Termination w.r.t. Q proof of Transformed_CSR_04_LengthOfFiniteLists_nokinds-noand

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

0 QTRS

↳1 QTRSRRRProof (⇔, 78 ms)

↳2 QTRS

↳3 QTRSRRRProof (⇔, 29 ms)

↳4 QTRS

↳5 DependencyPairsProof (⇔, 1 ms)

↳6 QDP

↳7 DependencyGraphProof (⇔, 0 ms)

↳8 AND

    ↳9 QDP

        ↳10 UsableRulesReductionPairsProof (⇔, 0 ms)

        ↳11 QDP

        ↳12 MRRProof (⇔, 13 ms)

        ↳13 QDP

        ↳14 DependencyGraphProof (⇔, 0 ms)

        ↳15 AND

            ↳16 QDP

                ↳17 UsableRulesProof (⇔, 0 ms)

                ↳18 QDP

                ↳19 QDPSizeChangeProof (⇔, 0 ms)

                ↳20 YES

            ↳21 QDP

                ↳22 QDPOrderProof (⇔, 29 ms)

                ↳23 QDP

                ↳24 PisEmptyProof (⇔, 0 ms)

                ↳25 YES

            ↳26 QDP

                ↳27 TransformationProof (⇔, 0 ms)

                ↳28 QDP

                ↳29 TransformationProof (⇔, 0 ms)

                ↳30 QDP

                ↳31 TransformationProof (⇔, 0 ms)

                ↳32 QDP

                ↳33 DependencyGraphProof (⇔, 0 ms)

                ↳34 QDP

                ↳35 TransformationProof (⇔, 0 ms)

                ↳36 QDP

                ↳37 DependencyGraphProof (⇔, 0 ms)

                ↳38 QDP

                ↳39 TransformationProof (⇔, 0 ms)

                ↳40 QDP

                ↳41 TransformationProof (⇔, 0 ms)

                ↳42 QDP

                ↳43 DependencyGraphProof (⇔, 0 ms)

                ↳44 QDP

                ↳45 MRRProof (⇔, 0 ms)

                ↳46 QDP

                ↳47 MRRProof (⇔, 0 ms)

                ↳48 QDP

                ↳49 MRRProof (⇔, 11 ms)

                ↳50 QDP

                ↳51 TransformationProof (⇔, 0 ms)

                ↳52 QDP

                ↳53 TransformationProof (⇔, 0 ms)

                ↳54 QDP

                ↳55 DependencyGraphProof (⇔, 0 ms)

                ↳56 QDP

                ↳57 TransformationProof (⇔, 0 ms)

                ↳58 QDP

                ↳59 DependencyGraphProof (⇔, 0 ms)

                ↳60 QDP

                ↳61 TransformationProof (⇔, 0 ms)

                ↳62 QDP

                ↳63 QDPOrderProof (⇔, 0 ms)

                ↳64 QDP

                ↳65 QDPOrderProof (⇔, 46 ms)

                ↳66 QDP

                ↳67 QDPOrderProof (⇔, 0 ms)

                ↳68 QDP

                ↳69 NonTerminationLoopProof (⇒, 0 ms)

                ↳70 NO

            ↳71 QDP

                ↳72 QDPOrderProof (⇔, 24 ms)

                ↳73 QDP

                ↳74 DependencyGraphProof (⇔, 0 ms)

                ↳75 TRUE

    ↳76 QDP

        ↳77 UsableRulesReductionPairsProof (⇔, 0 ms)

        ↳78 QDP

        ↳79 TransformationProof (⇔, 0 ms)

        ↳80 QDP

        ↳81 TransformationProof (⇔, 0 ms)

        ↳82 QDP

        ↳83 TransformationProof (⇔, 0 ms)

        ↳84 QDP

        ↳85 DependencyGraphProof (⇔, 0 ms)

        ↳86 QDP

        ↳87 TransformationProof (⇔, 0 ms)

        ↳88 QDP

        ↳89 DependencyGraphProof (⇔, 0 ms)

        ↳90 QDP

        ↳91 TransformationProof (⇔, 0 ms)

        ↳92 QDP

        ↳93 TransformationProof (⇔, 0 ms)

        ↳94 QDP

        ↳95 DependencyGraphProof (⇔, 0 ms)

        ↳96 QDP

        ↳97 TransformationProof (⇔, 27 ms)

        ↳98 QDP

        ↳99 DependencyGraphProof (⇔, 0 ms)

        ↳100 QDP

        ↳101 TransformationProof (⇔, 0 ms)

        ↳102 QDP

        ↳103 DependencyGraphProof (⇔, 0 ms)

        ↳104 QDP

        ↳105 TransformationProof (⇔, 0 ms)

        ↳106 QDP

        ↳107 TransformationProof (⇔, 0 ms)

        ↳108 QDP

        ↳109 DependencyGraphProof (⇔, 0 ms)

        ↳110 QDP

        ↳111 TransformationProof (⇔, 1 ms)

        ↳112 QDP

        ↳113 DependencyGraphProof (⇔, 0 ms)

        ↳114 QDP

        ↳115 TransformationProof (⇔, 0 ms)

        ↳116 QDP

        ↳117 MRRProof (⇔, 38 ms)

        ↳118 QDP

        ↳119 MRRProof (⇔, 0 ms)

        ↳120 QDP

        ↳121 QDPOrderProof (⇔, 38 ms)

        ↳122 QDP

        ↳123 QDPOrderProof (⇔, 61 ms)

        ↳124 QDP

        ↳125 QDPOrderProof (⇔, 0 ms)

        ↳126 QDP

        ↳127 NonTerminationLoopProof (⇒, 0 ms)

        ↳128 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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

Q is empty.

(1) 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)) = 1 + 2·x₁ + 2·x₂
POL(U42(x₁)) = x₁
POL(U51(x₁, x₂)) = x₁ + x₂
POL(U52(x₁)) = x₁
POL(U61(x₁, x₂, x₃)) = 2·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(isNatIList(x₁)) = 1 + 2·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

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
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(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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(U11(x₁)) = 1 + 2·x₁
POL(U21(x₁)) = x₁
POL(U31(x₁)) = 1 + 2·x₁
POL(U41(x₁, x₂)) = x₁ + 2·x₂
POL(U42(x₁)) = x₁
POL(U51(x₁, x₂)) = x₁ + x₂
POL(U52(x₁)) = x₁
POL(U61(x₁, x₂, x₃)) = x₁ + x₂ + 2·x₃
POL(U62(x₁, x₂)) = x₁ + x₂
POL(activate(x₁)) = x₁
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = 2 + 2·x₁
POL(isNatIList(x₁)) = 2 + 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) = 2
POL(n__s(x₁)) = x₁
POL(n__zeros) = 0
POL(nil) = 2
POL(s(x₁)) = x₁
POL(tt) = 2
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
U31(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
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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

Q is empty.

(5) DependencyPairsProof (EQUIVALENT transformation)

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

(6) 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(activate(X))
ACTIVATE(n__length(X)) → ACTIVATE(X)
ACTIVATE(n__s(X)) → S(activate(X))
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → CONS(activate(X1), X2)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__nil) → NIL

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(7) DependencyGraphProof (EQUIVALENT transformation)

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

(8) Complex Obligation (AND)

(9) Obligation:

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

ACTIVATE(n__length(X)) → LENGTH(activate(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)
ACTIVATE(n__length(X)) → ACTIVATE(X)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(10) 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.

No dependency pairs are removed.

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)) = 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 + x₂
POL(U51¹(x₁, x₂)) = x₁ + 2·x₂
POL(U52(x₁)) = x₁
POL(U61(x₁, x₂, x₃)) = x₁ + 2·x₂ + 2·x₃
POL(U61¹(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + 2·x₃
POL(U62(x₁, x₂)) = x₁ + 2·x₂
POL(U62¹(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) = 0
POL(n__s(x₁)) = x₁
POL(n__zeros) = 0
POL(nil) = 0
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

(11) Obligation:

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

ACTIVATE(n__length(X)) → LENGTH(activate(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)
ACTIVATE(n__length(X)) → ACTIVATE(X)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
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.

(12) 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:

ACTIVATE(n__length(X)) → LENGTH(activate(X))
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → ACTIVATE(X)
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)

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(ACTIVATE(x₁)) = 2·x₁
POL(ISNAT(x₁)) = 1 + 2·x₁
POL(ISNATLIST(x₁)) = 2 + 2·x₁
POL(LENGTH(x₁)) = 2 + 2·x₁
POL(U21(x₁)) = x₁
POL(U51(x₁, x₂)) = x₁ + 2·x₂
POL(U51¹(x₁, x₂)) = 2 + x₁ + 2·x₂
POL(U52(x₁)) = x₁
POL(U61(x₁, x₂, x₃)) = 2 + x₁ + 2·x₂ + 2·x₃
POL(U61¹(x₁, x₂, x₃)) = 2 + x₁ + 2·x₂ + 2·x₃
POL(U62(x₁, x₂)) = 2 + x₁ + 2·x₂
POL(U62¹(x₁, x₂)) = 2 + x₁ + 2·x₂
POL(activate(x₁)) = x₁
POL(cons(x₁, x₂)) = x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatList(x₁)) = 2·x₁
POL(length(x₁)) = 2 + 2·x₁
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = x₁ + 2·x₂
POL(n__length(x₁)) = 2 + 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

(13) 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))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)

The TRS R consists of the following rules:

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

(14) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 1 less node.

(15) Complex Obligation (AND)

(16) Obligation:

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

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

The TRS R consists of the following rules:

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

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

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

(19) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

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

(20) YES

(21) 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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
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.

(22) 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) ) = 2x₁

POL( activate(x₁) ) = x₁

POL( n__zeros ) = 2

POL( zeros ) = 2

POL( n__0 ) = 1

POL( 0 ) = 1

POL( n__length(x₁) ) = 1

POL( length(x₁) ) = 1

POL( n__s(x₁) ) = 2x₁ + 2

POL( s(x₁) ) = 2x₁ + 2

POL( n__cons(x₁, x₂) ) = 2

POL( cons(x₁, x₂) ) = 2

POL( n__nil ) = 0

POL( nil ) = 0

POL( U61(x₁, ..., x₃) ) = 2x₁ + 1

POL( isNatList(x₁) ) = 0

POL( U51(x₁, x₂) ) = 0

POL( isNat(x₁) ) = 2

POL( U21(x₁) ) = 2

POL( U62(x₁, x₂) ) = 2x₁

POL( tt ) = 2

POL( U52(x₁) ) = max{0, 2x₁ - 2}

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

(23) 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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
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) PisEmptyProof (EQUIVALENT transformation)

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

(25) YES

(26) 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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
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.

(27) TransformationProof (EQUIVALENT transformation)

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

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

(28) 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(activate(x0)))
U51¹(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
U51¹(tt, n__nil) → ISNATLIST(nil)
U51¹(tt, x0) → ISNATLIST(x0)

The TRS R consists of the following rules:

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

(29) TransformationProof (EQUIVALENT transformation)

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

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

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

The TRS R consists of the following rules:

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

(31) TransformationProof (EQUIVALENT transformation)

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

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

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

The TRS R consists of the following rules:

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

(33) DependencyGraphProof (EQUIVALENT transformation)

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

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

The TRS R consists of the following rules:

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

(35) TransformationProof (EQUIVALENT transformation)

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

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

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

The TRS R consists of the following rules:

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

(37) DependencyGraphProof (EQUIVALENT transformation)

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

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

The TRS R consists of the following rules:

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

(39) TransformationProof (EQUIVALENT transformation)

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

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

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

The TRS R consists of the following rules:

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

(41) TransformationProof (EQUIVALENT transformation)

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

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

(42) Obligation:

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

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

The TRS R consists of the following rules:

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

(43) DependencyGraphProof (EQUIVALENT transformation)

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

(44) Obligation:

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

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

The TRS R consists of the following rules:

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

(45) MRRProof (EQUIVALENT transformation)

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

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(ISNATLIST(x₁)) = 2 + x₁
POL(U21(x₁)) = x₁
POL(U51(x₁, x₂)) = x₁ + 2·x₂
POL(U51¹(x₁, x₂)) = 2 + x₁ + 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)) = 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) = 2
POL(n__s(x₁)) = x₁
POL(n__zeros) = 0
POL(nil) = 2
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

(46) Obligation:

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

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

(47) MRRProof (EQUIVALENT transformation)

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

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(ISNATLIST(x₁)) = 1 + x₁
POL(U21(x₁)) = x₁
POL(U51(x₁, x₂)) = x₁ + 2·x₂
POL(U51¹(x₁, x₂)) = x₁ + 2·x₂
POL(U52(x₁)) = x₁
POL(U61(x₁, x₂, x₃)) = 1 + x₁ + x₂ + 2·x₃
POL(U62(x₁, x₂)) = 1 + x₁ + x₂
POL(activate(x₁)) = x₁
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = 1 + x₁
POL(isNatList(x₁)) = 1 + 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) = 1
POL(zeros) = 0

(48) Obligation:

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

U51¹(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(s(activate(x0))), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U51¹(isNat(cons(activate(x0), x1)), activate(y1))
U51¹(tt, n__nil) → ISNATLIST(nil)
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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
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.

(49) MRRProof (EQUIVALENT transformation)

U51¹(tt, n__nil) → ISNATLIST(nil)

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(ISNATLIST(x₁)) = 2 + x₁
POL(U21(x₁)) = x₁
POL(U51(x₁, x₂)) = x₁ + 2·x₂
POL(U51¹(x₁, x₂)) = 1 + 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)) = 1 + x₁
POL(isNatList(x₁)) = 1 + 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) = 1
POL(n__s(x₁)) = x₁
POL(n__zeros) = 0
POL(nil) = 1
POL(s(x₁)) = x₁
POL(tt) = 1
POL(zeros) = 0

(50) Obligation:

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

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

The TRS R consists of the following rules:

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

(51) TransformationProof (EQUIVALENT transformation)

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

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

(52) Obligation:

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

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

The TRS R consists of the following rules:

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

(53) TransformationProof (EQUIVALENT transformation)

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

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

(54) Obligation:

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

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

The TRS R consists of the following rules:

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

(55) DependencyGraphProof (EQUIVALENT transformation)

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

(56) Obligation:

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

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

The TRS R consists of the following rules:

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

(57) TransformationProof (EQUIVALENT transformation)

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

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

(58) Obligation:

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

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

The TRS R consists of the following rules:

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

(59) DependencyGraphProof (EQUIVALENT transformation)

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

(60) Obligation:

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

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

The TRS R consists of the following rules:

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

(61) TransformationProof (EQUIVALENT transformation)

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

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

(62) Obligation:

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

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

The TRS R consists of the following rules:

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

(63) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

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

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

POL( U51¹(x₁, x₂) ) = 2x₂ + 2

POL( ISNATLIST(x₁) ) = x₁ + 2

POL( cons(x₁, x₂) ) = 2x₂

POL( s(x₁) ) = 2x₁ + 2

POL( U51(x₁, x₂) ) = max{0, -2}

POL( U61(x₁, ..., x₃) ) = 2x₁ + 2

POL( U62(x₁, x₂) ) = 2x₁ + 2

POL( length(x₁) ) = 2

POL( U21(x₁) ) = x₁

POL( isNat(x₁) ) = 2

POL( U52(x₁) ) = 2x₁

POL( isNatList(x₁) ) = 0

POL( activate(x₁) ) = x₁

POL( n__zeros ) = 0

POL( zeros ) = 0

POL( n__0 ) = 2

POL( 0 ) = 2

POL( n__length(x₁) ) = 2

POL( n__s(x₁) ) = 2x₁ + 2

POL( n__cons(x₁, x₂) ) = 2x₂

POL( n__nil ) = 0

POL( nil ) = 0

POL( tt ) = 2

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

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

(64) Obligation:

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

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

The TRS R consists of the following rules:

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

(65) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

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

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

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

POL( ISNATLIST(x₁) ) = x₁

POL( cons(x₁, x₂) ) = 2x₁ + x₂

POL( U51(x₁, x₂) ) = max{0, -2}

POL( U61(x₁, ..., x₃) ) = x₁ + 2

POL( U62(x₁, x₂) ) = x₁ + 1

POL( length(x₁) ) = 2

POL( s(x₁) ) = 2

POL( U21(x₁) ) = x₁

POL( isNat(x₁) ) = 1

POL( U52(x₁) ) = 2x₁

POL( isNatList(x₁) ) = 0

POL( activate(x₁) ) = x₁

POL( n__zeros ) = 2

POL( zeros ) = 2

POL( n__0 ) = 0

POL( 0 ) = 0

POL( n__length(x₁) ) = 2

POL( n__s(x₁) ) = 2

POL( n__cons(x₁, x₂) ) = 2x₁ + x₂

POL( n__nil ) = 0

POL( nil ) = 0

POL( tt ) = 1

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

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

(66) Obligation:

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

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

The TRS R consists of the following rules:

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

(67) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U51¹(isNat(cons(activate(x0), x1)), activate(y1))

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

POL(U51¹(x₁, x₂)) = 1 +
[ 0, 0 ]

· x₁ +
[ 0, 1 ]

· x₂

POL(tt) =
/ 0 \

\ 1 /

POL(n__cons(x₁, x₂)) =
/ 1 \

\ 0 /

+
/ 1 0 \

\ 1 1 /

· x₁ +
/ 0 0 \

\ 0 1 /

· x₂

POL(ISNATLIST(x₁)) = 1 +
[ 0, 1 ]

· x₁

POL(cons(x₁, x₂)) =
/ 1 \

\ 0 /

+
/ 1 0 \

\ 1 1 /

· x₁ +
/ 0 0 \

\ 0 1 /

· x₂

POL(activate(x₁)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 0 1 /

· x₁

POL(isNat(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 1 1 /

· x₁

POL(n__zeros) =
/ 1 \

\ 1 /

POL(0) =
/ 0 \

\ 0 /

POL(n__0) =
/ 0 \

\ 0 /

POL(zeros) =
/ 1 \

\ 1 /

POL(n__length(x₁)) =
/ 1 \

\ 0 /

+
/ 1 1 \

\ 1 0 /

· x₁

POL(length(x₁)) =
/ 1 \

\ 0 /

+
/ 1 1 \

\ 1 0 /

· x₁

POL(n__s(x₁)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(s(x₁)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(n__nil) =
/ 0 \

\ 0 /

POL(nil) =
/ 0 \

\ 0 /

POL(U21(x₁)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(U61(x₁, x₂, x₃)) =
/ 0 \

\ 0 /

+
/ 1 1 \

\ 1 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃

POL(isNatList(x₁)) =
/ 1 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(U51(x₁, x₂)) =
/ 1 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(U52(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 1 /

· x₁

POL(U62(x₁, x₂)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
cons(X1, X2) → n__cons(X1, X2)
isNat(n__s(V1)) → U21(isNat(activate(V1)))
0 → n__0
s(X) → n__s(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))
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)))
zeros → cons(0, n__zeros)
zeros → n__zeros
nil → n__nil

(68) Obligation:

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

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

The TRS R consists of the following rules:

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

(69) NonTerminationLoopProof (COMPLETE transformation)

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

s = 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.

(70) NO

(71) 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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
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) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

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

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

POL( LENGTH(x₁) ) = 0

POL( U61¹(x₁, ..., x₃) ) = max{0, 2x₁ - 1}

POL( U62¹(x₁, x₂) ) = 0

POL( U51(x₁, x₂) ) = 0

POL( U61(x₁, ..., x₃) ) = 2x₁

POL( U62(x₁, x₂) ) = 2

POL( cons(x₁, x₂) ) = x₂

POL( isNatList(x₁) ) = 0

POL( length(x₁) ) = max{0, -2}

POL( s(x₁) ) = 2x₁ + 2

POL( isNat(x₁) ) = max{0, x₁ - 1}

POL( U21(x₁) ) = x₁ + 1

POL( U52(x₁) ) = x₁

POL( activate(x₁) ) = 2x₁ + 1

POL( n__zeros ) = 0

POL( zeros ) = 0

POL( n__0 ) = 2

POL( 0 ) = 2

POL( n__length(x₁) ) = 0

POL( n__s(x₁) ) = 2x₁ + 2

POL( n__cons(x₁, x₂) ) = x₂

POL( n__nil ) = 0

POL( nil ) = 0

POL( tt ) = 1

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

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

(73) Obligation:

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

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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
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) DependencyGraphProof (EQUIVALENT transformation)

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

(75) TRUE

(76) Obligation:

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

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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(77) 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 + 2·x₂
POL(U51(x₁, x₂)) = x₁ + x₂
POL(U52(x₁)) = x₁
POL(U61(x₁, x₂, x₃)) = x₁ + x₂ + 2·x₃
POL(U62(x₁, x₂)) = x₁ + x₂
POL(activate(x₁)) = x₁
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = x₁
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(n__length(x₁)) = 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

(78) 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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__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.

(79) TransformationProof (EQUIVALENT transformation)

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

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

(80) 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(activate(x0)))
U41¹(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U41¹(tt, n__nil) → ISNATILIST(nil)
U41¹(tt, x0) → ISNATILIST(x0)

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__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.

(81) TransformationProof (EQUIVALENT transformation)

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

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

(82) 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(activate(x0)))
U41¹(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U41¹(tt, n__nil) → ISNATILIST(nil)
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__zeros, 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(activate(x0))), activate(y1))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(activate(x0))), activate(y1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
ISNATILIST(n__cons(n__nil, y1)) → U41¹(isNat(nil), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__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.

(83) TransformationProof (EQUIVALENT transformation)

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

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

(84) 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(activate(x0)))
U41¹(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U41¹(tt, n__nil) → ISNATILIST(nil)
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__zeros, 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(activate(x0))), activate(y1))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(activate(x0))), activate(y1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
ISNATILIST(n__cons(n__nil, y1)) → U41¹(isNat(nil), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
U41¹(tt, n__zeros) → ISNATILIST(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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__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.

(85) DependencyGraphProof (EQUIVALENT transformation)

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__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.

(87) TransformationProof (EQUIVALENT transformation)

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

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

(88) 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(activate(x0)))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(activate(x0))), activate(y1))
U41¹(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(activate(x0))), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
U41¹(tt, n__nil) → ISNATILIST(nil)
ISNATILIST(n__cons(n__nil, y1)) → U41¹(isNat(nil), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
U41¹(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(cons(0, n__zeros)), activate(y0))
ISNATILIST(n__cons(n__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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__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.

(89) DependencyGraphProof (EQUIVALENT transformation)

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__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.

(91) TransformationProof (EQUIVALENT transformation)

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

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

(92) 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(activate(x0))), activate(y1))
U41¹(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(activate(x0))), activate(y1))
U41¹(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__nil, y1)) → U41¹(isNat(nil), activate(y1))
U41¹(tt, n__nil) → ISNATILIST(nil)
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(cons(0, n__zeros)), activate(y0))
U41¹(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__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.

(93) TransformationProof (EQUIVALENT transformation)

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

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

(94) Obligation:

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

ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(activate(x0))), activate(y1))
U41¹(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(activate(x0))), activate(y1))
U41¹(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__nil, y1)) → U41¹(isNat(nil), activate(y1))
U41¹(tt, n__nil) → ISNATILIST(nil)
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(cons(0, n__zeros)), activate(y0))
U41¹(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__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.

(95) DependencyGraphProof (EQUIVALENT transformation)

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

(96) Obligation:

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__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.

(97) TransformationProof (EQUIVALENT transformation)

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

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

(98) Obligation:

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__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.

(99) DependencyGraphProof (EQUIVALENT transformation)

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

(100) Obligation:

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

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

(101) TransformationProof (EQUIVALENT transformation)

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

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

(102) Obligation:

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

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

(103) DependencyGraphProof (EQUIVALENT transformation)

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

(104) Obligation:

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

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

(105) TransformationProof (EQUIVALENT transformation)

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

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

(106) Obligation:

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

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

(107) TransformationProof (EQUIVALENT transformation)

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

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

(108) Obligation:

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

U41¹(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(activate(x0))), activate(y1))
U41¹(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(activate(x0))), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U41¹(tt, n__zeros) → ISNATILIST(cons(n__0, n__zeros))
ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(n__cons(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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__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.

(109) DependencyGraphProof (EQUIVALENT transformation)

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

(110) Obligation:

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

ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(activate(x0))), activate(y1))
U41¹(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(activate(x0))), activate(y1))
U41¹(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__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.

(111) TransformationProof (EQUIVALENT transformation)

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

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

(112) Obligation:

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

ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(activate(x0))), activate(y1))
U41¹(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(activate(x0))), activate(y1))
U41¹(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U41¹(tt, n__zeros) → ISNATILIST(cons(n__0, n__zeros))
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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__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.

(113) DependencyGraphProof (EQUIVALENT transformation)

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

(114) Obligation:

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

U41¹(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(activate(x0))), activate(y1))
U41¹(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(activate(x0))), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), 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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__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.

(115) TransformationProof (EQUIVALENT transformation)

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

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

(116) Obligation:

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

U41¹(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(activate(x0))), activate(y1))
U41¹(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(activate(x0))), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__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.

(117) MRRProof (EQUIVALENT transformation)

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

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(ISNATILIST(x₁)) = 1 + x₁
POL(U21(x₁)) = x₁
POL(U41¹(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(U51(x₁, x₂)) = x₁ + 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)) = 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)) = x₁ + 2·x₂
POL(n__length(x₁)) = 2 + 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

(118) Obligation:

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

ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(activate(x0))), activate(y1))
U41¹(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(activate(x0))), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__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.

(119) MRRProof (EQUIVALENT transformation)

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

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(ISNATILIST(x₁)) = 2 + 2·x₁
POL(U21(x₁)) = x₁
POL(U41¹(x₁, x₂)) = 2 + 2·x₁ + 2·x₂
POL(U51(x₁, x₂)) = 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) = 2
POL(n__s(x₁)) = x₁
POL(n__zeros) = 0
POL(nil) = 2
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

(120) Obligation:

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

U41¹(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(activate(x0))), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__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.

(121) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

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

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

POL( U41¹(x₁, x₂) ) = x₁ + 2x₂

POL( ISNATILIST(x₁) ) = 2x₁ + 1

POL( cons(x₁, x₂) ) = 2x₁ + 2x₂

POL( s(x₁) ) = 1

POL( U51(x₁, x₂) ) = x₂

POL( U61(x₁, ..., x₃) ) = x₁ + x₂ + 2x₃ + 1

POL( U62(x₁, x₂) ) = max{0, 2x₁ - 1}

POL( length(x₁) ) = x₁ + 2

POL( U21(x₁) ) = max{0, 2x₁ - 1}

POL( isNat(x₁) ) = 1

POL( U52(x₁) ) = x₁

POL( isNatList(x₁) ) = x₁

POL( activate(x₁) ) = x₁

POL( n__zeros ) = 0

POL( zeros ) = 0

POL( n__0 ) = 0

POL( 0 ) = 0

POL( n__length(x₁) ) = x₁ + 2

POL( n__s(x₁) ) = 1

POL( n__cons(x₁, x₂) ) = 2x₁ + 2x₂

POL( n__nil ) = 0

POL( nil ) = 0

POL( tt ) = 1

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

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

(122) Obligation:

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

U41¹(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__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.

(123) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

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

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

POL( U41¹(x₁, x₂) ) = x₁ + 2x₂

POL( ISNATILIST(x₁) ) = x₁ + 2

POL( cons(x₁, x₂) ) = 2x₁ + 2x₂

POL( s(x₁) ) = 2

POL( U51(x₁, x₂) ) = 2

POL( U61(x₁, ..., x₃) ) = x₁ + 2x₃

POL( U62(x₁, x₂) ) = x₁

POL( length(x₁) ) = x₁ + 2

POL( U21(x₁) ) = 2

POL( isNat(x₁) ) = 2x₁ + 2

POL( U52(x₁) ) = max{0, 2x₁ - 2}

POL( isNatList(x₁) ) = 2

POL( activate(x₁) ) = x₁

POL( n__zeros ) = 0

POL( zeros ) = 0

POL( n__0 ) = 0

POL( 0 ) = 0

POL( n__length(x₁) ) = x₁ + 2

POL( n__s(x₁) ) = 2

POL( n__cons(x₁, x₂) ) = 2x₁ + 2x₂

POL( n__nil ) = 0

POL( nil ) = 0

POL( tt ) = 2

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

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

(124) Obligation:

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

U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__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.

(125) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

POL(U41¹(x₁, x₂)) = 0 +
[ 0, 1 ]

· x₁ +
[ 1, 0 ]

· x₂

POL(tt) =
/ 1 \

\ 1 /

POL(n__cons(x₁, x₂)) =
/ 0 \

\ 1 /

+
/ 0 1 \

\ 0 0 /

· x₁ +
/ 1 0 \

\ 1 0 /

· x₂

POL(ISNATILIST(x₁)) = 1 +
[ 1, 0 ]

· x₁

POL(cons(x₁, x₂)) =
/ 0 \

\ 1 /

+
/ 0 1 \

\ 0 0 /

· x₁ +
/ 1 0 \

\ 1 0 /

· x₂

POL(activate(x₁)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 0 1 /

· x₁

POL(isNat(x₁)) =
/ 1 \

\ 1 /

+
/ 1 1 \

\ 0 0 /

· x₁

POL(n__zeros) =
/ 0 \

\ 1 /

POL(0) =
/ 1 \

\ 0 /

POL(n__0) =
/ 1 \

\ 0 /

POL(zeros) =
/ 0 \

\ 1 /

POL(n__length(x₁)) =
/ 0 \

\ 1 /

+
/ 1 1 \

\ 0 0 /

· x₁

POL(length(x₁)) =
/ 0 \

\ 1 /

+
/ 1 1 \

\ 0 0 /

· x₁

POL(n__s(x₁)) =
/ 1 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(s(x₁)) =
/ 1 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(n__nil) =
/ 0 \

\ 0 /

POL(nil) =
/ 0 \

\ 0 /

POL(U21(x₁)) =
/ 1 \

\ 0 /

+
/ 0 1 \

\ 0 1 /

· x₁

POL(U61(x₁, x₂, x₃)) =
/ 0 \

\ 1 /

+
/ 1 1 \

\ 0 1 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃

POL(isNatList(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(U51(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(U52(x₁)) =
/ 0 \

\ 0 /

+
/ 1 1 \

\ 1 1 /

· x₁

POL(U62(x₁, x₂)) =
/ 1 \

\ 1 /

+
/ 0 1 \

\ 0 1 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
cons(X1, X2) → n__cons(X1, X2)
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
0 → n__0
s(X) → n__s(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))
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)))
zeros → cons(0, n__zeros)
zeros → n__zeros
nil → n__nil

(126) Obligation:

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

U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
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(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__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.

(127) NonTerminationLoopProof (COMPLETE transformation)

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

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

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Matcher: [ ]
Semiunifier: [ ]

Rewriting sequence

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

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

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

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

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

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