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

0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 DependencyGraphProof (⇔)
↳4 QDP
↳5 QDPOrderProof (⇔)
↳6 QDP
↳7 DependencyGraphProof (⇔)
↳8 QDP
↳9 QDPOrderProof (⇔)
↳10 QDP
↳11 DependencyGraphProof (⇔)
↳12 AND
  ↳13 QDP
    ↳14 QDPOrderProof (⇔)
    ↳15 QDP
    ↳16 QDPOrderProof (⇔)
    ↳17 QDP
    ↳18 QDPOrderProof (⇔)
    ↳19 QDP
    ↳20 DependencyGraphProof (⇔)
    ↳21 AND
      ↳22 QDP
        ↳23 UsableRulesProof (⇔)
        ↳24 QDP
        ↳25 QDPSizeChangeProof (⇔)
        ↳26 TRUE
      ↳27 QDP
        ↳28 QDPOrderProof (⇔)
        ↳29 QDP
        ↳30 QDPOrderProof (⇔)
        ↳31 QDP
        ↳32 QDPOrderProof (⇔)
        ↳33 QDP
        ↳34 QDPOrderProof (⇔)
        ↳35 QDP
        ↳36 QDPOrderProof (⇔)
        ↳37 QDP
        ↳38 DependencyGraphProof (⇔)
        ↳39 QDP
        ↳40 QDPOrderProof (⇔)
        ↳41 QDP
        ↳42 DependencyGraphProof (⇔)
        ↳43 QDP
        ↳44 QDPOrderProof (⇔)
        ↳45 QDP
        ↳46 QDPOrderProof (⇔)
        ↳47 QDP
        ↳48 DependencyGraphProof (⇔)
        ↳49 TRUE
  ↳50 QDP
    ↳51 QDPOrderProof (⇔)
    ↳52 QDP
    ↳53 DependencyGraphProof (⇔)
    ↳54 TRUE

(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].


The following pairs can be oriented strictly and are deleted.


MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
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 interpretation [POLO]:

POL(0) = 0   
POL(A__AND(x1, x2)) = x2   
POL(A__ISNAT(x1)) = 0   
POL(A__ISNATILIST(x1)) = 0   
POL(A__ISNATLIST(x1)) = 0   
POL(A__LENGTH(x1)) = x1   
POL(A__TAKE(x1, x2)) = x2   
POL(A__ULENGTH(x1, x2)) = x2   
POL(A__UTAKE2(x1, x2, x3, x4)) = x3 + x4   
POL(MARK(x1)) = x1   
POL(a__and(x1, x2)) = x1 + x2   
POL(a__isNat(x1)) = 0   
POL(a__isNatIList(x1)) = 0   
POL(a__isNatList(x1)) = 0   
POL(a__length(x1)) = x1   
POL(a__take(x1, x2)) = 1 + x1 + x2   
POL(a__uLength(x1, x2)) = x1 + x2   
POL(a__uTake1(x1)) = x1   
POL(a__uTake2(x1, x2, x3, x4)) = 1 + x1 + x2 + x3 + x4   
POL(a__zeros) = 0   
POL(and(x1, x2)) = x1 + x2   
POL(cons(x1, x2)) = x1 + x2   
POL(isNat(x1)) = 0   
POL(isNatIList(x1)) = 0   
POL(isNatList(x1)) = 0   
POL(length(x1)) = x1   
POL(mark(x1)) = x1   
POL(nil) = 0   
POL(s(x1)) = x1   
POL(take(x1, x2)) = 1 + x1 + x2   
POL(tt) = 0   
POL(uLength(x1, x2)) = x1 + x2   
POL(uTake1(x1)) = x1   
POL(uTake2(x1, x2, x3, x4)) = 1 + x1 + x2 + x3 + x4   
POL(zeros) = 0   

The following usable rules [FROCOS05] were oriented:

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

(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))
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)
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(uTake1(X)) → MARK(X)
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.

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 8 less nodes.

(8) 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(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(uTake1(X)) → MARK(X)
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.

(9) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(length(X)) → A__LENGTH(mark(X))
MARK(length(X)) → MARK(X)
MARK(uLength(X1, X2)) → A__ULENGTH(mark(X1), X2)
MARK(uLength(X1, X2)) → MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(A__AND(x1, x2)) = x2   
POL(A__ISNAT(x1)) = 0   
POL(A__ISNATILIST(x1)) = 0   
POL(A__ISNATLIST(x1)) = 0   
POL(A__LENGTH(x1)) = x1   
POL(A__ULENGTH(x1, x2)) = x2   
POL(MARK(x1)) = x1   
POL(a__and(x1, x2)) = x1 + x2   
POL(a__isNat(x1)) = 0   
POL(a__isNatIList(x1)) = 0   
POL(a__isNatList(x1)) = 0   
POL(a__length(x1)) = 1 + x1   
POL(a__take(x1, x2)) = x2   
POL(a__uLength(x1, x2)) = 1 + x1 + x2   
POL(a__uTake1(x1)) = x1   
POL(a__uTake2(x1, x2, x3, x4)) = x3 + x4   
POL(a__zeros) = 0   
POL(and(x1, x2)) = x1 + x2   
POL(cons(x1, x2)) = x1 + x2   
POL(isNat(x1)) = 0   
POL(isNatIList(x1)) = 0   
POL(isNatList(x1)) = 0   
POL(length(x1)) = 1 + x1   
POL(mark(x1)) = x1   
POL(nil) = 0   
POL(s(x1)) = x1   
POL(take(x1, x2)) = x2   
POL(tt) = 0   
POL(uLength(x1, x2)) = 1 + x1 + x2   
POL(uTake1(x1)) = x1   
POL(uTake2(x1, x2, x3, x4)) = x3 + x4   
POL(zeros) = 0   

The following usable rules [FROCOS05] were oriented:

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

(10) 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__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(uTake1(X)) → MARK(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.

(11) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 4 less nodes.

(12) Complex Obligation (AND)

(13) 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(uTake1(X)) → MARK(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.

(14) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


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 interpretation [POLO]:

POL(0) = 0   
POL(A__AND(x1, x2)) = x2   
POL(A__ISNAT(x1)) = 0   
POL(A__ISNATILIST(x1)) = 0   
POL(A__ISNATLIST(x1)) = 0   
POL(MARK(x1)) = x1   
POL(a__and(x1, x2)) = x1 + x2   
POL(a__isNat(x1)) = 0   
POL(a__isNatIList(x1)) = 0   
POL(a__isNatList(x1)) = 0   
POL(a__length(x1)) = 0   
POL(a__take(x1, x2)) = x1 + x2   
POL(a__uLength(x1, x2)) = 0   
POL(a__uTake1(x1)) = x1   
POL(a__uTake2(x1, x2, x3, x4)) = 1 + x2 + x3   
POL(a__zeros) = 1   
POL(and(x1, x2)) = x1 + x2   
POL(cons(x1, x2)) = 1 + x1   
POL(isNat(x1)) = 0   
POL(isNatIList(x1)) = 0   
POL(isNatList(x1)) = 0   
POL(length(x1)) = 0   
POL(mark(x1)) = x1   
POL(nil) = 0   
POL(s(x1)) = x1   
POL(take(x1, x2)) = x1 + x2   
POL(tt) = 0   
POL(uLength(x1, x2)) = 0   
POL(uTake1(x1)) = x1   
POL(uTake2(x1, x2, x3, x4)) = 1 + x2 + x3   
POL(zeros) = 1   

The following usable rules [FROCOS05] were oriented:

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

(15) 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(uTake1(X)) → MARK(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.

(16) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


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__ISNAT(N)
A__ISNATLIST(take(N, IL)) → A__ISNATILIST(IL)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(A__AND(x1, x2)) = x2   
POL(A__ISNAT(x1)) = x1   
POL(A__ISNATILIST(x1)) = x1   
POL(A__ISNATLIST(x1)) = x1   
POL(MARK(x1)) = x1   
POL(a__and(x1, x2)) = x1 + x2   
POL(a__isNat(x1)) = x1   
POL(a__isNatIList(x1)) = x1   
POL(a__isNatList(x1)) = x1   
POL(a__length(x1)) = x1   
POL(a__take(x1, x2)) = 1 + x1 + x2   
POL(a__uLength(x1, x2)) = x2   
POL(a__uTake1(x1)) = x1   
POL(a__uTake2(x1, x2, x3, x4)) = 1 + x2 + x3 + x4   
POL(a__zeros) = 0   
POL(and(x1, x2)) = x1 + x2   
POL(cons(x1, x2)) = x1 + x2   
POL(isNat(x1)) = x1   
POL(isNatIList(x1)) = x1   
POL(isNatList(x1)) = x1   
POL(length(x1)) = x1   
POL(mark(x1)) = x1   
POL(nil) = 0   
POL(s(x1)) = x1   
POL(take(x1, x2)) = 1 + x1 + x2   
POL(tt) = 0   
POL(uLength(x1, x2)) = x2   
POL(uTake1(x1)) = x1   
POL(uTake2(x1, x2, x3, x4)) = 1 + x2 + x3 + x4   
POL(zeros) = 0   

The following usable rules [FROCOS05] were oriented:

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

(17) 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__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(uTake1(X)) → MARK(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.

(18) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


A__ISNAT(length(L)) → A__ISNATLIST(L)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(A__AND(x1, x2)) = x2   
POL(A__ISNAT(x1)) = x1   
POL(A__ISNATILIST(x1)) = x1   
POL(A__ISNATLIST(x1)) = x1   
POL(MARK(x1)) = x1   
POL(a__and(x1, x2)) = x1 + x2   
POL(a__isNat(x1)) = x1   
POL(a__isNatIList(x1)) = x1   
POL(a__isNatList(x1)) = x1   
POL(a__length(x1)) = 1 + x1   
POL(a__take(x1, x2)) = x1 + x2   
POL(a__uLength(x1, x2)) = 1 + x2   
POL(a__uTake1(x1)) = x1   
POL(a__uTake2(x1, x2, x3, x4)) = x2 + x3 + x4   
POL(a__zeros) = 0   
POL(and(x1, x2)) = x1 + x2   
POL(cons(x1, x2)) = x1 + x2   
POL(isNat(x1)) = x1   
POL(isNatIList(x1)) = x1   
POL(isNatList(x1)) = x1   
POL(length(x1)) = 1 + x1   
POL(mark(x1)) = x1   
POL(nil) = 0   
POL(s(x1)) = x1   
POL(take(x1, x2)) = x1 + x2   
POL(tt) = 0   
POL(uLength(x1, x2)) = 1 + x2   
POL(uTake1(x1)) = x1   
POL(uTake2(x1, x2, x3, x4)) = x2 + x3 + x4   
POL(zeros) = 0   

The following usable rules [FROCOS05] were oriented:

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

(19) 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__ISNATLIST(cons(N, 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)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(isNat(X)) → A__ISNAT(X)
MARK(uTake1(X)) → MARK(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.

(20) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 3 less nodes.

(21) Complex Obligation (AND)

(22) 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.

(23) 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.

(24) 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.

(25) 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

(26) TRUE

(27) 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__ISNATLIST(L)
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(uTake1(X)) → MARK(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.

(28) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(uTake1(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(A__AND(x1, x2)) = x2   
POL(A__ISNATILIST(x1)) = 0   
POL(A__ISNATLIST(x1)) = 0   
POL(MARK(x1)) = x1   
POL(a__and(x1, x2)) = x1 + x2   
POL(a__isNat(x1)) = 0   
POL(a__isNatIList(x1)) = 0   
POL(a__isNatList(x1)) = 0   
POL(a__length(x1)) = 0   
POL(a__take(x1, x2)) = 1 + x1   
POL(a__uLength(x1, x2)) = 0   
POL(a__uTake1(x1)) = 1 + x1   
POL(a__uTake2(x1, x2, x3, x4)) = x2   
POL(a__zeros) = 0   
POL(and(x1, x2)) = x1 + x2   
POL(cons(x1, x2)) = 0   
POL(isNat(x1)) = 0   
POL(isNatIList(x1)) = 0   
POL(isNatList(x1)) = 0   
POL(length(x1)) = 0   
POL(mark(x1)) = x1   
POL(nil) = 0   
POL(s(x1)) = x1   
POL(take(x1, x2)) = 1 + x1   
POL(tt) = 0   
POL(uLength(x1, x2)) = 0   
POL(uTake1(x1)) = 1 + x1   
POL(uTake2(x1, x2, x3, x4)) = x2   
POL(zeros) = 0   

The following usable rules [FROCOS05] were oriented:

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

(29) 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__ISNATLIST(L)
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(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.

(30) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


A__ISNATILIST(IL) → A__ISNATLIST(IL)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(A__AND(x1, x2)) = x2   
POL(A__ISNATILIST(x1)) = 1 + x1   
POL(A__ISNATLIST(x1)) = x1   
POL(MARK(x1)) = x1   
POL(a__and(x1, x2)) = x1 + x2   
POL(a__isNat(x1)) = x1   
POL(a__isNatIList(x1)) = 1 + x1   
POL(a__isNatList(x1)) = x1   
POL(a__length(x1)) = x1   
POL(a__take(x1, x2)) = 1 + x1 + x2   
POL(a__uLength(x1, x2)) = x2   
POL(a__uTake1(x1)) = 1   
POL(a__uTake2(x1, x2, x3, x4)) = 1 + x2 + x3 + x4   
POL(a__zeros) = 0   
POL(and(x1, x2)) = x1 + x2   
POL(cons(x1, x2)) = x1 + x2   
POL(isNat(x1)) = x1   
POL(isNatIList(x1)) = 1 + x1   
POL(isNatList(x1)) = x1   
POL(length(x1)) = x1   
POL(mark(x1)) = x1   
POL(nil) = 0   
POL(s(x1)) = x1   
POL(take(x1, x2)) = 1 + x1 + x2   
POL(tt) = 0   
POL(uLength(x1, x2)) = x2   
POL(uTake1(x1)) = 1   
POL(uTake2(x1, x2, x3, x4)) = 1 + x2 + x3 + x4   
POL(zeros) = 0   

The following usable rules [FROCOS05] were oriented:

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

(31) 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__ISNATLIST(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
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(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.

(32) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(s(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(A__AND(x1, x2)) =
/0\
\0/
 +
/00\
\00/
·x1 +
/10\
\00/
·x2

POL(tt) =
/0\
\1/

POL(MARK(x1)) =
/0\
\0/
 +
/10\
\00/
·x1

POL(and(x1, x2)) =
/0\
\0/
 +
/10\
\00/
·x1 +
/10\
\01/
·x2

POL(mark(x1)) =
/0\
\0/
 +
/10\
\01/
·x1

POL(isNatIList(x1)) =
/0\
\1/
 +
/00\
\10/
·x1

POL(A__ISNATILIST(x1)) =
/0\
\0/
 +
/00\
\00/
·x1

POL(A__ISNATLIST(x1)) =
/0\
\0/
 +
/00\
\00/
·x1

POL(cons(x1, x2)) =
/0\
\0/
 +
/00\
\00/
·x1 +
/10\
\11/
·x2

POL(a__isNat(x1)) =
/0\
\1/
 +
/00\
\01/
·x1

POL(a__isNatList(x1)) =
/0\
\0/
 +
/00\
\10/
·x1

POL(a__isNatIList(x1)) =
/0\
\1/
 +
/00\
\10/
·x1

POL(isNatList(x1)) =
/0\
\0/
 +
/00\
\10/
·x1

POL(s(x1)) =
/1\
\0/
 +
/10\
\01/
·x1

POL(a__zeros) =
/0\
\0/

POL(zeros) =
/0\
\0/

POL(0) =
/0\
\0/

POL(a__length(x1)) =
/0\
\0/
 +
/01\
\10/
·x1

POL(length(x1)) =
/0\
\0/
 +
/01\
\10/
·x1

POL(nil) =
/1\
\0/

POL(a__uTake1(x1)) =
/0\
\0/
 +
/01\
\00/
·x1

POL(uTake1(x1)) =
/0\
\0/
 +
/01\
\00/
·x1

POL(a__take(x1, x2)) =
/1\
\0/
 +
/00\
\10/
·x1 +
/10\
\01/
·x2

POL(take(x1, x2)) =
/1\
\0/
 +
/00\
\10/
·x1 +
/10\
\01/
·x2

POL(uTake2(x1, x2, x3, x4)) =
/1\
\1/
 +
/10\
\00/
·x1 +
/00\
\10/
·x2 +
/00\
\00/
·x3 +
/10\
\11/
·x4

POL(a__uTake2(x1, x2, x3, x4)) =
/1\
\1/
 +
/10\
\00/
·x1 +
/00\
\10/
·x2 +
/00\
\00/
·x3 +
/10\
\11/
·x4

POL(a__and(x1, x2)) =
/0\
\0/
 +
/10\
\00/
·x1 +
/10\
\01/
·x2

POL(isNat(x1)) =
/0\
\1/
 +
/00\
\01/
·x1

POL(uLength(x1, x2)) =
/0\
\0/
 +
/01\
\00/
·x1 +
/01\
\10/
·x2

POL(a__uLength(x1, x2)) =
/0\
\0/
 +
/01\
\00/
·x1 +
/01\
\10/
·x2

The following usable rules [FROCOS05] were oriented:

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

(33) 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__ISNATLIST(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
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)

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.

(34) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


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)) = -I + 0A·x1 + 1A·x2

POL(tt) = 0A

POL(MARK(x1)) = -I + 1A·x1

POL(and(x1, x2)) = -I + 1A·x1 + 0A·x2

POL(mark(x1)) = -I + 0A·x1

POL(isNatIList(x1)) = 0A + 1A·x1

POL(A__ISNATILIST(x1)) = 1A + 2A·x1

POL(A__ISNATLIST(x1)) = -I + 2A·x1

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

POL(a__isNat(x1)) = -I + 0A·x1

POL(a__isNatList(x1)) = -I + 1A·x1

POL(a__isNatIList(x1)) = 0A + 1A·x1

POL(isNatList(x1)) = -I + 1A·x1

POL(a__zeros) = 3A

POL(zeros) = 3A

POL(s(x1)) = -I + 0A·x1

POL(0) = 1A

POL(a__length(x1)) = 0A + 1A·x1

POL(length(x1)) = 0A + 1A·x1

POL(nil) = 2A

POL(a__uTake1(x1)) = 2A + -I·x1

POL(uTake1(x1)) = 2A + -I·x1

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

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

POL(uTake2(x1, x2, x3, x4)) = 0A + -I·x1 + 5A·x2 + 1A·x3 + 0A·x4

POL(a__uTake2(x1, x2, x3, x4)) = 0A + -I·x1 + 5A·x2 + 1A·x3 + 0A·x4

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

POL(isNat(x1)) = -I + 0A·x1

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

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

The following usable rules [FROCOS05] were oriented:

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

(35) 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__ISNATLIST(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
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)

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.

(36) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


A__ISNATLIST(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(A__AND(x1, x2)) =
/0\
\0/
 +
/00\
\00/
·x1 +
/01\
\00/
·x2

POL(tt) =
/1\
\0/

POL(MARK(x1)) =
/0\
\0/
 +
/01\
\00/
·x1

POL(and(x1, x2)) =
/1\
\0/
 +
/00\
\00/
·x1 +
/00\
\11/
·x2

POL(mark(x1)) =
/1\
\0/
 +
/00\
\11/
·x1

POL(isNatIList(x1)) =
/0\
\0/
 +
/00\
\11/
·x1

POL(A__ISNATILIST(x1)) =
/0\
\0/
 +
/01\
\00/
·x1

POL(A__ISNATLIST(x1)) =
/0\
\0/
 +
/11\
\00/
·x1

POL(cons(x1, x2)) =
/1\
\0/
 +
/00\
\00/
·x1 +
/00\
\11/
·x2

POL(a__isNat(x1)) =
/1\
\0/
 +
/00\
\01/
·x1

POL(a__isNatList(x1)) =
/1\
\0/
 +
/00\
\11/
·x1

POL(a__isNatIList(x1)) =
/1\
\0/
 +
/00\
\11/
·x1

POL(isNatList(x1)) =
/0\
\0/
 +
/00\
\11/
·x1

POL(a__zeros) =
/1\
\0/

POL(zeros) =
/0\
\0/

POL(s(x1)) =
/0\
\0/
 +
/10\
\01/
·x1

POL(0) =
/0\
\0/

POL(a__length(x1)) =
/1\
\0/
 +
/00\
\11/
·x1

POL(length(x1)) =
/1\
\0/
 +
/00\
\11/
·x1

POL(nil) =
/0\
\0/

POL(a__uTake1(x1)) =
/0\
\0/
 +
/00\
\00/
·x1

POL(uTake1(x1)) =
/0\
\0/
 +
/00\
\00/
·x1

POL(a__take(x1, x2)) =
/0\
\1/
 +
/00\
\00/
·x1 +
/10\
\01/
·x2

POL(take(x1, x2)) =
/0\
\1/
 +
/00\
\00/
·x1 +
/10\
\01/
·x2

POL(uTake2(x1, x2, x3, x4)) =
/1\
\0/
 +
/00\
\00/
·x1 +
/00\
\00/
·x2 +
/00\
\00/
·x3 +
/00\
\11/
·x4

POL(a__uTake2(x1, x2, x3, x4)) =
/1\
\0/
 +
/00\
\10/
·x1 +
/00\
\00/
·x2 +
/00\
\00/
·x3 +
/00\
\11/
·x4

POL(a__and(x1, x2)) =
/1\
\0/
 +
/00\
\00/
·x1 +
/00\
\11/
·x2

POL(isNat(x1)) =
/0\
\0/
 +
/00\
\01/
·x1

POL(uLength(x1, x2)) =
/1\
\0/
 +
/00\
\00/
·x1 +
/00\
\11/
·x2

POL(a__uLength(x1, x2)) =
/1\
\0/
 +
/00\
\10/
·x1 +
/00\
\11/
·x2

The following usable rules [FROCOS05] were oriented:

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

(37) 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)

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.

(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:

MARK(and(X1, X2)) → A__AND(mark(X1), mark(X2))
A__AND(tt, T) → MARK(T)
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)

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.

(40) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


A__ISNATILIST(cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(MARK(x1)) =
/0\
\0/
 +
/10\
\00/
·x1

POL(and(x1, x2)) =
/0\
\0/
 +
/00\
\00/
·x1 +
/10\
\00/
·x2

POL(A__AND(x1, x2)) =
/0\
\0/
 +
/00\
\00/
·x1 +
/10\
\00/
·x2

POL(mark(x1)) =
/0\
\1/
 +
/10\
\00/
·x1

POL(tt) =
/0\
\1/

POL(isNatIList(x1)) =
/0\
\0/
 +
/11\
\00/
·x1

POL(A__ISNATILIST(x1)) =
/0\
\0/
 +
/11\
\00/
·x1

POL(cons(x1, x2)) =
/0\
\1/
 +
/00\
\00/
·x1 +
/11\
\00/
·x2

POL(a__isNat(x1)) =
/0\
\1/
 +
/11\
\00/
·x1

POL(a__isNatIList(x1)) =
/0\
\1/
 +
/11\
\00/
·x1

POL(a__zeros) =
/0\
\1/

POL(zeros) =
/0\
\0/

POL(s(x1)) =
/0\
\0/
 +
/10\
\01/
·x1

POL(0) =
/0\
\0/

POL(a__length(x1)) =
/0\
\0/
 +
/10\
\01/
·x1

POL(length(x1)) =
/0\
\0/
 +
/10\
\01/
·x1

POL(nil) =
/0\
\0/

POL(a__uTake1(x1)) =
/0\
\0/
 +
/00\
\00/
·x1

POL(uTake1(x1)) =
/0\
\0/
 +
/00\
\00/
·x1

POL(a__take(x1, x2)) =
/0\
\0/
 +
/00\
\00/
·x1 +
/10\
\01/
·x2

POL(take(x1, x2)) =
/0\
\0/
 +
/00\
\00/
·x1 +
/10\
\01/
·x2

POL(uTake2(x1, x2, x3, x4)) =
/0\
\0/
 +
/00\
\00/
·x1 +
/00\
\00/
·x2 +
/00\
\00/
·x3 +
/11\
\00/
·x4

POL(a__uTake2(x1, x2, x3, x4)) =
/0\
\1/
 +
/00\
\00/
·x1 +
/00\
\00/
·x2 +
/00\
\00/
·x3 +
/11\
\00/
·x4

POL(a__and(x1, x2)) =
/0\
\0/
 +
/00\
\01/
·x1 +
/10\
\00/
·x2

POL(isNat(x1)) =
/0\
\1/
 +
/11\
\00/
·x1

POL(uLength(x1, x2)) =
/0\
\0/
 +
/00\
\00/
·x1 +
/10\
\00/
·x2

POL(a__uLength(x1, x2)) =
/0\
\1/
 +
/00\
\00/
·x1 +
/10\
\00/
·x2

POL(a__isNatList(x1)) =
/0\
\1/
 +
/11\
\00/
·x1

POL(isNatList(x1)) =
/0\
\1/
 +
/11\
\00/
·x1

The following usable rules [FROCOS05] were oriented:

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

(41) 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(X2)
MARK(isNatIList(X)) → A__ISNATILIST(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.

(42) DependencyGraphProof (EQUIVALENT transformation)

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

(43) 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)

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.

(44) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(and(X1, X2)) → MARK(X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(A__AND(x1, x2)) =
/0\
\0/
 +
/00\
\00/
·x1 +
/11\
\11/
·x2

POL(tt) =
/0\
\0/

POL(MARK(x1)) =
/0\
\0/
 +
/11\
\11/
·x1

POL(and(x1, x2)) =
/0\
\1/
 +
/00\
\00/
·x1 +
/11\
\00/
·x2

POL(mark(x1)) =
/0\
\1/
 +
/11\
\00/
·x1

POL(a__zeros) =
/0\
\1/

POL(zeros) =
/0\
\0/

POL(s(x1)) =
/0\
\0/
 +
/10\
\01/
·x1

POL(0) =
/0\
\0/

POL(a__length(x1)) =
/0\
\0/
 +
/10\
\01/
·x1

POL(length(x1)) =
/0\
\0/
 +
/10\
\01/
·x1

POL(nil) =
/0\
\0/

POL(a__uTake1(x1)) =
/0\
\0/
 +
/00\
\00/
·x1

POL(uTake1(x1)) =
/0\
\0/
 +
/00\
\00/
·x1

POL(cons(x1, x2)) =
/0\
\1/
 +
/00\
\00/
·x1 +
/11\
\00/
·x2

POL(a__take(x1, x2)) =
/1\
\0/
 +
/00\
\00/
·x1 +
/10\
\01/
·x2

POL(take(x1, x2)) =
/1\
\0/
 +
/00\
\00/
·x1 +
/10\
\01/
·x2

POL(uTake2(x1, x2, x3, x4)) =
/1\
\1/
 +
/00\
\00/
·x1 +
/00\
\00/
·x2 +
/00\
\00/
·x3 +
/11\
\00/
·x4

POL(a__uTake2(x1, x2, x3, x4)) =
/1\
\1/
 +
/00\
\00/
·x1 +
/00\
\00/
·x2 +
/00\
\00/
·x3 +
/11\
\00/
·x4

POL(a__isNatIList(x1)) =
/0\
\1/
 +
/11\
\00/
·x1

POL(isNatIList(x1)) =
/0\
\0/
 +
/11\
\00/
·x1

POL(a__and(x1, x2)) =
/0\
\1/
 +
/00\
\00/
·x1 +
/11\
\00/
·x2

POL(a__isNat(x1)) =
/0\
\1/
 +
/11\
\00/
·x1

POL(isNat(x1)) =
/0\
\0/
 +
/11\
\00/
·x1

POL(uLength(x1, x2)) =
/0\
\1/
 +
/00\
\00/
·x1 +
/11\
\00/
·x2

POL(a__uLength(x1, x2)) =
/0\
\1/
 +
/00\
\00/
·x1 +
/11\
\00/
·x2

POL(a__isNatList(x1)) =
/0\
\1/
 +
/11\
\00/
·x1

POL(isNatList(x1)) =
/0\
\0/
 +
/11\
\00/
·x1

The following usable rules [FROCOS05] were oriented:

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

(45) 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))

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.

(46) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


A__AND(tt, T) → MARK(T)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(A__AND(x1, x2)) =
/1\
\1/
 +
/00\
\00/
·x1 +
/11\
\10/
·x2

POL(tt) =
/0\
\0/

POL(MARK(x1)) =
/0\
\1/
 +
/11\
\10/
·x1

POL(and(x1, x2)) =
/1\
\1/
 +
/00\
\00/
·x1 +
/00\
\01/
·x2

POL(mark(x1)) =
/1\
\0/
 +
/00\
\01/
·x1

POL(a__zeros) =
/1\
\0/

POL(zeros) =
/0\
\0/

POL(s(x1)) =
/0\
\0/
 +
/10\
\01/
·x1

POL(0) =
/0\
\0/

POL(a__length(x1)) =
/0\
\0/
 +
/10\
\01/
·x1

POL(length(x1)) =
/0\
\0/
 +
/10\
\01/
·x1

POL(nil) =
/0\
\0/

POL(a__uTake1(x1)) =
/0\
\0/
 +
/00\
\00/
·x1

POL(uTake1(x1)) =
/0\
\0/
 +
/00\
\00/
·x1

POL(cons(x1, x2)) =
/1\
\0/
 +
/00\
\00/
·x1 +
/00\
\11/
·x2

POL(a__take(x1, x2)) =
/0\
\1/
 +
/00\
\00/
·x1 +
/10\
\01/
·x2

POL(take(x1, x2)) =
/0\
\1/
 +
/00\
\00/
·x1 +
/10\
\01/
·x2

POL(uTake2(x1, x2, x3, x4)) =
/1\
\1/
 +
/00\
\00/
·x1 +
/00\
\00/
·x2 +
/00\
\00/
·x3 +
/00\
\11/
·x4

POL(a__uTake2(x1, x2, x3, x4)) =
/1\
\1/
 +
/00\
\00/
·x1 +
/00\
\00/
·x2 +
/00\
\00/
·x3 +
/00\
\11/
·x4

POL(a__isNatIList(x1)) =
/1\
\0/
 +
/00\
\11/
·x1

POL(isNatIList(x1)) =
/0\
\0/
 +
/00\
\11/
·x1

POL(a__and(x1, x2)) =
/1\
\1/
 +
/00\
\00/
·x1 +
/00\
\01/
·x2

POL(a__isNat(x1)) =
/1\
\0/
 +
/00\
\11/
·x1

POL(isNat(x1)) =
/1\
\0/
 +
/00\
\11/
·x1

POL(uLength(x1, x2)) =
/0\
\0/
 +
/00\
\00/
·x1 +
/00\
\01/
·x2

POL(a__uLength(x1, x2)) =
/1\
\0/
 +
/00\
\00/
·x1 +
/00\
\01/
·x2

POL(a__isNatList(x1)) =
/1\
\0/
 +
/00\
\11/
·x1

POL(isNatList(x1)) =
/0\
\0/
 +
/00\
\11/
·x1

The following usable rules [FROCOS05] were oriented:

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

(47) Obligation:

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

MARK(and(X1, X2)) → A__AND(mark(X1), mark(X2))

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.

(48) DependencyGraphProof (EQUIVALENT transformation)

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

(49) TRUE

(50) 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.

(51) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


A__LENGTH(cons(N, L)) → A__ULENGTH(a__and(a__isNat(N), a__isNatList(L)), L)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(A__ULENGTH(x1, x2)) =
/0\
\1/
 +
/10\
\00/
·x1 +
/10\
\10/
·x2

POL(tt) =
/1\
\1/

POL(A__LENGTH(x1)) =
/1\
\1/
 +
/10\
\10/
·x1

POL(mark(x1)) =
/0\
\0/
 +
/10\
\01/
·x1

POL(cons(x1, x2)) =
/0\
\0/
 +
/00\
\00/
·x1 +
/11\
\01/
·x2

POL(a__and(x1, x2)) =
/0\
\0/
 +
/00\
\00/
·x1 +
/10\
\01/
·x2

POL(a__isNat(x1)) =
/1\
\1/
 +
/01\
\00/
·x1

POL(a__isNatList(x1)) =
/0\
\1/
 +
/01\
\00/
·x1

POL(a__zeros) =
/0\
\0/

POL(zeros) =
/0\
\0/

POL(s(x1)) =
/0\
\1/
 +
/00\
\01/
·x1

POL(0) =
/0\
\0/

POL(a__length(x1)) =
/0\
\0/
 +
/00\
\11/
·x1

POL(length(x1)) =
/0\
\0/
 +
/00\
\11/
·x1

POL(nil) =
/0\
\1/

POL(a__uTake1(x1)) =
/0\
\0/
 +
/00\
\10/
·x1

POL(uTake1(x1)) =
/0\
\0/
 +
/00\
\10/
·x1

POL(a__take(x1, x2)) =
/0\
\1/
 +
/01\
\00/
·x1 +
/10\
\01/
·x2

POL(take(x1, x2)) =
/0\
\1/
 +
/01\
\00/
·x1 +
/10\
\01/
·x2

POL(uTake2(x1, x2, x3, x4)) =
/0\
\1/
 +
/01\
\00/
·x1 +
/01\
\00/
·x2 +
/00\
\00/
·x3 +
/11\
\01/
·x4

POL(a__uTake2(x1, x2, x3, x4)) =
/0\
\1/
 +
/01\
\00/
·x1 +
/01\
\00/
·x2 +
/00\
\00/
·x3 +
/11\
\01/
·x4

POL(a__isNatIList(x1)) =
/1\
\1/
 +
/01\
\00/
·x1

POL(isNatIList(x1)) =
/1\
\1/
 +
/01\
\00/
·x1

POL(and(x1, x2)) =
/0\
\0/
 +
/00\
\00/
·x1 +
/10\
\01/
·x2

POL(isNat(x1)) =
/1\
\1/
 +
/01\
\00/
·x1

POL(uLength(x1, x2)) =
/0\
\0/
 +
/00\
\10/
·x1 +
/00\
\11/
·x2

POL(a__uLength(x1, x2)) =
/0\
\0/
 +
/00\
\10/
·x1 +
/00\
\11/
·x2

POL(isNatList(x1)) =
/0\
\1/
 +
/01\
\00/
·x1

The following usable rules [FROCOS05] were oriented:

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

(52) Obligation:

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

A__ULENGTH(tt, L) → A__LENGTH(mark(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.

(53) DependencyGraphProof (EQUIVALENT transformation)

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

(54) TRUE