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

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

0 QTRS

↳1 QTRSRRRProof (⇔, 184 ms)

↳2 QTRS

↳3 QTRSRRRProof (⇔, 14 ms)

↳4 QTRS

↳5 QTRSRRRProof (⇔, 12 ms)

↳6 QTRS

↳7 QTRSRRRProof (⇔, 19 ms)

↳8 QTRS

↳9 DependencyPairsProof (⇔, 40 ms)

↳10 QDP

↳11 DependencyGraphProof (⇔, 0 ms)

↳12 AND

    ↳13 QDP

        ↳14 UsableRulesReductionPairsProof (⇔, 27 ms)

        ↳15 QDP

        ↳16 DependencyGraphProof (⇔, 0 ms)

        ↳17 AND

            ↳18 QDP

                ↳19 QDPOrderProof (⇔, 68 ms)

                ↳20 QDP

                ↳21 PisEmptyProof (⇔, 0 ms)

                ↳22 YES

            ↳23 QDP

                ↳24 Narrowing (⇔, 0 ms)

                ↳25 QDP

                ↳26 Narrowing (⇔, 0 ms)

                ↳27 QDP

                ↳28 Narrowing (⇔, 0 ms)

                ↳29 QDP

                ↳30 DependencyGraphProof (⇔, 0 ms)

                ↳31 QDP

                ↳32 Narrowing (⇔, 0 ms)

                ↳33 QDP

                ↳34 DependencyGraphProof (⇔, 0 ms)

                ↳35 QDP

                ↳36 Narrowing (⇔, 0 ms)

                ↳37 QDP

                ↳38 Narrowing (⇔, 0 ms)

                ↳39 QDP

                ↳40 DependencyGraphProof (⇔, 0 ms)

                ↳41 QDP

                ↳42 Narrowing (⇔, 0 ms)

                ↳43 QDP

                ↳44 DependencyGraphProof (⇔, 0 ms)

                ↳45 QDP

                ↳46 Narrowing (⇔, 0 ms)

                ↳47 QDP

                ↳48 Narrowing (⇔, 0 ms)

                ↳49 QDP

                ↳50 Narrowing (⇔, 0 ms)

                ↳51 QDP

                ↳52 DependencyGraphProof (⇔, 0 ms)

                ↳53 QDP

                ↳54 Narrowing (⇔, 0 ms)

                ↳55 QDP

                ↳56 DependencyGraphProof (⇔, 0 ms)

                ↳57 QDP

                ↳58 Narrowing (⇔, 0 ms)

                ↳59 QDP

                ↳60 DependencyGraphProof (⇔, 0 ms)

                ↳61 QDP

                ↳62 Narrowing (⇔, 0 ms)

                ↳63 QDP

                ↳64 Narrowing (⇔, 0 ms)

                ↳65 QDP

                ↳66 DependencyGraphProof (⇔, 0 ms)

                ↳67 QDP

                ↳68 Narrowing (⇔, 0 ms)

                ↳69 QDP

                ↳70 DependencyGraphProof (⇔, 0 ms)

                ↳71 QDP

                ↳72 Narrowing (⇔, 0 ms)

                ↳73 QDP

                ↳74 MRRProof (⇔, 29 ms)

                ↳75 QDP

                ↳76 MRRProof (⇔, 54 ms)

                ↳77 QDP

                ↳78 QDPOrderProof (⇔, 19 ms)

                ↳79 QDP

                ↳80 NonTerminationLoopProof (⇔, 0 ms)

                ↳81 NO

            ↳82 QDP

                ↳83 QDPOrderProof (⇔, 175 ms)

                ↳84 QDP

                ↳85 DependencyGraphProof (⇔, 0 ms)

                ↳86 TRUE

    ↳87 QDP

        ↳88 UsableRulesReductionPairsProof (⇔, 41 ms)

        ↳89 QDP

        ↳90 Narrowing (⇔, 0 ms)

        ↳91 QDP

        ↳92 Narrowing (⇔, 0 ms)

        ↳93 QDP

        ↳94 Narrowing (⇔, 0 ms)

        ↳95 QDP

        ↳96 DependencyGraphProof (⇔, 0 ms)

        ↳97 QDP

        ↳98 Narrowing (⇔, 0 ms)

        ↳99 QDP

        ↳100 DependencyGraphProof (⇔, 0 ms)

        ↳101 QDP

        ↳102 Narrowing (⇔, 0 ms)

        ↳103 QDP

        ↳104 Narrowing (⇔, 0 ms)

        ↳105 QDP

        ↳106 DependencyGraphProof (⇔, 0 ms)

        ↳107 QDP

        ↳108 Narrowing (⇔, 0 ms)

        ↳109 QDP

        ↳110 DependencyGraphProof (⇔, 0 ms)

        ↳111 QDP

        ↳112 Narrowing (⇔, 0 ms)

        ↳113 QDP

        ↳114 Narrowing (⇔, 0 ms)

        ↳115 QDP

        ↳116 Narrowing (⇔, 0 ms)

        ↳117 QDP

        ↳118 DependencyGraphProof (⇔, 0 ms)

        ↳119 QDP

        ↳120 Narrowing (⇔, 0 ms)

        ↳121 QDP

        ↳122 DependencyGraphProof (⇔, 0 ms)

        ↳123 QDP

        ↳124 Narrowing (⇔, 0 ms)

        ↳125 QDP

        ↳126 DependencyGraphProof (⇔, 0 ms)

        ↳127 QDP

        ↳128 Narrowing (⇔, 0 ms)

        ↳129 QDP

        ↳130 Narrowing (⇔, 0 ms)

        ↳131 QDP

        ↳132 DependencyGraphProof (⇔, 0 ms)

        ↳133 QDP

        ↳134 Narrowing (⇔, 0 ms)

        ↳135 QDP

        ↳136 DependencyGraphProof (⇔, 0 ms)

        ↳137 QDP

        ↳138 Narrowing (⇔, 0 ms)

        ↳139 QDP

        ↳140 MRRProof (⇔, 20 ms)

        ↳141 QDP

        ↳142 MRRProof (⇔, 22 ms)

        ↳143 QDP

        ↳144 QDPOrderProof (⇔, 0 ms)

        ↳145 QDP

        ↳146 NonTerminationLoopProof (⇔, 0 ms)

        ↳147 NO

(0) Obligation:

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

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, 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)

The following rules are removed from R:

U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))

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,JAR06].

The following pairs can be oriented strictly and are deleted.

ISNAT(n__s(V1)) → ISNAT(activate(V1))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ISNAT(x₁) ) = x₁ + 2

POL( activate(x₁) ) = x₁

POL( n__zeros ) = 2

POL( zeros ) = 2

POL( n__0 ) = 2

POL( 0 ) = 2

POL( n__length(x₁) ) = 0

POL( length(x₁) ) = max{0, -2}

POL( n__s(x₁) ) = 2x₁ + 1

POL( s(x₁) ) = 2x₁ + 1

POL( n__cons(x₁, x₂) ) = 0

POL( cons(x₁, x₂) ) = max{0, -2}

POL( n__nil ) = 2

POL( nil ) = 2

POL( U61(x₁, ..., x₃) ) = max{0, 2x₁ - 1}

POL( isNatList(x₁) ) = max{0, -2}

POL( U51(x₁, x₂) ) = max{0, -2}

POL( isNat(x₁) ) = max{0, x₁ - 1}

POL( U21(x₁) ) = max{0, 2x₁ - 1}

POL( U62(x₁, x₂) ) = 1

POL( tt ) = 1

POL( U52(x₁) ) = x₁

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

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

(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) YES

(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,JAR06].

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)) = 1 + x₂
POL(U51¹(x₁, x₂)) = x₂
POL(U52(x₁)) = 1
POL(U61(x₁, x₂, x₃)) = 1 + x₂
POL(U62(x₁, x₂)) = 1 + x₂
POL(activate(x₁)) = x₁
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 1
POL(isNatList(x₁)) = 1 + 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] with respect to the argument filtering of the ordering [JAR06] were oriented:

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

(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) NonTerminationLoopProof (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) NO

(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) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U61¹(x1, x2, x3) = U61¹(x1)
tt = tt
U62¹(x1, x2) = U62¹(x1)
isNat(x1) = isNat
activate(x1) = activate(x1)
LENGTH(x1) = LENGTH
cons(x1, x2) = x2
isNatList(x1) = isNatList
n__zeros = n__zeros
zeros = zeros
n__0 = n__0
0 = 0
n__length(x1) = n__length(x1)
length(x1) = length(x1)
n__s(x1) = n__s
s(x1) = s
n__cons(x1, x2) = x2
n__nil = n__nil
nil = nil
U21(x1) = x1
U51(x1, x2) = U51
U61(x1, x2, x3) = U61(x1, x2)
U52(x1) = x1
U62(x1, x2) = U62(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[n_{_length₁}, length₁] > [activate₁, U61₂, U62₁] > [tt, isNat, LENGTH, s] > [U61^1₁, U62^1₁]
[n_{_length₁}, length₁] > [activate₁, U61₂, U62₁] > [tt, isNat, LENGTH, s] > [isNatList, U51]
[n_{_length₁}, length₁] > [activate₁, U61₂, U62₁] > [tt, isNat, LENGTH, s] > n_{_s}
[n_{_length₁}, length₁] > [activate₁, U61₂, U62₁] > zeros > n_{_zeros}
[n_{_length₁}, length₁] > [activate₁, U61₂, U62₁] > 0 > n_₀
[n_{_length₁}, length₁] > [activate₁, U61₂, U62₁] > nil > n_{_nil}

Status:

U61^1₁: multiset
tt: multiset
U62^1₁: multiset
isNat: []
activate₁: multiset
LENGTH: []
isNatList: multiset
n_{_zeros}: multiset
zeros: multiset
n_₀: multiset
0: multiset
n_{_length₁}: multiset
length₁: multiset
n_{_s}: multiset
s: multiset
n_{_nil}: multiset
nil: multiset
U51: multiset
U61₂: multiset
U62₁: multiset

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

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

(84) Obligation:

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

U61¹(tt, L, N) → U62¹(isNat(activate(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.

(85) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(86) TRUE

(87) 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.

(88) UsableRulesReductionPairsProof (EQUIVALENT transformation)

U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))

Used ordering: POLO with Polynomial interpretation [POLO]:

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

(89) 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.

(90) 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)

(91) 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.

(92) 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))

(93) 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.

(94) 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)

(95) 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.

(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__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.

(98) 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))

(99) 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.

(100) DependencyGraphProof (EQUIVALENT transformation)

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

(101) 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.

(102) 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))

(103) 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.

(104) 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)

(105) 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.

(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:

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.

(108) 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))

(109) 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.

(110) DependencyGraphProof (EQUIVALENT transformation)

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

(111) Obligation:

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

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.

(112) 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))

(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__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.

(114) 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))

(115) 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.

(116) 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))

(117) 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.

(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:

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.

(120) 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)

(121) 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.

(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:

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.

(124) 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))

(125) 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.

(126) DependencyGraphProof (EQUIVALENT transformation)

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

(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__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.

(128) 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))

(129) 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.

(130) 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))

(131) 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.

(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:

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.

(134) 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))

(135) 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.

(136) DependencyGraphProof (EQUIVALENT transformation)

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

(137) 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.

(138) 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))

(139) 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.

(140) 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

(141) 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.

(142) 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

(143) 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.

(144) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(ISNATILIST(x₁)) = x₁
POL(U21(x₁)) = 0
POL(U41¹(x₁, x₂)) = x₂
POL(U51(x₁, x₂)) = 1 + x₂
POL(U52(x₁)) = 1
POL(U61(x₁, x₂, x₃)) = 1 + x₂
POL(U62(x₁, x₂)) = 1 + x₂
POL(activate(x₁)) = x₁
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 1
POL(isNatList(x₁)) = 1 + 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] with respect to the argument filtering of the ordering [JAR06] were oriented:

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

(145) 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.

(146) NonTerminationLoopProof (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.