Termination w.r.t. Q proof of Transformed_CSR_04_Ex4_DLMMU04

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

0 QTRS

↳1 DependencyPairsProof (⇔, 10 ms)

↳2 QDP

↳3 DependencyGraphProof (⇔, 0 ms)

↳4 QDP

↳5 QDPOrderProof (⇔, 139 ms)

↳6 QDP

↳7 DependencyGraphProof (⇔, 0 ms)

↳8 AND

    ↳9 QDP

        ↳10 QDPOrderProof (⇔, 53 ms)

        ↳11 QDP

        ↳12 DependencyGraphProof (⇔, 0 ms)

        ↳13 AND

            ↳14 QDP

                ↳15 QDPOrderProof (⇔, 100 ms)

                ↳16 QDP

                ↳17 PisEmptyProof (⇔, 0 ms)

                ↳18 YES

            ↳19 QDP

                ↳20 TransformationProof (⇔, 0 ms)

                ↳21 QDP

                ↳22 TransformationProof (⇔, 0 ms)

                ↳23 QDP

                ↳24 DependencyGraphProof (⇔, 0 ms)

                ↳25 QDP

                ↳26 TransformationProof (⇔, 0 ms)

                ↳27 QDP

                ↳28 DependencyGraphProof (⇔, 0 ms)

                ↳29 QDP

                ↳30 TransformationProof (⇔, 0 ms)

                ↳31 QDP

                ↳32 DependencyGraphProof (⇔, 0 ms)

                ↳33 QDP

                ↳34 TransformationProof (⇔, 0 ms)

                ↳35 QDP

                ↳36 TransformationProof (⇔, 0 ms)

                ↳37 QDP

                ↳38 DependencyGraphProof (⇔, 0 ms)

                ↳39 QDP

                ↳40 TransformationProof (⇔, 0 ms)

                ↳41 QDP

                ↳42 TransformationProof (⇔, 0 ms)

                ↳43 QDP

                ↳44 QDPOrderProof (⇔, 62 ms)

                ↳45 QDP

                ↳46 QDPOrderProof (⇔, 25 ms)

                ↳47 QDP

                ↳48 DependencyGraphProof (⇔, 0 ms)

                ↳49 AND

                    ↳50 QDP

                        ↳51 UsableRulesProof (⇔, 0 ms)

                        ↳52 QDP

                        ↳53 MNOCProof (⇔, 0 ms)

                        ↳54 QDP

                    ↳55 QDP

            ↳56 QDP

                ↳57 QDPOrderProof (⇔, 126 ms)

                ↳58 QDP

                ↳59 DependencyGraphProof (⇔, 0 ms)

                ↳60 TRUE

    ↳61 QDP

        ↳62 TransformationProof (⇔, 0 ms)

        ↳63 QDP

        ↳64 TransformationProof (⇔, 0 ms)

        ↳65 QDP

        ↳66 DependencyGraphProof (⇔, 0 ms)

        ↳67 QDP

        ↳68 TransformationProof (⇔, 0 ms)

        ↳69 QDP

        ↳70 DependencyGraphProof (⇔, 0 ms)

        ↳71 QDP

        ↳72 TransformationProof (⇔, 0 ms)

        ↳73 QDP

        ↳74 DependencyGraphProof (⇔, 0 ms)

        ↳75 QDP

        ↳76 TransformationProof (⇔, 0 ms)

        ↳77 QDP

        ↳78 TransformationProof (⇔, 0 ms)

        ↳79 QDP

        ↳80 DependencyGraphProof (⇔, 0 ms)

        ↳81 QDP

        ↳82 TransformationProof (⇔, 0 ms)

        ↳83 QDP

        ↳84 TransformationProof (⇔, 0 ms)

        ↳85 QDP

        ↳86 QDPOrderProof (⇔, 50 ms)

        ↳87 QDP

        ↳88 QDPOrderProof (⇔, 85 ms)

        ↳89 QDP

        ↳90 DependencyGraphProof (⇔, 0 ms)

        ↳91 AND

            ↳92 QDP

                ↳93 UsableRulesProof (⇔, 0 ms)

                ↳94 QDP

                ↳95 MNOCProof (⇔, 0 ms)

                ↳96 QDP

                ↳97 TransformationProof (⇔, 0 ms)

                ↳98 QDP

                ↳99 UsableRulesProof (⇔, 0 ms)

                ↳100 QDP

                ↳101 QReductionProof (⇔, 0 ms)

                ↳102 QDP

                ↳103 TransformationProof (⇔, 0 ms)

                ↳104 QDP

                ↳105 NonTerminationLoopProof (⇒, 0 ms)

                ↳106 NO

            ↳107 QDP

(0) Obligation:

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

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

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

(2) Obligation:

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

ISNATILIST(IL) → ISNATLIST(activate(IL))
ISNATILIST(IL) → ACTIVATE(IL)
ISNAT(n__s(N)) → ISNAT(activate(N))
ISNAT(n__s(N)) → ACTIVATE(N)
ISNAT(n__length(L)) → ISNATLIST(activate(L))
ISNAT(n__length(L)) → ACTIVATE(L)
ISNATILIST(n__cons(N, IL)) → AND(isNat(activate(N)), isNatIList(activate(IL)))
ISNATILIST(n__cons(N, IL)) → ISNAT(activate(N))
ISNATILIST(n__cons(N, IL)) → ACTIVATE(N)
ISNATILIST(n__cons(N, IL)) → ISNATILIST(activate(IL))
ISNATILIST(n__cons(N, IL)) → ACTIVATE(IL)
ISNATLIST(n__cons(N, L)) → AND(isNat(activate(N)), isNatList(activate(L)))
ISNATLIST(n__cons(N, L)) → ISNAT(activate(N))
ISNATLIST(n__cons(N, L)) → ACTIVATE(N)
ISNATLIST(n__cons(N, L)) → ISNATLIST(activate(L))
ISNATLIST(n__cons(N, L)) → ACTIVATE(L)
ISNATLIST(n__take(N, IL)) → AND(isNat(activate(N)), isNatIList(activate(IL)))
ISNATLIST(n__take(N, IL)) → ISNAT(activate(N))
ISNATLIST(n__take(N, IL)) → ACTIVATE(N)
ISNATLIST(n__take(N, IL)) → ISNATILIST(activate(IL))
ISNATLIST(n__take(N, IL)) → ACTIVATE(IL)
ZEROS → CONS(0, n__zeros)
ZEROS → 0¹
TAKE(0, IL) → UTAKE1(isNatIList(IL))
TAKE(0, IL) → ISNATILIST(IL)
UTAKE1(tt) → NIL
TAKE(s(M), cons(N, IL)) → UTAKE2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
TAKE(s(M), cons(N, IL)) → AND(isNat(M), and(isNat(N), isNatIList(activate(IL))))
TAKE(s(M), cons(N, IL)) → ISNAT(M)
TAKE(s(M), cons(N, IL)) → AND(isNat(N), isNatIList(activate(IL)))
TAKE(s(M), cons(N, IL)) → ISNAT(N)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
UTAKE2(tt, M, N, IL) → CONS(activate(N), n__take(activate(M), activate(IL)))
UTAKE2(tt, M, N, IL) → ACTIVATE(N)
UTAKE2(tt, M, N, IL) → ACTIVATE(M)
UTAKE2(tt, M, N, IL) → ACTIVATE(IL)
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(activate(L))), activate(L))
LENGTH(cons(N, L)) → AND(isNat(N), isNatList(activate(L)))
LENGTH(cons(N, L)) → ISNAT(N)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
LENGTH(cons(N, L)) → ACTIVATE(L)
ULENGTH(tt, L) → S(length(activate(L)))
ULENGTH(tt, L) → LENGTH(activate(L))
ULENGTH(tt, L) → ACTIVATE(L)
ACTIVATE(n__0) → 0¹
ACTIVATE(n__s(X)) → S(X)
ACTIVATE(n__length(X)) → LENGTH(X)
ACTIVATE(n__zeros) → ZEROS
ACTIVATE(n__cons(X1, X2)) → CONS(X1, X2)
ACTIVATE(n__nil) → NIL
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 17 less nodes.

(4) Obligation:

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

ISNATLIST(n__cons(N, L)) → ISNAT(activate(N))
ISNAT(n__s(N)) → ISNAT(activate(N))
ISNAT(n__s(N)) → ACTIVATE(N)
ACTIVATE(n__length(X)) → LENGTH(X)
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(activate(L))), activate(L))
ULENGTH(tt, L) → LENGTH(activate(L))
LENGTH(cons(N, L)) → ISNAT(N)
ISNAT(n__length(L)) → ISNATLIST(activate(L))
ISNATLIST(n__cons(N, L)) → ACTIVATE(N)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
TAKE(0, IL) → ISNATILIST(IL)
ISNATILIST(IL) → ISNATLIST(activate(IL))
ISNATLIST(n__cons(N, L)) → ISNATLIST(activate(L))
ISNATLIST(n__cons(N, L)) → ACTIVATE(L)
ISNATLIST(n__take(N, IL)) → ISNAT(activate(N))
ISNAT(n__length(L)) → ACTIVATE(L)
ISNATLIST(n__take(N, IL)) → ACTIVATE(N)
ISNATLIST(n__take(N, IL)) → ISNATILIST(activate(IL))
ISNATILIST(IL) → ACTIVATE(IL)
ISNATILIST(n__cons(N, IL)) → ISNAT(activate(N))
ISNATILIST(n__cons(N, IL)) → ACTIVATE(N)
ISNATILIST(n__cons(N, IL)) → ISNATILIST(activate(IL))
ISNATILIST(n__cons(N, IL)) → ACTIVATE(IL)
ISNATLIST(n__take(N, IL)) → ACTIVATE(IL)
TAKE(s(M), cons(N, IL)) → UTAKE2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
UTAKE2(tt, M, N, IL) → ACTIVATE(N)
UTAKE2(tt, M, N, IL) → ACTIVATE(M)
UTAKE2(tt, M, N, IL) → ACTIVATE(IL)
TAKE(s(M), cons(N, IL)) → ISNAT(M)
TAKE(s(M), cons(N, IL)) → ISNAT(N)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
LENGTH(cons(N, L)) → ACTIVATE(L)
ULENGTH(tt, L) → ACTIVATE(L)

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(5) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
ISNATLIST(n__take(N, IL)) → ISNAT(activate(N))
ISNATLIST(n__take(N, IL)) → ACTIVATE(N)
ISNATLIST(n__take(N, IL)) → ISNATILIST(activate(IL))
ISNATLIST(n__take(N, IL)) → ACTIVATE(IL)

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

POL( ISNAT(x₁) ) = x₁ + 2

POL( ISNATILIST(x₁) ) = x₁ + 2

POL( ISNATLIST(x₁) ) = x₁ + 2

POL( LENGTH(x₁) ) = x₁ + 2

POL( ULENGTH(x₁, x₂) ) = x₁ + x₂

POL( UTAKE2(x₁, ..., x₄) ) = x₁ + 2x₂ + 2x₃ + x₄

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

POL( cons(x₁, x₂) ) = 2x₁ + x₂

POL( isNat(x₁) ) = 2

POL( isNatList(x₁) ) = 2

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

POL( uTake2(x₁, ..., x₄) ) = 2x₂ + 2x₃ + x₄ + 1

POL( isNatIList(x₁) ) = 2

POL( n__take(x₁, x₂) ) = 2x₁ + x₂ + 1

POL( length(x₁) ) = x₁

POL( s(x₁) ) = x₁

POL( activate(x₁) ) = x₁

POL( n__0 ) = 0

POL( 0 ) = 0

POL( n__s(x₁) ) = x₁

POL( n__length(x₁) ) = x₁

POL( n__zeros ) = 0

POL( zeros ) = 0

POL( n__cons(x₁, x₂) ) = 2x₁ + x₂

POL( n__nil ) = 0

POL( nil ) = 0

POL( take(x₁, x₂) ) = 2x₁ + x₂ + 1

POL( tt ) = 2

POL( uTake1(x₁) ) = 0

POL( ACTIVATE(x₁) ) = x₁ + 2

POL( TAKE(x₁, x₂) ) = 2x₁ + x₂ + 2

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

activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
length(X) → n__length(X)
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
take(X1, X2) → n__take(X1, X2)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
cons(X1, X2) → n__cons(X1, X2)
uLength(tt, L) → s(length(activate(L)))
s(X) → n__s(X)
zeros → cons(0, n__zeros)
zeros → n__zeros
0 → n__0
nil → n__nil

(6) Obligation:

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

ISNATLIST(n__cons(N, L)) → ISNAT(activate(N))
ISNAT(n__s(N)) → ISNAT(activate(N))
ISNAT(n__s(N)) → ACTIVATE(N)
ACTIVATE(n__length(X)) → LENGTH(X)
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(activate(L))), activate(L))
ULENGTH(tt, L) → LENGTH(activate(L))
LENGTH(cons(N, L)) → ISNAT(N)
ISNAT(n__length(L)) → ISNATLIST(activate(L))
ISNATLIST(n__cons(N, L)) → ACTIVATE(N)
TAKE(0, IL) → ISNATILIST(IL)
ISNATILIST(IL) → ISNATLIST(activate(IL))
ISNATLIST(n__cons(N, L)) → ISNATLIST(activate(L))
ISNATLIST(n__cons(N, L)) → ACTIVATE(L)
ISNAT(n__length(L)) → ACTIVATE(L)
ISNATILIST(IL) → ACTIVATE(IL)
ISNATILIST(n__cons(N, IL)) → ISNAT(activate(N))
ISNATILIST(n__cons(N, IL)) → ACTIVATE(N)
ISNATILIST(n__cons(N, IL)) → ISNATILIST(activate(IL))
ISNATILIST(n__cons(N, IL)) → ACTIVATE(IL)
TAKE(s(M), cons(N, IL)) → UTAKE2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
UTAKE2(tt, M, N, IL) → ACTIVATE(N)
UTAKE2(tt, M, N, IL) → ACTIVATE(M)
UTAKE2(tt, M, N, IL) → ACTIVATE(IL)
TAKE(s(M), cons(N, IL)) → ISNAT(M)
TAKE(s(M), cons(N, IL)) → ISNAT(N)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
LENGTH(cons(N, L)) → ACTIVATE(L)
ULENGTH(tt, L) → ACTIVATE(L)

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
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 14 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

ISNAT(n__s(N)) → ISNAT(activate(N))
ISNAT(n__s(N)) → ACTIVATE(N)
ACTIVATE(n__length(X)) → LENGTH(X)
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(activate(L))), activate(L))
ULENGTH(tt, L) → LENGTH(activate(L))
LENGTH(cons(N, L)) → ISNAT(N)
ISNAT(n__length(L)) → ISNATLIST(activate(L))
ISNATLIST(n__cons(N, L)) → ISNAT(activate(N))
ISNAT(n__length(L)) → ACTIVATE(L)
ISNATLIST(n__cons(N, L)) → ACTIVATE(N)
ISNATLIST(n__cons(N, L)) → ISNATLIST(activate(L))
ISNATLIST(n__cons(N, L)) → ACTIVATE(L)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
LENGTH(cons(N, L)) → ACTIVATE(L)
ULENGTH(tt, L) → ACTIVATE(L)

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(10) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

ISNAT(n__s(N)) → ACTIVATE(N)
ACTIVATE(n__length(X)) → LENGTH(X)
ISNAT(n__length(L)) → ISNATLIST(activate(L))
ISNAT(n__length(L)) → ACTIVATE(L)
ISNATLIST(n__cons(N, L)) → ACTIVATE(N)
ISNATLIST(n__cons(N, L)) → ACTIVATE(L)
LENGTH(cons(N, L)) → ACTIVATE(L)
ULENGTH(tt, L) → ACTIVATE(L)

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

POL( ISNAT(x₁) ) = 2x₁ + 1

POL( ISNATLIST(x₁) ) = x₁ + 1

POL( LENGTH(x₁) ) = x₁ + 1

POL( ULENGTH(x₁, x₂) ) = 2x₂ + 1

POL( and(x₁, x₂) ) = x₂

POL( cons(x₁, x₂) ) = 2x₁ + 2x₂

POL( isNat(x₁) ) = x₁ + 2

POL( isNatList(x₁) ) = x₁ + 2

POL( uLength(x₁, x₂) ) = 2x₂ + 2

POL( uTake2(x₁, ..., x₄) ) = 2x₃ + 2x₄

POL( isNatIList(x₁) ) = x₁ + 2

POL( n__take(x₁, x₂) ) = x₂

POL( length(x₁) ) = x₁ + 2

POL( s(x₁) ) = x₁

POL( activate(x₁) ) = x₁

POL( n__0 ) = 0

POL( 0 ) = 0

POL( n__s(x₁) ) = x₁

POL( n__length(x₁) ) = x₁ + 2

POL( n__zeros ) = 0

POL( zeros ) = 0

POL( n__cons(x₁, x₂) ) = 2x₁ + 2x₂

POL( n__nil ) = 0

POL( nil ) = 0

POL( take(x₁, x₂) ) = x₂

POL( tt ) = 1

POL( uTake1(x₁) ) = max{0, -2}

POL( ACTIVATE(x₁) ) = 2x₁

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

activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
and(tt, T) → T
length(X) → n__length(X)
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
take(X1, X2) → n__take(X1, X2)
take(0, IL) → uTake1(isNatIList(IL))
isNatIList(n__zeros) → tt
uTake1(tt) → nil
isNatIList(IL) → isNatList(activate(IL))
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
cons(X1, X2) → n__cons(X1, X2)
uLength(tt, L) → s(length(activate(L)))
s(X) → n__s(X)
zeros → cons(0, n__zeros)
zeros → n__zeros
0 → n__0
nil → n__nil

(11) Obligation:

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

ISNAT(n__s(N)) → ISNAT(activate(N))
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(activate(L))), activate(L))
ULENGTH(tt, L) → LENGTH(activate(L))
LENGTH(cons(N, L)) → ISNAT(N)
ISNATLIST(n__cons(N, L)) → ISNAT(activate(N))
ISNATLIST(n__cons(N, L)) → ISNATLIST(activate(L))
LENGTH(cons(N, L)) → ISNATLIST(activate(L))

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(12) DependencyGraphProof (EQUIVALENT transformation)

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

(13) Complex Obligation (AND)

(14) Obligation:

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

ISNAT(n__s(N)) → ISNAT(activate(N))

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(15) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

ISNAT(n__s(N)) → ISNAT(activate(N))

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(ISNAT(x₁)) = 0A + 0A · x₁

POL(n__s(x₁)) = 2A + 1A · x₁

POL(activate(x₁)) = 1A + 0A · x₁

POL(n__0) = 0A

POL(0) = 0A

POL(s(x₁)) = 2A + 1A · x₁

POL(n__length(x₁)) = 2A + 0A · x₁

POL(length(x₁)) = 2A + 0A · x₁

POL(n__zeros) = 0A

POL(zeros) = 1A

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

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

POL(n__nil) = 2A

POL(nil) = 2A

POL(n__take(x₁, x₂)) = 4A + 3A · x₁ + 1A · x₂

POL(take(x₁, x₂)) = 4A + 3A · x₁ + 1A · x₂

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

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

POL(isNat(x₁)) = 2A + 4A · x₁

POL(isNatList(x₁)) = 2A + 1A · x₁

POL(tt) = 3A

POL(uTake1(x₁)) = 2A + -I · x₁

POL(isNatIList(x₁)) = 5A + 2A · x₁

POL(uTake2(x₁, x₂, x₃, x₄)) = 5A + 0A · x₁ + 4A · x₂ + 0A · x₃ + 2A · x₄

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

activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X
length(X) → n__length(X)
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
isNat(n__0) → tt
isNatList(n__nil) → tt
and(tt, T) → T
isNat(n__s(N)) → isNat(activate(N))
take(X1, X2) → n__take(X1, X2)
take(0, IL) → uTake1(isNatIList(IL))
isNatIList(n__zeros) → tt
uTake1(tt) → nil
isNatIList(IL) → isNatList(activate(IL))
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNat(n__length(L)) → isNatList(activate(L))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
cons(X1, X2) → n__cons(X1, X2)
uLength(tt, L) → s(length(activate(L)))
s(X) → n__s(X)
zeros → cons(0, n__zeros)
zeros → n__zeros
0 → n__0
nil → n__nil

(16) Obligation:

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

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(17) PisEmptyProof (EQUIVALENT transformation)

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

(18) YES

(19) Obligation:

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

ISNATLIST(n__cons(N, L)) → ISNATLIST(activate(L))

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(20) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATLIST(n__cons(N, L)) → ISNATLIST(activate(L)) at position [0] we obtained the following new rules [LPAR04]:

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

(21) Obligation:

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

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

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(22) TransformationProof (EQUIVALENT transformation)

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

ISNATLIST(n__cons(y0, n__0)) → ISNATLIST(n__0) → ISNATLIST(n__cons(y0, n__0)) → ISNATLIST(n__0)

(23) Obligation:

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

ISNATLIST(n__cons(y0, n__s(x0))) → ISNATLIST(s(x0))
ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(x0))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(zeros)
ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(cons(x0, x1))
ISNATLIST(n__cons(y0, n__nil)) → ISNATLIST(nil)
ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)
ISNATLIST(n__cons(y0, n__0)) → ISNATLIST(n__0)

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(24) DependencyGraphProof (EQUIVALENT transformation)

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

(25) Obligation:

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

ISNATLIST(n__cons(y0, n__s(x0))) → ISNATLIST(s(x0))
ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(x0))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(zeros)
ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(cons(x0, x1))
ISNATLIST(n__cons(y0, n__nil)) → ISNATLIST(nil)
ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(26) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATLIST(n__cons(y0, n__s(x0))) → ISNATLIST(s(x0)) at position [0] we obtained the following new rules [LPAR04]:

ISNATLIST(n__cons(y0, n__s(x0))) → ISNATLIST(n__s(x0)) → ISNATLIST(n__cons(y0, n__s(x0))) → ISNATLIST(n__s(x0))

(27) Obligation:

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

ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(x0))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(zeros)
ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(cons(x0, x1))
ISNATLIST(n__cons(y0, n__nil)) → ISNATLIST(nil)
ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)
ISNATLIST(n__cons(y0, n__s(x0))) → ISNATLIST(n__s(x0))

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(28) DependencyGraphProof (EQUIVALENT transformation)

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

(29) Obligation:

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

ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(x0))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(zeros)
ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(cons(x0, x1))
ISNATLIST(n__cons(y0, n__nil)) → ISNATLIST(nil)
ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(30) TransformationProof (EQUIVALENT transformation)

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

ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(cons(0, n__zeros)) → ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__zeros) → ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__zeros)

(31) Obligation:

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

ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(x0))
ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(cons(x0, x1))
ISNATLIST(n__cons(y0, n__nil)) → ISNATLIST(nil)
ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__zeros)

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(32) DependencyGraphProof (EQUIVALENT transformation)

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

(33) Obligation:

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

ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(x0))
ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(cons(x0, x1))
ISNATLIST(n__cons(y0, n__nil)) → ISNATLIST(nil)
ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(cons(0, n__zeros))

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(34) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(cons(x0, x1)) at position [0] we obtained the following new rules [LPAR04]:

ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(n__cons(x0, x1)) → ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(n__cons(x0, x1))

(35) Obligation:

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

ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(x0))
ISNATLIST(n__cons(y0, n__nil)) → ISNATLIST(nil)
ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(n__cons(x0, x1))

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(36) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATLIST(n__cons(y0, n__nil)) → ISNATLIST(nil) at position [0] we obtained the following new rules [LPAR04]:

ISNATLIST(n__cons(y0, n__nil)) → ISNATLIST(n__nil) → ISNATLIST(n__cons(y0, n__nil)) → ISNATLIST(n__nil)

(37) Obligation:

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

ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(x0))
ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(n__cons(x0, x1))
ISNATLIST(n__cons(y0, n__nil)) → ISNATLIST(n__nil)

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(38) DependencyGraphProof (EQUIVALENT transformation)

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

(39) Obligation:

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

ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(x0))
ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(n__cons(x0, x1))

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(40) TransformationProof (EQUIVALENT transformation)

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

ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(0, n__zeros)) → ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(cons(n__0, n__zeros)) → ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(cons(n__0, n__zeros))

(41) Obligation:

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

ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(x0))
ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)
ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(n__cons(x0, x1))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(cons(n__0, n__zeros))

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(42) TransformationProof (EQUIVALENT transformation)

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

ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(n__0, n__zeros)) → ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(n__0, n__zeros))

(43) Obligation:

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

ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(x0))
ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)
ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(n__cons(x0, x1))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(44) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(x0))

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

POL( ISNATLIST(x₁) ) = x₁

POL( length(x₁) ) = 2

POL( cons(x₁, x₂) ) = 2x₂

POL( uLength(x₁, x₂) ) = 2

POL( and(x₁, x₂) ) = x₂

POL( isNat(x₁) ) = 1

POL( isNatList(x₁) ) = 1

POL( activate(x₁) ) = x₁ + 2

POL( n__length(x₁) ) = 2

POL( take(x₁, x₂) ) = 0

POL( 0 ) = 0

POL( uTake1(x₁) ) = 0

POL( isNatIList(x₁) ) = 1

POL( s(x₁) ) = x₁

POL( uTake2(x₁, ..., x₄) ) = 0

POL( n__take(x₁, x₂) ) = max{0, -1}

POL( n__cons(x₁, x₂) ) = 2x₂

POL( n__0 ) = 1

POL( tt ) = 1

POL( n__s(x₁) ) = x₁

POL( n__zeros ) = 1

POL( zeros ) = 2

POL( n__nil ) = 0

POL( nil ) = 0

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

length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
length(X) → n__length(X)
take(0, IL) → uTake1(isNatIList(IL))
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
take(X1, X2) → n__take(X1, X2)
activate(n__length(X)) → length(X)
isNat(n__0) → tt
isNatList(n__nil) → tt
and(tt, T) → T
isNat(n__s(N)) → isNat(activate(N))
activate(n__take(X1, X2)) → take(X1, X2)
isNatIList(n__zeros) → tt
uTake1(tt) → nil
isNatIList(IL) → isNatList(activate(IL))
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNat(n__length(L)) → isNatList(activate(L))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
cons(X1, X2) → n__cons(X1, X2)
uLength(tt, L) → s(length(activate(L)))
s(X) → n__s(X)
nil → n__nil

(45) Obligation:

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

ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)
ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(n__cons(x0, x1))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(46) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)
ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(n__cons(x0, x1))

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

POL( ISNATLIST(x₁) ) = x₁ + 1

POL( take(x₁, x₂) ) = 2

POL( 0 ) = 0

POL( uTake1(x₁) ) = max{0, -2}

POL( isNatIList(x₁) ) = 1

POL( s(x₁) ) = 2

POL( cons(x₁, x₂) ) = 2x₂ + 2

POL( uTake2(x₁, ..., x₄) ) = x₁ + 1

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

POL( isNat(x₁) ) = 1

POL( activate(x₁) ) = x₁ + 2

POL( n__take(x₁, x₂) ) = 0

POL( isNatList(x₁) ) = 1

POL( n__cons(x₁, x₂) ) = 2x₂ + 2

POL( uLength(x₁, x₂) ) = 2x₁ + 2

POL( length(x₁) ) = 2x₁ + 2

POL( n__0 ) = 0

POL( n__length(x₁) ) = 2x₁

POL( tt ) = 1

POL( n__s(x₁) ) = 2

POL( n__zeros ) = 2

POL( zeros ) = 2

POL( n__nil ) = 0

POL( nil ) = 0

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

take(0, IL) → uTake1(isNatIList(IL))
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
take(X1, X2) → n__take(X1, X2)
activate(n__length(X)) → length(X)
length(X) → n__length(X)
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
isNat(n__0) → tt
isNatList(n__nil) → tt
and(tt, T) → T
isNat(n__s(N)) → isNat(activate(N))
activate(n__take(X1, X2)) → take(X1, X2)
isNatIList(n__zeros) → tt
uTake1(tt) → nil
isNatIList(IL) → isNatList(activate(IL))
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNat(n__length(L)) → isNatList(activate(L))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
cons(X1, X2) → n__cons(X1, X2)
uLength(tt, L) → s(length(activate(L)))
s(X) → n__s(X)
nil → n__nil

(47) Obligation:

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

ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(48) DependencyGraphProof (EQUIVALENT transformation)

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

(49) Complex Obligation (AND)

(50) Obligation:

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

ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(51) 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.

(52) Obligation:

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

ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

0 → n__0

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

(53) MNOCProof (EQUIVALENT transformation)

We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.

(54) Obligation:

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

ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

0 → n__0

The set Q consists of the following terms:

0

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

(55) Obligation:

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

ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(x0, x1))

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(56) Obligation:

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

LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(activate(L))), activate(L))
ULENGTH(tt, L) → LENGTH(activate(L))

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(57) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

ULENGTH(tt, L) → LENGTH(activate(L))

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(LENGTH(x₁)) = 3A + 0A · x₁

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

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

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

POL(isNat(x₁)) = 5A + 5A · x₁

POL(isNatList(x₁)) = 3A + 1A · x₁

POL(activate(x₁)) = 2A + 0A · x₁

POL(tt) = 4A

POL(n__0) = 5A

POL(n__s(x₁)) = 5A + 1A · x₁

POL(n__length(x₁)) = 5A + 2A · x₁

POL(0) = 5A

POL(s(x₁)) = 5A + 1A · x₁

POL(length(x₁)) = 5A + 2A · x₁

POL(n__zeros) = 0A

POL(zeros) = 2A

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

POL(n__nil) = 3A

POL(nil) = 3A

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

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

POL(isNatIList(x₁)) = 4A + 1A · x₁

POL(uLength(x₁, x₂)) = 2A + 2A · x₁ + 3A · x₂

POL(uTake1(x₁)) = 5A + 0A · x₁

POL(uTake2(x₁, x₂, x₃, x₄)) = -I + 1A · x₁ + 1A · x₂ + -I · x₃ + 2A · x₄

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

isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
and(tt, T) → T
length(X) → n__length(X)
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
take(X1, X2) → n__take(X1, X2)
take(0, IL) → uTake1(isNatIList(IL))
isNatIList(n__zeros) → tt
uTake1(tt) → nil
isNatIList(IL) → isNatList(activate(IL))
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
cons(X1, X2) → n__cons(X1, X2)
uLength(tt, L) → s(length(activate(L)))
s(X) → n__s(X)
zeros → cons(0, n__zeros)
zeros → n__zeros
0 → n__0
nil → n__nil

(58) Obligation:

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

LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(activate(L))), activate(L))

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(59) DependencyGraphProof (EQUIVALENT transformation)

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

(60) TRUE

(61) Obligation:

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

ISNATILIST(n__cons(N, IL)) → ISNATILIST(activate(IL))

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(62) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATILIST(n__cons(N, IL)) → ISNATILIST(activate(IL)) at position [0] we obtained the following new rules [LPAR04]:

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

(63) Obligation:

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

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

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(64) TransformationProof (EQUIVALENT transformation)

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

ISNATILIST(n__cons(y0, n__0)) → ISNATILIST(n__0) → ISNATILIST(n__cons(y0, n__0)) → ISNATILIST(n__0)

(65) Obligation:

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

ISNATILIST(n__cons(y0, n__s(x0))) → ISNATILIST(s(x0))
ISNATILIST(n__cons(y0, n__length(x0))) → ISNATILIST(length(x0))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(zeros)
ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(cons(x0, x1))
ISNATILIST(n__cons(y0, n__nil)) → ISNATILIST(nil)
ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(y0, x0)) → ISNATILIST(x0)
ISNATILIST(n__cons(y0, n__0)) → ISNATILIST(n__0)

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(66) DependencyGraphProof (EQUIVALENT transformation)

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

(67) Obligation:

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

ISNATILIST(n__cons(y0, n__s(x0))) → ISNATILIST(s(x0))
ISNATILIST(n__cons(y0, n__length(x0))) → ISNATILIST(length(x0))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(zeros)
ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(cons(x0, x1))
ISNATILIST(n__cons(y0, n__nil)) → ISNATILIST(nil)
ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(y0, x0)) → ISNATILIST(x0)

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(68) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATILIST(n__cons(y0, n__s(x0))) → ISNATILIST(s(x0)) at position [0] we obtained the following new rules [LPAR04]:

ISNATILIST(n__cons(y0, n__s(x0))) → ISNATILIST(n__s(x0)) → ISNATILIST(n__cons(y0, n__s(x0))) → ISNATILIST(n__s(x0))

(69) Obligation:

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

ISNATILIST(n__cons(y0, n__length(x0))) → ISNATILIST(length(x0))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(zeros)
ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(cons(x0, x1))
ISNATILIST(n__cons(y0, n__nil)) → ISNATILIST(nil)
ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(y0, x0)) → ISNATILIST(x0)
ISNATILIST(n__cons(y0, n__s(x0))) → ISNATILIST(n__s(x0))

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(70) DependencyGraphProof (EQUIVALENT transformation)

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

(71) Obligation:

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

ISNATILIST(n__cons(y0, n__length(x0))) → ISNATILIST(length(x0))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(zeros)
ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(cons(x0, x1))
ISNATILIST(n__cons(y0, n__nil)) → ISNATILIST(nil)
ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(y0, x0)) → ISNATILIST(x0)

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(72) TransformationProof (EQUIVALENT transformation)

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

ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(cons(0, n__zeros)) → ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__zeros) → ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__zeros)

(73) Obligation:

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

ISNATILIST(n__cons(y0, n__length(x0))) → ISNATILIST(length(x0))
ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(cons(x0, x1))
ISNATILIST(n__cons(y0, n__nil)) → ISNATILIST(nil)
ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(y0, x0)) → ISNATILIST(x0)
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__zeros)

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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 1 SCC with 1 less node.

(75) Obligation:

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

ISNATILIST(n__cons(y0, n__length(x0))) → ISNATILIST(length(x0))
ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(cons(x0, x1))
ISNATILIST(n__cons(y0, n__nil)) → ISNATILIST(nil)
ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(y0, x0)) → ISNATILIST(x0)
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(cons(0, n__zeros))

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(76) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(cons(x0, x1)) at position [0] we obtained the following new rules [LPAR04]:

ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(n__cons(x0, x1)) → ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(n__cons(x0, x1))

(77) Obligation:

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

ISNATILIST(n__cons(y0, n__length(x0))) → ISNATILIST(length(x0))
ISNATILIST(n__cons(y0, n__nil)) → ISNATILIST(nil)
ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(y0, x0)) → ISNATILIST(x0)
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(n__cons(x0, x1))

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(78) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATILIST(n__cons(y0, n__nil)) → ISNATILIST(nil) at position [0] we obtained the following new rules [LPAR04]:

ISNATILIST(n__cons(y0, n__nil)) → ISNATILIST(n__nil) → ISNATILIST(n__cons(y0, n__nil)) → ISNATILIST(n__nil)

(79) Obligation:

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

ISNATILIST(n__cons(y0, n__length(x0))) → ISNATILIST(length(x0))
ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(y0, x0)) → ISNATILIST(x0)
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(n__cons(x0, x1))
ISNATILIST(n__cons(y0, n__nil)) → ISNATILIST(n__nil)

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(80) DependencyGraphProof (EQUIVALENT transformation)

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

(81) Obligation:

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

ISNATILIST(n__cons(y0, n__length(x0))) → ISNATILIST(length(x0))
ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(y0, x0)) → ISNATILIST(x0)
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(n__cons(x0, x1))

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(82) TransformationProof (EQUIVALENT transformation)

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

ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(0, n__zeros)) → ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(0, n__zeros))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(cons(n__0, n__zeros)) → ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(cons(n__0, n__zeros))

(83) Obligation:

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

ISNATILIST(n__cons(y0, n__length(x0))) → ISNATILIST(length(x0))
ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(y0, x0)) → ISNATILIST(x0)
ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(n__cons(x0, x1))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(0, n__zeros))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(cons(n__0, n__zeros))

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(84) TransformationProof (EQUIVALENT transformation)

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

ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(n__0, n__zeros)) → ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(n__0, n__zeros))

(85) Obligation:

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

ISNATILIST(n__cons(y0, n__length(x0))) → ISNATILIST(length(x0))
ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(y0, x0)) → ISNATILIST(x0)
ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(n__cons(x0, x1))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(0, n__zeros))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(86) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

ISNATILIST(n__cons(y0, n__length(x0))) → ISNATILIST(length(x0))

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

POL( ISNATILIST(x₁) ) = x₁

POL( length(x₁) ) = 2

POL( cons(x₁, x₂) ) = 2x₂

POL( uLength(x₁, x₂) ) = 2

POL( and(x₁, x₂) ) = x₂

POL( isNat(x₁) ) = 1

POL( isNatList(x₁) ) = 1

POL( activate(x₁) ) = x₁ + 2

POL( n__length(x₁) ) = 2

POL( take(x₁, x₂) ) = 0

POL( 0 ) = 0

POL( uTake1(x₁) ) = 0

POL( isNatIList(x₁) ) = 1

POL( s(x₁) ) = x₁

POL( uTake2(x₁, ..., x₄) ) = 0

POL( n__take(x₁, x₂) ) = max{0, -1}

POL( n__cons(x₁, x₂) ) = 2x₂

POL( n__0 ) = 1

POL( tt ) = 1

POL( n__s(x₁) ) = x₁

POL( n__zeros ) = 1

POL( zeros ) = 2

POL( n__nil ) = 0

POL( nil ) = 0

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

length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
length(X) → n__length(X)
take(0, IL) → uTake1(isNatIList(IL))
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
take(X1, X2) → n__take(X1, X2)
activate(n__length(X)) → length(X)
isNat(n__0) → tt
isNatList(n__nil) → tt
and(tt, T) → T
isNat(n__s(N)) → isNat(activate(N))
activate(n__take(X1, X2)) → take(X1, X2)
isNatIList(n__zeros) → tt
uTake1(tt) → nil
isNatIList(IL) → isNatList(activate(IL))
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNat(n__length(L)) → isNatList(activate(L))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
cons(X1, X2) → n__cons(X1, X2)
uLength(tt, L) → s(length(activate(L)))
s(X) → n__s(X)
nil → n__nil

(87) Obligation:

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

ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(y0, x0)) → ISNATILIST(x0)
ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(n__cons(x0, x1))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(0, n__zeros))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(88) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

ISNATILIST(n__cons(y0, x0)) → ISNATILIST(x0)
ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(n__cons(x0, x1))

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

POL( ISNATILIST(x₁) ) = x₁ + 1

POL( take(x₁, x₂) ) = 2

POL( 0 ) = 0

POL( uTake1(x₁) ) = max{0, -2}

POL( isNatIList(x₁) ) = 1

POL( s(x₁) ) = 2

POL( cons(x₁, x₂) ) = 2x₂ + 2

POL( uTake2(x₁, ..., x₄) ) = x₁ + 1

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

POL( isNat(x₁) ) = 1

POL( activate(x₁) ) = x₁ + 2

POL( n__take(x₁, x₂) ) = 0

POL( isNatList(x₁) ) = 1

POL( n__cons(x₁, x₂) ) = 2x₂ + 2

POL( uLength(x₁, x₂) ) = 2x₁ + 2

POL( length(x₁) ) = 2x₁ + 2

POL( n__0 ) = 0

POL( n__length(x₁) ) = 2x₁

POL( tt ) = 1

POL( n__s(x₁) ) = 2

POL( n__zeros ) = 2

POL( zeros ) = 2

POL( n__nil ) = 0

POL( nil ) = 0

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

take(0, IL) → uTake1(isNatIList(IL))
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
take(X1, X2) → n__take(X1, X2)
activate(n__length(X)) → length(X)
length(X) → n__length(X)
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
isNat(n__0) → tt
isNatList(n__nil) → tt
and(tt, T) → T
isNat(n__s(N)) → isNat(activate(N))
activate(n__take(X1, X2)) → take(X1, X2)
isNatIList(n__zeros) → tt
uTake1(tt) → nil
isNatIList(IL) → isNatList(activate(IL))
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNat(n__length(L)) → isNatList(activate(L))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
cons(X1, X2) → n__cons(X1, X2)
uLength(tt, L) → s(length(activate(L)))
s(X) → n__s(X)
nil → n__nil

(89) Obligation:

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

ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(take(x0, x1))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(0, n__zeros))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(90) DependencyGraphProof (EQUIVALENT transformation)

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

(91) Complex Obligation (AND)

(92) Obligation:

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

ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(0, n__zeros))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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

(93) UsableRulesProof (EQUIVALENT transformation)

(94) Obligation:

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

ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(0, n__zeros))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

0 → n__0

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

(95) MNOCProof (EQUIVALENT transformation)

We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.

(96) Obligation:

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

ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(0, n__zeros))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

0 → n__0

The set Q consists of the following terms:

0

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

(97) TransformationProof (EQUIVALENT transformation)

By rewriting [LPAR04] the rule ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(0, n__zeros)) at position [0,0] we obtained the following new rules [LPAR04]:

ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(n__0, n__zeros)) → ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(n__0, n__zeros))

(98) Obligation:

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

ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

0 → n__0

The set Q consists of the following terms:

0

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

(99) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(100) Obligation:

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

ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(n__0, n__zeros))

R is empty.
The set Q consists of the following terms:

0

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

(101) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

0

(102) Obligation:

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

ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(n__0, n__zeros))

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

(103) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(n__0, n__zeros)) we obtained the following new rules [LPAR04]:

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

(104) Obligation:

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

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

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

(105) NonTerminationLoopProof (COMPLETE transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = ISNATILIST(n__cons(n__0, n__zeros)) evaluates to t =ISNATILIST(n__cons(n__0, n__zeros))

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

Matcher: [ ]
Semiunifier: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from ISNATILIST(n__cons(n__0, n__zeros)) to ISNATILIST(n__cons(n__0, n__zeros)).

(106) NO

(107) Obligation:

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

ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(take(x0, x1))

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

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