Termination w.r.t. Q proof of Transformed_CSR_04_Ex4_DLMMU04

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

0 QTRS

↳1 DependencyPairsProof (⇔, 0 ms)

↳2 QDP

↳3 DependencyGraphProof (⇔, 0 ms)

↳4 QDP

↳5 QDPOrderProof (⇔, 54 ms)

↳6 QDP

↳7 DependencyGraphProof (⇔, 0 ms)

↳8 AND

    ↳9 QDP

        ↳10 QDPOrderProof (⇔, 120 ms)

        ↳11 QDP

        ↳12 DependencyGraphProof (⇔, 0 ms)

        ↳13 TRUE

    ↳14 QDP

        ↳15 QDPOrderProof (⇔, 54 ms)

        ↳16 QDP

        ↳17 DependencyGraphProof (⇔, 0 ms)

        ↳18 QDP

        ↳19 QDPOrderProof (⇔, 26 ms)

        ↳20 QDP

        ↳21 QDPOrderProof (⇔, 38 ms)

        ↳22 QDP

        ↳23 DependencyGraphProof (⇔, 0 ms)

        ↳24 AND

            ↳25 QDP

                ↳26 UsableRulesProof (⇔, 0 ms)

                ↳27 QDP

                ↳28 QDPSizeChangeProof (⇔, 0 ms)

                ↳29 YES

            ↳30 QDP

                ↳31 QDPOrderProof (⇔, 36 ms)

                ↳32 QDP

                ↳33 QDPOrderProof (⇔, 111 ms)

                ↳34 QDP

                ↳35 QDPOrderProof (⇔, 60 ms)

                ↳36 QDP

                ↳37 QDPOrderProof (⇔, 71 ms)

                ↳38 QDP

                ↳39 DependencyGraphProof (⇔, 0 ms)

                ↳40 QDP

                ↳41 UsableRulesProof (⇔, 0 ms)

                ↳42 QDP

                ↳43 QDPSizeChangeProof (⇔, 0 ms)

                ↳44 YES

(0) Obligation:

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

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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:

A__AND(tt, T) → MARK(T)
A__ISNATILIST(IL) → A__ISNATLIST(IL)
A__ISNAT(s(N)) → A__ISNAT(N)
A__ISNAT(length(L)) → A__ISNATLIST(L)
A__ISNATILIST(cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATILIST(cons(N, IL)) → A__ISNAT(N)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
A__ISNATLIST(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
A__ISNATLIST(cons(N, L)) → A__ISNAT(N)
A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
A__ISNATLIST(take(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATLIST(take(N, IL)) → A__ISNAT(N)
A__ISNATLIST(take(N, IL)) → A__ISNATILIST(IL)
A__TAKE(0, IL) → A__UTAKE1(a__isNatIList(IL))
A__TAKE(0, IL) → A__ISNATILIST(IL)
A__TAKE(s(M), cons(N, IL)) → A__UTAKE2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
A__TAKE(s(M), cons(N, IL)) → A__AND(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL)))
A__TAKE(s(M), cons(N, IL)) → A__ISNAT(M)
A__TAKE(s(M), cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__TAKE(s(M), cons(N, IL)) → A__ISNAT(N)
A__TAKE(s(M), cons(N, IL)) → A__ISNATILIST(IL)
A__UTAKE2(tt, M, N, IL) → MARK(N)
A__LENGTH(cons(N, L)) → A__ULENGTH(a__and(a__isNat(N), a__isNatList(L)), L)
A__LENGTH(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
A__LENGTH(cons(N, L)) → A__ISNAT(N)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
A__ULENGTH(tt, L) → A__LENGTH(mark(L))
A__ULENGTH(tt, L) → MARK(L)
MARK(and(X1, X2)) → A__AND(mark(X1), mark(X2))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X2)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(isNat(X)) → A__ISNAT(X)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(length(X)) → MARK(X)
MARK(zeros) → A__ZEROS
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(uTake1(X)) → A__UTAKE1(mark(X))
MARK(uTake1(X)) → MARK(X)
MARK(uTake2(X1, X2, X3, X4)) → A__UTAKE2(mark(X1), X2, X3, X4)
MARK(uTake2(X1, X2, X3, X4)) → MARK(X1)
MARK(uLength(X1, X2)) → A__ULENGTH(mark(X1), X2)
MARK(uLength(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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 3 less nodes.

(4) Obligation:

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

MARK(and(X1, X2)) → A__AND(mark(X1), mark(X2))
A__AND(tt, T) → MARK(T)
MARK(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X2)
MARK(isNatIList(X)) → A__ISNATILIST(X)
A__ISNATILIST(IL) → A__ISNATLIST(IL)
A__ISNATLIST(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
A__ISNATLIST(cons(N, L)) → A__ISNAT(N)
A__ISNAT(s(N)) → A__ISNAT(N)
A__ISNAT(length(L)) → A__ISNATLIST(L)
A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
A__ISNATLIST(take(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATLIST(take(N, IL)) → A__ISNAT(N)
A__ISNATLIST(take(N, IL)) → A__ISNATILIST(IL)
A__ISNATILIST(cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATILIST(cons(N, IL)) → A__ISNAT(N)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(isNat(X)) → A__ISNAT(X)
MARK(length(X)) → A__LENGTH(mark(X))
A__LENGTH(cons(N, L)) → A__ULENGTH(a__and(a__isNat(N), a__isNatList(L)), L)
A__ULENGTH(tt, L) → A__LENGTH(mark(L))
A__LENGTH(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
A__LENGTH(cons(N, L)) → A__ISNAT(N)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
A__ULENGTH(tt, L) → MARK(L)
MARK(length(X)) → MARK(X)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
A__TAKE(0, IL) → A__ISNATILIST(IL)
A__TAKE(s(M), cons(N, IL)) → A__UTAKE2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
A__UTAKE2(tt, M, N, IL) → MARK(N)
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(uTake1(X)) → MARK(X)
MARK(uTake2(X1, X2, X3, X4)) → A__UTAKE2(mark(X1), X2, X3, X4)
MARK(uTake2(X1, X2, X3, X4)) → MARK(X1)
MARK(uLength(X1, X2)) → A__ULENGTH(mark(X1), X2)
MARK(uLength(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
A__TAKE(s(M), cons(N, IL)) → A__AND(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL)))
A__TAKE(s(M), cons(N, IL)) → A__ISNAT(M)
A__TAKE(s(M), cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__TAKE(s(M), cons(N, IL)) → A__ISNAT(N)
A__TAKE(s(M), cons(N, IL)) → A__ISNATILIST(IL)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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.

A__LENGTH(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
A__LENGTH(cons(N, L)) → A__ISNAT(N)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
A__ULENGTH(tt, L) → MARK(L)
MARK(length(X)) → MARK(X)
MARK(uLength(X1, X2)) → MARK(X1)

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

POL( A__AND(x₁, x₂) ) = 2x₂ + 1

POL( A__LENGTH(x₁) ) = x₁ + 2

POL( A__TAKE(x₁, x₂) ) = x₁ + x₂ + 1

POL( A__ULENGTH(x₁, x₂) ) = x₂ + 2

POL( A__UTAKE2(x₁, ..., x₄) ) = x₁ + x₃ + 1

POL( mark(x₁) ) = x₁

POL( and(x₁, x₂) ) = 2x₁ + 2x₂

POL( a__and(x₁, x₂) ) = 2x₁ + 2x₂

POL( tt ) = 0

POL( isNatIList(x₁) ) = 0

POL( a__isNatIList(x₁) ) = 0

POL( a__isNatList(x₁) ) = 0

POL( cons(x₁, x₂) ) = 2x₁ + x₂

POL( a__isNat(x₁) ) = 0

POL( take(x₁, x₂) ) = x₁ + x₂

POL( isNatList(x₁) ) = 0

POL( isNat(x₁) ) = 0

POL( s(x₁) ) = x₁

POL( length(x₁) ) = x₁ + 1

POL( a__length(x₁) ) = x₁ + 1

POL( zeros ) = 0

POL( a__zeros ) = 0

POL( a__take(x₁, x₂) ) = x₁ + x₂

POL( uTake1(x₁) ) = x₁

POL( a__uTake1(x₁) ) = x₁

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

POL( a__uTake2(x₁, ..., x₄) ) = x₁ + x₂ + 2x₃ + x₄

POL( uLength(x₁, x₂) ) = x₁ + x₂ + 1

POL( a__uLength(x₁, x₂) ) = x₁ + x₂ + 1

POL( 0 ) = 0

POL( nil ) = 0

POL( MARK(x₁) ) = x₁ + 1

POL( A__ISNATILIST(x₁) ) = 1

POL( A__ISNATLIST(x₁) ) = 1

POL( A__ISNAT(x₁) ) = 1

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

mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
a__and(tt, T) → mark(T)
mark(isNatIList(X)) → a__isNatIList(X)
a__isNatIList(IL) → a__isNatList(IL)
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__isNat(0) → tt
a__isNat(X) → isNat(X)
a__isNatList(nil) → tt
a__isNatList(X) → isNatList(X)
a__isNatIList(zeros) → tt
a__isNatIList(X) → isNatIList(X)
a__and(X1, X2) → and(X1, X2)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(X) → length(X)
a__take(X1, X2) → take(X1, X2)
a__uTake1(tt) → nil
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)
a__uLength(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__zeros → cons(0, zeros)
a__zeros → zeros

(6) Obligation:

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

MARK(and(X1, X2)) → A__AND(mark(X1), mark(X2))
A__AND(tt, T) → MARK(T)
MARK(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X2)
MARK(isNatIList(X)) → A__ISNATILIST(X)
A__ISNATILIST(IL) → A__ISNATLIST(IL)
A__ISNATLIST(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
A__ISNATLIST(cons(N, L)) → A__ISNAT(N)
A__ISNAT(s(N)) → A__ISNAT(N)
A__ISNAT(length(L)) → A__ISNATLIST(L)
A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
A__ISNATLIST(take(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATLIST(take(N, IL)) → A__ISNAT(N)
A__ISNATLIST(take(N, IL)) → A__ISNATILIST(IL)
A__ISNATILIST(cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATILIST(cons(N, IL)) → A__ISNAT(N)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(isNat(X)) → A__ISNAT(X)
MARK(length(X)) → A__LENGTH(mark(X))
A__LENGTH(cons(N, L)) → A__ULENGTH(a__and(a__isNat(N), a__isNatList(L)), L)
A__ULENGTH(tt, L) → A__LENGTH(mark(L))
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
A__TAKE(0, IL) → A__ISNATILIST(IL)
A__TAKE(s(M), cons(N, IL)) → A__UTAKE2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
A__UTAKE2(tt, M, N, IL) → MARK(N)
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(uTake1(X)) → MARK(X)
MARK(uTake2(X1, X2, X3, X4)) → A__UTAKE2(mark(X1), X2, X3, X4)
MARK(uTake2(X1, X2, X3, X4)) → MARK(X1)
MARK(uLength(X1, X2)) → A__ULENGTH(mark(X1), X2)
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
A__TAKE(s(M), cons(N, IL)) → A__AND(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL)))
A__TAKE(s(M), cons(N, IL)) → A__ISNAT(M)
A__TAKE(s(M), cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__TAKE(s(M), cons(N, IL)) → A__ISNAT(N)
A__TAKE(s(M), cons(N, IL)) → A__ISNATILIST(IL)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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 2 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

A__ULENGTH(tt, L) → A__LENGTH(mark(L))
A__LENGTH(cons(N, L)) → A__ULENGTH(a__and(a__isNat(N), a__isNatList(L)), L)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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.

A__ULENGTH(tt, L) → A__LENGTH(mark(L))

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

POL(A__ULENGTH(x₁, x₂)) = 4A + 0A · x₁ + 2A · x₂

POL(tt) = 5A

POL(A__LENGTH(x₁)) = 4A + 1A · x₁

POL(mark(x₁)) = 1A + 0A · x₁

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

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

POL(a__isNat(x₁)) = 4A + 3A · x₁

POL(a__isNatList(x₁)) = 0A + 1A · x₁

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

POL(isNatIList(x₁)) = 5A + 1A · x₁

POL(a__isNatIList(x₁)) = 5A + 1A · x₁

POL(take(x₁, x₂)) = 5A + 4A · x₁ + 3A · x₂

POL(isNatList(x₁)) = 0A + 1A · x₁

POL(isNat(x₁)) = 4A + 3A · x₁

POL(s(x₁)) = 5A + 1A · x₁

POL(length(x₁)) = 3A + 0A · x₁

POL(a__length(x₁)) = 3A + 0A · x₁

POL(zeros) = 0A

POL(a__zeros) = 1A

POL(a__take(x₁, x₂)) = 5A + 4A · x₁ + 3A · x₂

POL(uTake1(x₁)) = 2A + 1A · x₁

POL(a__uTake1(x₁)) = 2A + 1A · x₁

POL(uTake2(x₁, x₂, x₃, x₄)) = 5A + 3A · x₁ + 5A · x₂ + -I · x₃ + 4A · x₄

POL(a__uTake2(x₁, x₂, x₃, x₄)) = 5A + 3A · x₁ + 5A · x₂ + -I · x₃ + 4A · x₄

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

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

POL(0) = 2A

POL(nil) = 5A

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

mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
a__and(tt, T) → mark(T)
mark(isNatIList(X)) → a__isNatIList(X)
a__isNatIList(IL) → a__isNatList(IL)
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__isNat(0) → tt
a__isNat(X) → isNat(X)
a__isNatList(nil) → tt
a__isNatList(X) → isNatList(X)
a__and(X1, X2) → and(X1, X2)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__isNatIList(zeros) → tt
a__isNatIList(X) → isNatIList(X)
a__length(X) → length(X)
a__take(X1, X2) → take(X1, X2)
a__uTake1(tt) → nil
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)
a__uLength(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__zeros → cons(0, zeros)
a__zeros → zeros

(11) Obligation:

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

A__LENGTH(cons(N, L)) → A__ULENGTH(a__and(a__isNat(N), a__isNatList(L)), L)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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 0 SCCs with 1 less node.

(13) TRUE

(14) Obligation:

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

A__AND(tt, T) → MARK(T)
MARK(and(X1, X2)) → A__AND(mark(X1), mark(X2))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X2)
MARK(isNatIList(X)) → A__ISNATILIST(X)
A__ISNATILIST(IL) → A__ISNATLIST(IL)
A__ISNATLIST(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
A__ISNATLIST(cons(N, L)) → A__ISNAT(N)
A__ISNAT(s(N)) → A__ISNAT(N)
A__ISNAT(length(L)) → A__ISNATLIST(L)
A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
A__ISNATLIST(take(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATLIST(take(N, IL)) → A__ISNAT(N)
A__ISNATLIST(take(N, IL)) → A__ISNATILIST(IL)
A__ISNATILIST(cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATILIST(cons(N, IL)) → A__ISNAT(N)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(isNat(X)) → A__ISNAT(X)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
A__TAKE(0, IL) → A__ISNATILIST(IL)
A__TAKE(s(M), cons(N, IL)) → A__UTAKE2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
A__UTAKE2(tt, M, N, IL) → MARK(N)
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(uTake1(X)) → MARK(X)
MARK(uTake2(X1, X2, X3, X4)) → A__UTAKE2(mark(X1), X2, X3, X4)
MARK(uTake2(X1, X2, X3, X4)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
A__TAKE(s(M), cons(N, IL)) → A__AND(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL)))
A__TAKE(s(M), cons(N, IL)) → A__ISNAT(M)
A__TAKE(s(M), cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__TAKE(s(M), cons(N, IL)) → A__ISNAT(N)
A__TAKE(s(M), cons(N, IL)) → A__ISNATILIST(IL)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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.

MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
A__TAKE(0, IL) → A__ISNATILIST(IL)
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(uTake1(X)) → MARK(X)
MARK(uTake2(X1, X2, X3, X4)) → A__UTAKE2(mark(X1), X2, X3, X4)
MARK(uTake2(X1, X2, X3, X4)) → MARK(X1)

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

POL( A__AND(x₁, x₂) ) = 2x₂ + 1

POL( A__TAKE(x₁, x₂) ) = x₁ + 2x₂ + 1

POL( A__UTAKE2(x₁, ..., x₄) ) = 2x₃ + 1

POL( mark(x₁) ) = x₁

POL( and(x₁, x₂) ) = x₁ + x₂

POL( a__and(x₁, x₂) ) = x₁ + x₂

POL( tt ) = 0

POL( isNatIList(x₁) ) = 0

POL( a__isNatIList(x₁) ) = 0

POL( a__isNatList(x₁) ) = 0

POL( cons(x₁, x₂) ) = 2x₁

POL( a__isNat(x₁) ) = 0

POL( take(x₁, x₂) ) = x₁ + 2x₂ + 2

POL( isNatList(x₁) ) = 0

POL( isNat(x₁) ) = 0

POL( s(x₁) ) = x₁

POL( length(x₁) ) = 1

POL( a__length(x₁) ) = 1

POL( zeros ) = 2

POL( a__zeros ) = 2

POL( a__take(x₁, x₂) ) = x₁ + 2x₂ + 2

POL( uTake1(x₁) ) = x₁ + 1

POL( a__uTake1(x₁) ) = x₁ + 1

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

POL( a__uTake2(x₁, ..., x₄) ) = 2x₁ + 2x₃ + 2

POL( uLength(x₁, x₂) ) = x₁ + 1

POL( a__uLength(x₁, x₂) ) = x₁ + 1

POL( 0 ) = 1

POL( nil ) = 1

POL( MARK(x₁) ) = 2x₁ + 1

POL( A__ISNATILIST(x₁) ) = 1

POL( A__ISNATLIST(x₁) ) = 1

POL( A__ISNAT(x₁) ) = 1

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

mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
a__and(tt, T) → mark(T)
mark(isNatIList(X)) → a__isNatIList(X)
a__isNatIList(IL) → a__isNatList(IL)
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__isNat(0) → tt
a__isNat(X) → isNat(X)
a__isNatList(nil) → tt
a__isNatList(X) → isNatList(X)
a__isNatIList(zeros) → tt
a__isNatIList(X) → isNatIList(X)
a__and(X1, X2) → and(X1, X2)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(X) → length(X)
a__take(X1, X2) → take(X1, X2)
a__uTake1(tt) → nil
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)
a__uLength(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__zeros → cons(0, zeros)
a__zeros → zeros

(16) Obligation:

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

A__AND(tt, T) → MARK(T)
MARK(and(X1, X2)) → A__AND(mark(X1), mark(X2))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X2)
MARK(isNatIList(X)) → A__ISNATILIST(X)
A__ISNATILIST(IL) → A__ISNATLIST(IL)
A__ISNATLIST(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
A__ISNATLIST(cons(N, L)) → A__ISNAT(N)
A__ISNAT(s(N)) → A__ISNAT(N)
A__ISNAT(length(L)) → A__ISNATLIST(L)
A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
A__ISNATLIST(take(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATLIST(take(N, IL)) → A__ISNAT(N)
A__ISNATLIST(take(N, IL)) → A__ISNATILIST(IL)
A__ISNATILIST(cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATILIST(cons(N, IL)) → A__ISNAT(N)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(isNat(X)) → A__ISNAT(X)
A__TAKE(s(M), cons(N, IL)) → A__UTAKE2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
A__UTAKE2(tt, M, N, IL) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
A__TAKE(s(M), cons(N, IL)) → A__AND(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL)))
A__TAKE(s(M), cons(N, IL)) → A__ISNAT(M)
A__TAKE(s(M), cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__TAKE(s(M), cons(N, IL)) → A__ISNAT(N)
A__TAKE(s(M), cons(N, IL)) → A__ISNATILIST(IL)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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

(17) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 7 less nodes.

(18) Obligation:

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

MARK(and(X1, X2)) → A__AND(mark(X1), mark(X2))
A__AND(tt, T) → MARK(T)
MARK(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X2)
MARK(isNatIList(X)) → A__ISNATILIST(X)
A__ISNATILIST(IL) → A__ISNATLIST(IL)
A__ISNATLIST(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
A__ISNATLIST(cons(N, L)) → A__ISNAT(N)
A__ISNAT(s(N)) → A__ISNAT(N)
A__ISNAT(length(L)) → A__ISNATLIST(L)
A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
A__ISNATLIST(take(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATLIST(take(N, IL)) → A__ISNAT(N)
A__ISNATLIST(take(N, IL)) → A__ISNATILIST(IL)
A__ISNATILIST(cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATILIST(cons(N, IL)) → A__ISNAT(N)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(isNat(X)) → A__ISNAT(X)
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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

(19) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

MARK(cons(X1, X2)) → MARK(X1)

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

POL( A__AND(x₁, x₂) ) = x₂ + 1

POL( mark(x₁) ) = x₁

POL( and(x₁, x₂) ) = 2x₁ + 2x₂

POL( a__and(x₁, x₂) ) = 2x₁ + 2x₂

POL( tt ) = 0

POL( isNatIList(x₁) ) = 0

POL( a__isNatIList(x₁) ) = 0

POL( a__isNatList(x₁) ) = 0

POL( cons(x₁, x₂) ) = x₁ + 1

POL( a__isNat(x₁) ) = 0

POL( take(x₁, x₂) ) = x₁ + x₂

POL( isNatList(x₁) ) = 0

POL( isNat(x₁) ) = 0

POL( s(x₁) ) = 2x₁

POL( length(x₁) ) = 0

POL( a__length(x₁) ) = 0

POL( zeros ) = 1

POL( a__zeros ) = 1

POL( a__take(x₁, x₂) ) = x₁ + x₂

POL( uTake1(x₁) ) = 0

POL( a__uTake1(x₁) ) = 0

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

POL( a__uTake2(x₁, ..., x₄) ) = 2x₂ + x₃ + 1

POL( uLength(x₁, x₂) ) = 0

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

POL( 0 ) = 0

POL( nil ) = 0

POL( MARK(x₁) ) = x₁ + 1

POL( A__ISNATILIST(x₁) ) = 1

POL( A__ISNATLIST(x₁) ) = 1

POL( A__ISNAT(x₁) ) = 1

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

mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
a__and(tt, T) → mark(T)
mark(isNatIList(X)) → a__isNatIList(X)
a__isNatIList(IL) → a__isNatList(IL)
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__isNat(0) → tt
a__isNat(X) → isNat(X)
a__isNatList(nil) → tt
a__isNatList(X) → isNatList(X)
a__isNatIList(zeros) → tt
a__isNatIList(X) → isNatIList(X)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__and(X1, X2) → and(X1, X2)
a__length(X) → length(X)
a__take(X1, X2) → take(X1, X2)
a__uTake1(tt) → nil
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)
a__uLength(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__zeros → cons(0, zeros)
a__zeros → zeros

(20) Obligation:

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

MARK(and(X1, X2)) → A__AND(mark(X1), mark(X2))
A__AND(tt, T) → MARK(T)
MARK(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X2)
MARK(isNatIList(X)) → A__ISNATILIST(X)
A__ISNATILIST(IL) → A__ISNATLIST(IL)
A__ISNATLIST(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
A__ISNATLIST(cons(N, L)) → A__ISNAT(N)
A__ISNAT(s(N)) → A__ISNAT(N)
A__ISNAT(length(L)) → A__ISNATLIST(L)
A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
A__ISNATLIST(take(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATLIST(take(N, IL)) → A__ISNAT(N)
A__ISNATLIST(take(N, IL)) → A__ISNATILIST(IL)
A__ISNATILIST(cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATILIST(cons(N, IL)) → A__ISNAT(N)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(isNat(X)) → A__ISNAT(X)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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

(21) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

A__ISNATILIST(IL) → A__ISNATLIST(IL)
A__ISNAT(length(L)) → A__ISNATLIST(L)
A__ISNATLIST(take(N, IL)) → A__ISNAT(N)
A__ISNATILIST(cons(N, IL)) → A__ISNAT(N)

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

POL( A__AND(x₁, x₂) ) = x₁ + x₂

POL( mark(x₁) ) = x₁

POL( and(x₁, x₂) ) = x₁ + x₂

POL( a__and(x₁, x₂) ) = x₁ + x₂

POL( tt ) = 0

POL( isNatIList(x₁) ) = 2x₁ + 2

POL( a__isNatIList(x₁) ) = 2x₁ + 2

POL( a__isNatList(x₁) ) = x₁

POL( cons(x₁, x₂) ) = x₁ + x₂

POL( a__isNat(x₁) ) = x₁

POL( take(x₁, x₂) ) = x₁ + 2x₂ + 2

POL( isNatList(x₁) ) = x₁

POL( isNat(x₁) ) = x₁

POL( s(x₁) ) = x₁

POL( length(x₁) ) = x₁ + 2

POL( a__length(x₁) ) = x₁ + 2

POL( zeros ) = 1

POL( a__zeros ) = 1

POL( a__take(x₁, x₂) ) = x₁ + 2x₂ + 2

POL( uTake1(x₁) ) = 0

POL( a__uTake1(x₁) ) = max{0, -2}

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

POL( a__uTake2(x₁, ..., x₄) ) = x₂ + x₃ + 2x₄ + 2

POL( uLength(x₁, x₂) ) = x₂ + 2

POL( a__uLength(x₁, x₂) ) = x₂ + 2

POL( 0 ) = 0

POL( nil ) = 0

POL( MARK(x₁) ) = x₁

POL( A__ISNATILIST(x₁) ) = 2x₁ + 2

POL( A__ISNATLIST(x₁) ) = x₁

POL( A__ISNAT(x₁) ) = x₁

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

mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
a__and(tt, T) → mark(T)
mark(isNatIList(X)) → a__isNatIList(X)
a__isNatIList(IL) → a__isNatList(IL)
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__isNat(0) → tt
a__isNat(X) → isNat(X)
a__isNatList(nil) → tt
a__isNatList(X) → isNatList(X)
a__isNatIList(zeros) → tt
a__isNatIList(X) → isNatIList(X)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__and(X1, X2) → and(X1, X2)
a__length(X) → length(X)
a__take(X1, X2) → take(X1, X2)
a__uTake1(tt) → nil
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)
a__uLength(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__zeros → cons(0, zeros)
a__zeros → zeros

(22) Obligation:

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

MARK(and(X1, X2)) → A__AND(mark(X1), mark(X2))
A__AND(tt, T) → MARK(T)
MARK(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X2)
MARK(isNatIList(X)) → A__ISNATILIST(X)
A__ISNATLIST(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
A__ISNATLIST(cons(N, L)) → A__ISNAT(N)
A__ISNAT(s(N)) → A__ISNAT(N)
A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
A__ISNATLIST(take(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATLIST(take(N, IL)) → A__ISNATILIST(IL)
A__ISNATILIST(cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(isNat(X)) → A__ISNAT(X)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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

(23) DependencyGraphProof (EQUIVALENT transformation)

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

(24) Complex Obligation (AND)

(25) Obligation:

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

A__ISNAT(s(N)) → A__ISNAT(N)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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

(26) 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.

(27) Obligation:

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

A__ISNAT(s(N)) → A__ISNAT(N)

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

(28) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

A__ISNAT(s(N)) → A__ISNAT(N)
The graph contains the following edges 1 > 1

(29) YES

(30) Obligation:

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

A__AND(tt, T) → MARK(T)
MARK(and(X1, X2)) → A__AND(mark(X1), mark(X2))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X2)
MARK(isNatIList(X)) → A__ISNATILIST(X)
A__ISNATILIST(cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(isNatList(X)) → A__ISNATLIST(X)
A__ISNATLIST(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
A__ISNATLIST(take(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATLIST(take(N, IL)) → A__ISNATILIST(IL)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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

(31) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

A__ISNATLIST(take(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATLIST(take(N, IL)) → A__ISNATILIST(IL)

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

POL( A__AND(x₁, x₂) ) = 2x₂ + 2

POL( mark(x₁) ) = x₁

POL( and(x₁, x₂) ) = 2x₁ + x₂

POL( a__and(x₁, x₂) ) = 2x₁ + x₂

POL( tt ) = 0

POL( isNatIList(x₁) ) = x₁

POL( a__isNatIList(x₁) ) = x₁

POL( a__isNatList(x₁) ) = x₁

POL( cons(x₁, x₂) ) = 2x₁ + x₂

POL( a__isNat(x₁) ) = x₁

POL( take(x₁, x₂) ) = 2x₁ + x₂ + 2

POL( isNatList(x₁) ) = x₁

POL( isNat(x₁) ) = x₁

POL( s(x₁) ) = x₁

POL( length(x₁) ) = x₁ + 2

POL( a__length(x₁) ) = x₁ + 2

POL( zeros ) = 0

POL( a__zeros ) = 0

POL( a__take(x₁, x₂) ) = 2x₁ + x₂ + 2

POL( uTake1(x₁) ) = 0

POL( a__uTake1(x₁) ) = max{0, -2}

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

POL( a__uTake2(x₁, ..., x₄) ) = 2x₂ + 2x₃ + x₄ + 2

POL( uLength(x₁, x₂) ) = x₂ + 2

POL( a__uLength(x₁, x₂) ) = x₂ + 2

POL( 0 ) = 0

POL( nil ) = 0

POL( MARK(x₁) ) = 2x₁ + 2

POL( A__ISNATILIST(x₁) ) = 2x₁ + 2

POL( A__ISNATLIST(x₁) ) = 2x₁ + 2

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

mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
a__and(tt, T) → mark(T)
mark(isNatIList(X)) → a__isNatIList(X)
a__isNatIList(IL) → a__isNatList(IL)
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__isNat(0) → tt
a__isNat(X) → isNat(X)
a__isNatIList(zeros) → tt
a__isNatIList(X) → isNatIList(X)
a__isNatList(nil) → tt
a__isNatList(X) → isNatList(X)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__and(X1, X2) → and(X1, X2)
a__length(X) → length(X)
a__take(X1, X2) → take(X1, X2)
a__uTake1(tt) → nil
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)
a__uLength(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__zeros → cons(0, zeros)
a__zeros → zeros

(32) Obligation:

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

A__AND(tt, T) → MARK(T)
MARK(and(X1, X2)) → A__AND(mark(X1), mark(X2))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X2)
MARK(isNatIList(X)) → A__ISNATILIST(X)
A__ISNATILIST(cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(isNatList(X)) → A__ISNATLIST(X)
A__ISNATLIST(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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

(33) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

MARK(and(X1, X2)) → MARK(X1)

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

POL(A__AND(x₁, x₂)) = 4A + -I · x₁ + 0A · x₂

POL(tt) = 0A

POL(MARK(x₁)) = 4A + 0A · x₁

POL(and(x₁, x₂)) = 5A + 1A · x₁ + 0A · x₂

POL(mark(x₁)) = -I + 0A · x₁

POL(isNatIList(x₁)) = 5A + 0A · x₁

POL(A__ISNATILIST(x₁)) = 5A + 0A · x₁

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

POL(a__isNat(x₁)) = 1A + 0A · x₁

POL(a__isNatIList(x₁)) = 5A + 0A · x₁

POL(isNatList(x₁)) = 5A + 0A · x₁

POL(A__ISNATLIST(x₁)) = 5A + 0A · x₁

POL(a__isNatList(x₁)) = 5A + 0A · x₁

POL(s(x₁)) = 2A + 0A · x₁

POL(a__and(x₁, x₂)) = 5A + 1A · x₁ + 0A · x₂

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

POL(isNat(x₁)) = 1A + 0A · x₁

POL(length(x₁)) = 5A + 1A · x₁

POL(a__length(x₁)) = 5A + 1A · x₁

POL(zeros) = 1A

POL(a__zeros) = 1A

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

POL(uTake1(x₁)) = 1A + -I · x₁

POL(a__uTake1(x₁)) = 1A + -I · x₁

POL(uTake2(x₁, x₂, x₃, x₄)) = 3A + -I · x₁ + 1A · x₂ + 1A · x₃ + 0A · x₄

POL(a__uTake2(x₁, x₂, x₃, x₄)) = 3A + -I · x₁ + 1A · x₂ + 1A · x₃ + 0A · x₄

POL(uLength(x₁, x₂)) = 5A + 0A · x₁ + 1A · x₂

POL(a__uLength(x₁, x₂)) = 5A + 0A · x₁ + 1A · x₂

POL(0) = 0A

POL(nil) = 0A

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

mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
a__and(tt, T) → mark(T)
mark(isNatIList(X)) → a__isNatIList(X)
a__isNatIList(IL) → a__isNatList(IL)
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__isNat(0) → tt
a__isNat(X) → isNat(X)
a__isNatIList(zeros) → tt
a__isNatIList(X) → isNatIList(X)
a__isNatList(nil) → tt
a__isNatList(X) → isNatList(X)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__and(X1, X2) → and(X1, X2)
a__length(X) → length(X)
a__take(X1, X2) → take(X1, X2)
a__uTake1(tt) → nil
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)
a__uLength(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__zeros → cons(0, zeros)
a__zeros → zeros

(34) Obligation:

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

A__AND(tt, T) → MARK(T)
MARK(and(X1, X2)) → A__AND(mark(X1), mark(X2))
MARK(and(X1, X2)) → MARK(X2)
MARK(isNatIList(X)) → A__ISNATILIST(X)
A__ISNATILIST(cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(isNatList(X)) → A__ISNATLIST(X)
A__ISNATLIST(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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

(35) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

MARK(and(X1, X2)) → MARK(X2)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)

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

POL( A__AND(x₁, x₂) ) = x₂ + 2

POL( mark(x₁) ) = x₁ + 1

POL( and(x₁, x₂) ) = x₂ + 1

POL( a__and(x₁, x₂) ) = x₂ + 1

POL( tt ) = 2

POL( isNatIList(x₁) ) = x₁

POL( a__isNatIList(x₁) ) = x₁ + 1

POL( a__isNatList(x₁) ) = x₁ + 1

POL( cons(x₁, x₂) ) = x₂ + 1

POL( a__isNat(x₁) ) = x₁ + 1

POL( take(x₁, x₂) ) = x₂ + 1

POL( isNatList(x₁) ) = x₁

POL( isNat(x₁) ) = x₁

POL( s(x₁) ) = x₁

POL( length(x₁) ) = x₁

POL( a__length(x₁) ) = x₁

POL( zeros ) = 1

POL( a__zeros ) = 2

POL( a__take(x₁, x₂) ) = x₂ + 1

POL( uTake1(x₁) ) = 0

POL( a__uTake1(x₁) ) = 1

POL( uTake2(x₁, ..., x₄) ) = x₄ + 2

POL( a__uTake2(x₁, ..., x₄) ) = x₄ + 2

POL( uLength(x₁, x₂) ) = x₂ + 1

POL( a__uLength(x₁, x₂) ) = x₂ + 1

POL( 0 ) = 2

POL( nil ) = 1

POL( MARK(x₁) ) = x₁ + 2

POL( A__ISNATILIST(x₁) ) = x₁ + 2

POL( A__ISNATLIST(x₁) ) = x₁ + 2

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

mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
a__and(tt, T) → mark(T)
mark(isNatIList(X)) → a__isNatIList(X)
a__isNatIList(IL) → a__isNatList(IL)
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__isNat(0) → tt
a__isNat(X) → isNat(X)
a__isNatIList(zeros) → tt
a__isNatIList(X) → isNatIList(X)
a__isNatList(nil) → tt
a__isNatList(X) → isNatList(X)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__and(X1, X2) → and(X1, X2)
a__length(X) → length(X)
a__take(X1, X2) → take(X1, X2)
a__uTake1(tt) → nil
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)
a__uLength(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__zeros → cons(0, zeros)
a__zeros → zeros

(36) Obligation:

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

A__AND(tt, T) → MARK(T)
MARK(and(X1, X2)) → A__AND(mark(X1), mark(X2))
MARK(isNatIList(X)) → A__ISNATILIST(X)
A__ISNATILIST(cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
MARK(isNatList(X)) → A__ISNATLIST(X)
A__ISNATLIST(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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

(37) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

MARK(and(X1, X2)) → A__AND(mark(X1), mark(X2))
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(isNatList(X)) → A__ISNATLIST(X)

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

POL( A__AND(x₁, x₂) ) = x₂ + 1

POL( mark(x₁) ) = x₁ + 1

POL( and(x₁, x₂) ) = x₂ + 2

POL( a__and(x₁, x₂) ) = x₂ + 2

POL( tt ) = 1

POL( isNatIList(x₁) ) = 2x₁

POL( a__isNatIList(x₁) ) = 2x₁ + 1

POL( a__isNatList(x₁) ) = 2x₁ + 1

POL( cons(x₁, x₂) ) = x₂ + 1

POL( a__isNat(x₁) ) = 2x₁ + 2

POL( take(x₁, x₂) ) = x₂ + 1

POL( isNatList(x₁) ) = 2x₁

POL( isNat(x₁) ) = 2x₁ + 1

POL( s(x₁) ) = x₁

POL( length(x₁) ) = x₁ + 1

POL( a__length(x₁) ) = x₁ + 1

POL( zeros ) = 0

POL( a__zeros ) = 1

POL( a__take(x₁, x₂) ) = x₂ + 1

POL( uTake1(x₁) ) = 0

POL( a__uTake1(x₁) ) = max{0, -2}

POL( uTake2(x₁, ..., x₄) ) = x₄ + 2

POL( a__uTake2(x₁, ..., x₄) ) = x₄ + 2

POL( uLength(x₁, x₂) ) = x₂ + 1

POL( a__uLength(x₁, x₂) ) = x₂ + 2

POL( 0 ) = 0

POL( nil ) = 0

POL( MARK(x₁) ) = x₁ + 1

POL( A__ISNATILIST(x₁) ) = 2x₁

POL( A__ISNATLIST(x₁) ) = 2x₁

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

mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
a__and(tt, T) → mark(T)
mark(isNatIList(X)) → a__isNatIList(X)
a__isNatIList(IL) → a__isNatList(IL)
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__isNat(0) → tt
a__isNat(X) → isNat(X)
a__isNatIList(zeros) → tt
a__isNatIList(X) → isNatIList(X)
a__isNatList(nil) → tt
a__isNatList(X) → isNatList(X)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__and(X1, X2) → and(X1, X2)
a__length(X) → length(X)
a__take(X1, X2) → take(X1, X2)
a__uTake1(tt) → nil
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)
a__uLength(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__zeros → cons(0, zeros)
a__zeros → zeros

(38) Obligation:

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

A__AND(tt, T) → MARK(T)
A__ISNATILIST(cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATLIST(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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

(39) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

(40) Obligation:

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

MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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

(41) UsableRulesProof (EQUIVALENT transformation)

(42) Obligation:

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

MARK(s(X)) → MARK(X)

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

(43) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

MARK(s(X)) → MARK(X)
The graph contains the following edges 1 > 1

(0) Obligation:

(1) DependencyPairsProof (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyGraphProof (EQUIVALENT transformation)

(4) Obligation:

(5) QDPOrderProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Complex Obligation (AND)

(9) Obligation:

(10) QDPOrderProof (EQUIVALENT transformation)

(11) Obligation:

(12) DependencyGraphProof (EQUIVALENT transformation)

(13) TRUE

(14) Obligation:

(15) QDPOrderProof (EQUIVALENT transformation)

(16) Obligation:

(17) DependencyGraphProof (EQUIVALENT transformation)

(18) Obligation:

(19) QDPOrderProof (EQUIVALENT transformation)

(20) Obligation:

(21) QDPOrderProof (EQUIVALENT transformation)

(22) Obligation:

(23) DependencyGraphProof (EQUIVALENT transformation)

(24) Complex Obligation (AND)

(25) Obligation:

(26) UsableRulesProof (EQUIVALENT transformation)

(27) Obligation:

(28) QDPSizeChangeProof (EQUIVALENT transformation)

(29) YES

(30) Obligation:

(31) QDPOrderProof (EQUIVALENT transformation)

(32) Obligation:

(33) QDPOrderProof (EQUIVALENT transformation)

(34) Obligation:

(35) QDPOrderProof (EQUIVALENT transformation)

(36) Obligation:

(37) QDPOrderProof (EQUIVALENT transformation)

(38) Obligation:

(39) DependencyGraphProof (EQUIVALENT transformation)

(40) Obligation:

(41) UsableRulesProof (EQUIVALENT transformation)

(42) Obligation:

(43) QDPSizeChangeProof (EQUIVALENT transformation)

(44) YES