁)) = 0
POL(U61(x₁, x₂, x₃)) = 1 + x₂
POL(U62(x₁, x₂)) = 1
POL(activate(x₁)) = x₁
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 0
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 1 + x₁
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = x₁ + x₂
POL(n__length(x₁)) = 1 + x₁
POL(n__nil) = 0
POL(n__s(x₁)) = 1
POL(n__zeros) = 0
POL(nil) = 0
POL(s(x₁)) = 1
POL(tt) = 0
POL(zeros) = 0

The following usable rules [FROCOS05] were oriented:

isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(X) → n__length(X)
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
s(X) → n__s(X)
0 → n__0
U52(tt) → tt
zeros → n__zeros
zeros → cons(0, n__zeros)
U62(tt, L) → s(length(activate(L)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U21(tt) → tt
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)

(79) Obligation:

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(80) 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 = U51¹(isNat(n__0), activate(n__zeros)) evaluates to t =U51¹(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

U51¹(isNat(n__0), activate(n__zeros)) → U51¹(isNat(n__0), n__zeros)
with rule activate(X) → X at position [1] and matcher [X / n__zeros]

U51¹(isNat(n__0), n__zeros) → U51¹(tt, n__zeros)
with rule isNat(n__0) → tt at position [0] and matcher [ ]

U51¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
with rule U51¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros)) at position [] and matcher [ ]

ISNATLIST(n__cons(n__0, n__zeros)) → U51¹(isNat(n__0), activate(n__zeros))
with rule ISNATLIST(n__cons(x0, y1)) → U51¹(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.

(81) FALSE

(82) Obligation:

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

U61¹(tt, L, N) → U62¹(isNat(activate(N)), activate(L))
U62¹(tt, L) → LENGTH(activate(L))
LENGTH(cons(N, L)) → U61¹(isNatList(activate(L)), activate(L), N)

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(83) Obligation:

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

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

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U21(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
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__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) UsableRulesReductionPairsProof (EQUIVALENT transformation)

POL(0) = 0
POL(ISNATILIST(x₁)) = x₁
POL(U21(x₁)) = x₁
POL(U41¹(x₁, x₂)) = x₁ + x₂
POL(U51(x₁, x₂)) = x₁ + 2·x₂
POL(U52(x₁)) = x₁
POL(U61(x₁, x₂, x₃)) = x₁ + 2·x₂ + 2·x₃
POL(U62(x₁, x₂)) = x₁ + 2·x₂
POL(activate(x₁)) = x₁
POL(cons(x₁, x₂)) = x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 2·x₁
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = x₁ + 2·x₂
POL(n__length(x₁)) = 2·x₁
POL(n__nil) = 2
POL(n__s(x₁)) = x₁
POL(n__zeros) = 0
POL(nil) = 2
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

(85) Obligation:

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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__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(V1, V2)) → U41¹(isNat(activate(V1)), activate(V2))
U41¹(tt, n__zeros) → ISNATILIST(zeros)
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:

activate(n__zeros) → zeros
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__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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(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__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))

(89) Obligation:

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

U41¹(tt, n__zeros) → ISNATILIST(zeros)
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, y1)) → U41¹(isNat(zeros), 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:

activate(n__zeros) → zeros
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__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(90) 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)

(91) Obligation:

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

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, y1)) → U41¹(isNat(zeros), 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(cons(0, n__zeros))
U41¹(tt, n__zeros) → ISNATILIST(n__zeros)

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(92) DependencyGraphProof (EQUIVALENT transformation)

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

(93) Obligation:

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

ISNATILIST(n__cons(n__zeros, y1)) → U41¹(isNat(zeros), 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))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(94) 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))

(95) Obligation:

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(96) DependencyGraphProof (EQUIVALENT transformation)

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

(97) Obligation:

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

ISNATILIST(n__cons(n__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))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(98) 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))

(99) Obligation:

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(100) 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)

(101) Obligation:

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(102) DependencyGraphProof (EQUIVALENT transformation)

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

(103) Obligation:

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

U41¹(tt, n__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:

activate(n__zeros) → zeros
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__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(104) 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))

(105) Obligation:

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

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:

activate(n__zeros) → zeros
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__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(106) DependencyGraphProof (EQUIVALENT transformation)

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

(107) Obligation:

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

ISNATILIST(n__cons(n__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__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:

activate(n__zeros) → zeros
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__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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:

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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__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))

(111) Obligation:

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

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, 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))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(n__s(x0)), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(112) 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))

(113) Obligation:

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

ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(x0)), activate(y1))
U41¹(tt, n__length(x0)) → ISNATILIST(length(x0))
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))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(n__s(x0)), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))
ISNATILIST(n__cons(n__cons(x0, x1), y2)) → U41¹(isNat(n__cons(x0, x1)), activate(y2))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(114) DependencyGraphProof (EQUIVALENT transformation)

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

(115) Obligation:

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

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__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__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))
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:

activate(n__zeros) → zeros
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__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(116) 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)

(117) Obligation:

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

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__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__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))
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))
U41¹(tt, n__nil) → ISNATILIST(n__nil)

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(118) DependencyGraphProof (EQUIVALENT transformation)

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

(119) Obligation:

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

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__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__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))
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:

activate(n__zeros) → zeros
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__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(120) 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))

(121) Obligation:

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

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(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__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))
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:

activate(n__zeros) → zeros
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__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(122) DependencyGraphProof (EQUIVALENT transformation)

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

(123) Obligation:

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

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__zeros, y0)) → U41¹(isNat(cons(0, n__zeros)), activate(y0))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))
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:

activate(n__zeros) → zeros
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__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(124) 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))

(125) Obligation:

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

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__zeros, y0)) → U41¹(isNat(cons(0, n__zeros)), activate(y0))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))
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))
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:

activate(n__zeros) → zeros
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__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(126) 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))

(127) Obligation:

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

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__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))
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))
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(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:

activate(n__zeros) → zeros
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__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(128) DependencyGraphProof (EQUIVALENT transformation)

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

(129) Obligation:

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(130) 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))

(131) Obligation:

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(132) DependencyGraphProof (EQUIVALENT transformation)

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

(133) Obligation:

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(134) 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))

(135) Obligation:

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(136) MRRProof (EQUIVALENT transformation)

ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(x0)), activate(y1))

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(ISNATILIST(x₁)) = 2·x₁
POL(U21(x₁)) = x₁
POL(U41¹(x₁, x₂)) = x₁ + 2·x₂
POL(U51(x₁, x₂)) = x₁ + 2·x₂
POL(U52(x₁)) = 2·x₁
POL(U61(x₁, x₂, x₃)) = 1 + x₁ + 2·x₂ + 2·x₃
POL(U62(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(activate(x₁)) = x₁
POL(cons(x₁, x₂)) = x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 1 + 2·x₁
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = x₁ + 2·x₂
POL(n__length(x₁)) = 1 + 2·x₁
POL(n__nil) = 0
POL(n__s(x₁)) = x₁
POL(n__zeros) = 0
POL(nil) = 0
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

(137) Obligation:

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

U41¹(tt, n__length(x0)) → ISNATILIST(length(x0))
U41¹(tt, x0) → ISNATILIST(x0)
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__0, y0)) → U41¹(isNat(n__0), activate(y0))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(n__s(x0)), activate(y1))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

(138) MRRProof (EQUIVALENT transformation)

U41¹(tt, n__length(x0)) → ISNATILIST(length(x0))

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(ISNATILIST(x₁)) = x₁
POL(U21(x₁)) = x₁
POL(U41¹(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U51(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U52(x₁)) = 2·x₁
POL(U61(x₁, x₂, x₃)) = 1 + 2·x₁ + 2·x₂ + 2·x₃
POL(U62(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(activate(x₁)) = x₁
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 1 + 2·x₁
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(n__length(x₁)) = 1 + 2·x₁
POL(n__nil) = 0
POL(n__s(x₁)) = x₁
POL(n__zeros) = 0
POL(nil) = 0
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

(139) Obligation:

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

U41¹(tt, x0) → ISNATILIST(x0)
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__0, y0)) → U41¹(isNat(n__0), activate(y0))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(n__s(x0)), activate(y1))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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.

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 interpretation [POLO]:

POL(0) = 0
POL(ISNATILIST(x₁)) = x₁
POL(U21(x₁)) = 0
POL(U41¹(x₁, x₂)) = x₂
POL(U51(x₁, x₂)) = 1
POL(U52(x₁)) = 0
POL(U61(x₁, x₂, x₃)) = 1
POL(U62(x₁, x₂)) = 1
POL(activate(x₁)) = x₁
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 0
POL(isNatList(x₁)) = 1 + x₁
POL(length(x₁)) = 1
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = x₁ + x₂
POL(n__length(x₁)) = 1
POL(n__nil) = 0
POL(n__s(x₁)) = 1
POL(n__zeros) = 0
POL(nil) = 0
POL(s(x₁)) = 1
POL(tt) = 0
POL(zeros) = 0

The following usable rules [FROCOS05] were oriented:

cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__s(X)) → s(X)
activate(X) → X
activate(n__nil) → nil
activate(n__zeros) → zeros
activate(n__length(X)) → length(X)
activate(n__0) → 0

(141) Obligation:

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

U41¹(tt, x0) → ISNATILIST(x0)
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__0, y0)) → U41¹(isNat(n__0), activate(y0))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
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__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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.