Termination w.r.t. Q proof of /tmp/tmpBL21i7/Ex4_DLMMU04

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

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 AND

  ↳5 QDP

    ↳6 UsableRulesProof (⇔)

    ↳7 QDP

    ↳8 QDPSizeChangeProof (⇔)

    ↳9 TRUE

  ↳10 QDP

    ↳11 UsableRulesProof (⇔)

    ↳12 QDP

    ↳13 QDPSizeChangeProof (⇔)

    ↳14 TRUE

  ↳15 QDP

    ↳16 UsableRulesProof (⇔)

    ↳17 QDP

    ↳18 QDPSizeChangeProof (⇔)

    ↳19 TRUE

  ↳20 QDP

    ↳21 UsableRulesProof (⇔)

    ↳22 QDP

    ↳23 QDPSizeChangeProof (⇔)

    ↳24 TRUE

  ↳25 QDP

    ↳26 UsableRulesProof (⇔)

    ↳27 QDP

    ↳28 QDPSizeChangeProof (⇔)

    ↳29 TRUE

  ↳30 QDP

    ↳31 UsableRulesProof (⇔)

    ↳32 QDP

    ↳33 QDPSizeChangeProof (⇔)

    ↳34 TRUE

  ↳35 QDP

    ↳36 UsableRulesProof (⇔)

    ↳37 QDP

    ↳38 QDPSizeChangeProof (⇔)

    ↳39 TRUE

  ↳40 QDP

    ↳41 UsableRulesProof (⇔)

    ↳42 QDP

    ↳43 QDPSizeChangeProof (⇔)

    ↳44 TRUE

  ↳45 QDP

    ↳46 UsableRulesProof (⇔)

    ↳47 QDP

    ↳48 QDPSizeChangeProof (⇔)

    ↳49 TRUE

  ↳50 QDP

    ↳51 UsableRulesProof (⇔)

    ↳52 QDP

    ↳53 QDPSizeChangeProof (⇔)

    ↳54 TRUE

  ↳55 QDP

    ↳56 UsableRulesProof (⇔)

    ↳57 QDP

    ↳58 QDPSizeChangeProof (⇔)

    ↳59 TRUE

  ↳60 QDP

    ↳61 QDPOrderProof (⇔)

    ↳62 QDP

    ↳63 QDPOrderProof (⇔)

    ↳64 QDP

    ↳65 QDPOrderProof (⇔)

    ↳66 QDP

    ↳67 QDPOrderProof (⇔)

    ↳68 QDP

    ↳69 DependencyGraphProof (⇔)

    ↳70 QDP

    ↳71 QDPOrderProof (⇔)

    ↳72 QDP

    ↳73 QDPOrderProof (⇔)

    ↳74 QDP

    ↳75 QDPOrderProof (⇔)

    ↳76 QDP

    ↳77 QDPOrderProof (⇔)

    ↳78 QDP

    ↳79 QDPOrderProof (⇔)

    ↳80 QDP

    ↳81 QDPOrderProof (⇔)

    ↳82 QDP

    ↳83 QDPOrderProof (⇔)

    ↳84 QDP

    ↳85 QDPOrderProof (⇔)

    ↳86 QDP

    ↳87 QDPOrderProof (⇔)

    ↳88 QDP

    ↳89 QDPOrderProof (⇔)

    ↳90 QDP

    ↳91 QDPOrderProof (⇔)

    ↳92 QDP

    ↳93 QDPOrderProof (⇔)

    ↳94 QDP

    ↳95 QDPOrderProof (⇔)

    ↳96 QDP

    ↳97 QDPOrderProof (⇔)

    ↳98 QDP

    ↳99 QDPOrderProof (⇔)

    ↳100 QDP

    ↳101 QDPOrderProof (⇔)

    ↳102 QDP

    ↳103 QDPOrderProof (⇔)

    ↳104 QDP

    ↳105 QDPOrderProof (⇔)

    ↳106 QDP

    ↳107 QDPOrderProof (⇔)

    ↳108 QDP

    ↳109 DependencyGraphProof (⇔)

    ↳110 TRUE

(0) Obligation:

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

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(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:

ACTIVE(and(tt, T)) → MARK(T)
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
ACTIVE(isNatIList(IL)) → ISNATLIST(IL)
ACTIVE(isNat(0)) → MARK(tt)
ACTIVE(isNat(s(N))) → MARK(isNat(N))
ACTIVE(isNat(s(N))) → ISNAT(N)
ACTIVE(isNat(length(L))) → MARK(isNatList(L))
ACTIVE(isNat(length(L))) → ISNATLIST(L)
ACTIVE(isNatIList(zeros)) → MARK(tt)
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
ACTIVE(isNatIList(cons(N, IL))) → AND(isNat(N), isNatIList(IL))
ACTIVE(isNatIList(cons(N, IL))) → ISNAT(N)
ACTIVE(isNatIList(cons(N, IL))) → ISNATILIST(IL)
ACTIVE(isNatList(nil)) → MARK(tt)
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
ACTIVE(isNatList(cons(N, L))) → AND(isNat(N), isNatList(L))
ACTIVE(isNatList(cons(N, L))) → ISNAT(N)
ACTIVE(isNatList(cons(N, L))) → ISNATLIST(L)
ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
ACTIVE(isNatList(take(N, IL))) → AND(isNat(N), isNatIList(IL))
ACTIVE(isNatList(take(N, IL))) → ISNAT(N)
ACTIVE(isNatList(take(N, IL))) → ISNATILIST(IL)
ACTIVE(zeros) → MARK(cons(0, zeros))
ACTIVE(zeros) → CONS(0, zeros)
ACTIVE(take(0, IL)) → MARK(uTake1(isNatIList(IL)))
ACTIVE(take(0, IL)) → UTAKE1(isNatIList(IL))
ACTIVE(take(0, IL)) → ISNATILIST(IL)
ACTIVE(uTake1(tt)) → MARK(nil)
ACTIVE(take(s(M), cons(N, IL))) → MARK(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
ACTIVE(take(s(M), cons(N, IL))) → UTAKE2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)
ACTIVE(take(s(M), cons(N, IL))) → AND(isNat(M), and(isNat(N), isNatIList(IL)))
ACTIVE(take(s(M), cons(N, IL))) → ISNAT(M)
ACTIVE(take(s(M), cons(N, IL))) → AND(isNat(N), isNatIList(IL))
ACTIVE(take(s(M), cons(N, IL))) → ISNAT(N)
ACTIVE(take(s(M), cons(N, IL))) → ISNATILIST(IL)
ACTIVE(uTake2(tt, M, N, IL)) → MARK(cons(N, take(M, IL)))
ACTIVE(uTake2(tt, M, N, IL)) → CONS(N, take(M, IL))
ACTIVE(uTake2(tt, M, N, IL)) → TAKE(M, IL)
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
ACTIVE(length(cons(N, L))) → ULENGTH(and(isNat(N), isNatList(L)), L)
ACTIVE(length(cons(N, L))) → AND(isNat(N), isNatList(L))
ACTIVE(length(cons(N, L))) → ISNAT(N)
ACTIVE(length(cons(N, L))) → ISNATLIST(L)
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))
ACTIVE(uLength(tt, L)) → S(length(L))
ACTIVE(uLength(tt, L)) → LENGTH(L)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
MARK(and(X1, X2)) → AND(mark(X1), mark(X2))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X2)
MARK(tt) → ACTIVE(tt)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(0) → ACTIVE(0)
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(s(X)) → S(mark(X))
MARK(s(X)) → MARK(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(length(X)) → LENGTH(mark(X))
MARK(length(X)) → MARK(X)
MARK(zeros) → ACTIVE(zeros)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(cons(X1, X2)) → CONS(mark(X1), X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(nil) → ACTIVE(nil)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(take(X1, X2)) → TAKE(mark(X1), mark(X2))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(uTake1(X)) → ACTIVE(uTake1(mark(X)))
MARK(uTake1(X)) → UTAKE1(mark(X))
MARK(uTake1(X)) → MARK(X)
MARK(uTake2(X1, X2, X3, X4)) → ACTIVE(uTake2(mark(X1), X2, X3, X4))
MARK(uTake2(X1, X2, X3, X4)) → UTAKE2(mark(X1), X2, X3, X4)
MARK(uTake2(X1, X2, X3, X4)) → MARK(X1)
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
MARK(uLength(X1, X2)) → ULENGTH(mark(X1), X2)
MARK(uLength(X1, X2)) → MARK(X1)
AND(mark(X1), X2) → AND(X1, X2)
AND(X1, mark(X2)) → AND(X1, X2)
AND(active(X1), X2) → AND(X1, X2)
AND(X1, active(X2)) → AND(X1, X2)
ISNATILIST(mark(X)) → ISNATILIST(X)
ISNATILIST(active(X)) → ISNATILIST(X)
ISNATLIST(mark(X)) → ISNATLIST(X)
ISNATLIST(active(X)) → ISNATLIST(X)
ISNAT(mark(X)) → ISNAT(X)
ISNAT(active(X)) → ISNAT(X)
S(mark(X)) → S(X)
S(active(X)) → S(X)
LENGTH(mark(X)) → LENGTH(X)
LENGTH(active(X)) → LENGTH(X)
CONS(mark(X1), X2) → CONS(X1, X2)
CONS(X1, mark(X2)) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)
TAKE(mark(X1), X2) → TAKE(X1, X2)
TAKE(X1, mark(X2)) → TAKE(X1, X2)
TAKE(active(X1), X2) → TAKE(X1, X2)
TAKE(X1, active(X2)) → TAKE(X1, X2)
UTAKE1(mark(X)) → UTAKE1(X)
UTAKE1(active(X)) → UTAKE1(X)
UTAKE2(mark(X1), X2, X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, mark(X2), X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, X2, mark(X3), X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, X2, X3, mark(X4)) → UTAKE2(X1, X2, X3, X4)
UTAKE2(active(X1), X2, X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, active(X2), X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, X2, active(X3), X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, X2, X3, active(X4)) → UTAKE2(X1, X2, X3, X4)
ULENGTH(mark(X1), X2) → ULENGTH(X1, X2)
ULENGTH(X1, mark(X2)) → ULENGTH(X1, X2)
ULENGTH(active(X1), X2) → ULENGTH(X1, X2)
ULENGTH(X1, active(X2)) → ULENGTH(X1, X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(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 12 SCCs with 44 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

ULENGTH(X1, mark(X2)) → ULENGTH(X1, X2)
ULENGTH(mark(X1), X2) → ULENGTH(X1, X2)
ULENGTH(active(X1), X2) → ULENGTH(X1, X2)
ULENGTH(X1, active(X2)) → ULENGTH(X1, X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

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

(6) 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.

(7) Obligation:

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

ULENGTH(X1, mark(X2)) → ULENGTH(X1, X2)
ULENGTH(mark(X1), X2) → ULENGTH(X1, X2)
ULENGTH(active(X1), X2) → ULENGTH(X1, X2)
ULENGTH(X1, active(X2)) → ULENGTH(X1, X2)

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

(8) 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:

ULENGTH(X1, mark(X2)) → ULENGTH(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
ULENGTH(mark(X1), X2) → ULENGTH(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
ULENGTH(active(X1), X2) → ULENGTH(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
ULENGTH(X1, active(X2)) → ULENGTH(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2

(9) TRUE

(10) Obligation:

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

UTAKE2(X1, mark(X2), X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(mark(X1), X2, X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, X2, mark(X3), X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, X2, X3, mark(X4)) → UTAKE2(X1, X2, X3, X4)
UTAKE2(active(X1), X2, X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, active(X2), X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, X2, active(X3), X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, X2, X3, active(X4)) → UTAKE2(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

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

(11) UsableRulesProof (EQUIVALENT transformation)

(12) Obligation:

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

UTAKE2(X1, mark(X2), X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(mark(X1), X2, X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, X2, mark(X3), X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, X2, X3, mark(X4)) → UTAKE2(X1, X2, X3, X4)
UTAKE2(active(X1), X2, X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, active(X2), X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, X2, active(X3), X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, X2, X3, active(X4)) → UTAKE2(X1, X2, X3, X4)

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

(13) 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:

UTAKE2(X1, mark(X2), X3, X4) → UTAKE2(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4
UTAKE2(mark(X1), X2, X3, X4) → UTAKE2(X1, X2, X3, X4)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4
UTAKE2(X1, X2, mark(X3), X4) → UTAKE2(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4
UTAKE2(X1, X2, X3, mark(X4)) → UTAKE2(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4
UTAKE2(active(X1), X2, X3, X4) → UTAKE2(X1, X2, X3, X4)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4
UTAKE2(X1, active(X2), X3, X4) → UTAKE2(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4
UTAKE2(X1, X2, active(X3), X4) → UTAKE2(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4
UTAKE2(X1, X2, X3, active(X4)) → UTAKE2(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4

(14) TRUE

(15) Obligation:

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

UTAKE1(active(X)) → UTAKE1(X)
UTAKE1(mark(X)) → UTAKE1(X)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

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

(16) UsableRulesProof (EQUIVALENT transformation)

(17) Obligation:

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

UTAKE1(active(X)) → UTAKE1(X)
UTAKE1(mark(X)) → UTAKE1(X)

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

(18) 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:

UTAKE1(active(X)) → UTAKE1(X)
The graph contains the following edges 1 > 1
UTAKE1(mark(X)) → UTAKE1(X)
The graph contains the following edges 1 > 1

(19) TRUE

(20) Obligation:

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

TAKE(X1, mark(X2)) → TAKE(X1, X2)
TAKE(mark(X1), X2) → TAKE(X1, X2)
TAKE(active(X1), X2) → TAKE(X1, X2)
TAKE(X1, active(X2)) → TAKE(X1, X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

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

(21) UsableRulesProof (EQUIVALENT transformation)

(22) Obligation:

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

TAKE(X1, mark(X2)) → TAKE(X1, X2)
TAKE(mark(X1), X2) → TAKE(X1, X2)
TAKE(active(X1), X2) → TAKE(X1, X2)
TAKE(X1, active(X2)) → TAKE(X1, X2)

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

(23) 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:

TAKE(X1, mark(X2)) → TAKE(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
TAKE(mark(X1), X2) → TAKE(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
TAKE(active(X1), X2) → TAKE(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
TAKE(X1, active(X2)) → TAKE(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2

(24) TRUE

(25) Obligation:

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

CONS(X1, mark(X2)) → CONS(X1, X2)
CONS(mark(X1), X2) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

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

(26) UsableRulesProof (EQUIVALENT transformation)

(27) Obligation:

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

CONS(X1, mark(X2)) → CONS(X1, X2)
CONS(mark(X1), X2) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)

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

(28) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

CONS(X1, mark(X2)) → CONS(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
CONS(mark(X1), X2) → CONS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
CONS(active(X1), X2) → CONS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
CONS(X1, active(X2)) → CONS(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2

(29) TRUE

(30) Obligation:

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

LENGTH(active(X)) → LENGTH(X)
LENGTH(mark(X)) → LENGTH(X)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

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

(31) UsableRulesProof (EQUIVALENT transformation)

(32) Obligation:

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

LENGTH(active(X)) → LENGTH(X)
LENGTH(mark(X)) → LENGTH(X)

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

(33) 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:

LENGTH(active(X)) → LENGTH(X)
The graph contains the following edges 1 > 1
LENGTH(mark(X)) → LENGTH(X)
The graph contains the following edges 1 > 1

(34) TRUE

(35) Obligation:

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

S(active(X)) → S(X)
S(mark(X)) → S(X)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

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

(36) UsableRulesProof (EQUIVALENT transformation)

(37) Obligation:

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

S(active(X)) → S(X)
S(mark(X)) → S(X)

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

(38) 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:

S(active(X)) → S(X)
The graph contains the following edges 1 > 1
S(mark(X)) → S(X)
The graph contains the following edges 1 > 1

(39) TRUE

(40) Obligation:

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

ISNAT(active(X)) → ISNAT(X)
ISNAT(mark(X)) → ISNAT(X)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

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

(41) UsableRulesProof (EQUIVALENT transformation)

(42) Obligation:

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

ISNAT(active(X)) → ISNAT(X)
ISNAT(mark(X)) → ISNAT(X)

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

(43) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

ISNAT(active(X)) → ISNAT(X)
The graph contains the following edges 1 > 1
ISNAT(mark(X)) → ISNAT(X)
The graph contains the following edges 1 > 1

(44) TRUE

(45) Obligation:

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

ISNATLIST(active(X)) → ISNATLIST(X)
ISNATLIST(mark(X)) → ISNATLIST(X)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

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

(46) UsableRulesProof (EQUIVALENT transformation)

(47) Obligation:

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

ISNATLIST(active(X)) → ISNATLIST(X)
ISNATLIST(mark(X)) → ISNATLIST(X)

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

(48) 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:

ISNATLIST(active(X)) → ISNATLIST(X)
The graph contains the following edges 1 > 1
ISNATLIST(mark(X)) → ISNATLIST(X)
The graph contains the following edges 1 > 1

(49) TRUE

(50) Obligation:

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

ISNATILIST(active(X)) → ISNATILIST(X)
ISNATILIST(mark(X)) → ISNATILIST(X)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

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

(51) UsableRulesProof (EQUIVALENT transformation)

(52) Obligation:

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

ISNATILIST(active(X)) → ISNATILIST(X)
ISNATILIST(mark(X)) → ISNATILIST(X)

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

(53) 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:

ISNATILIST(active(X)) → ISNATILIST(X)
The graph contains the following edges 1 > 1
ISNATILIST(mark(X)) → ISNATILIST(X)
The graph contains the following edges 1 > 1

(54) TRUE

(55) Obligation:

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

AND(X1, mark(X2)) → AND(X1, X2)
AND(mark(X1), X2) → AND(X1, X2)
AND(active(X1), X2) → AND(X1, X2)
AND(X1, active(X2)) → AND(X1, X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

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

(56) UsableRulesProof (EQUIVALENT transformation)

(57) Obligation:

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

AND(X1, mark(X2)) → AND(X1, X2)
AND(mark(X1), X2) → AND(X1, X2)
AND(active(X1), X2) → AND(X1, X2)
AND(X1, active(X2)) → AND(X1, X2)

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

(58) 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:

AND(X1, mark(X2)) → AND(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
AND(mark(X1), X2) → AND(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
AND(active(X1), X2) → AND(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
AND(X1, active(X2)) → AND(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2

(59) TRUE

(60) Obligation:

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

MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(and(tt, T)) → MARK(T)
MARK(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X2)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNat(s(N))) → MARK(isNat(N))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(isNat(length(L))) → MARK(isNatList(L))
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(s(X)) → MARK(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(length(X)) → MARK(X)
MARK(zeros) → ACTIVE(zeros)
ACTIVE(zeros) → MARK(cons(0, zeros))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
ACTIVE(take(0, IL)) → MARK(uTake1(isNatIList(IL)))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(uTake1(X)) → ACTIVE(uTake1(mark(X)))
ACTIVE(take(s(M), cons(N, IL))) → MARK(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
MARK(uTake1(X)) → MARK(X)
MARK(uTake2(X1, X2, X3, X4)) → ACTIVE(uTake2(mark(X1), X2, X3, X4))
ACTIVE(uTake2(tt, M, N, IL)) → MARK(cons(N, take(M, IL)))
MARK(uTake2(X1, X2, X3, X4)) → MARK(X1)
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
MARK(uLength(X1, X2)) → MARK(X1)
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

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

(61) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(uTake1(X)) → ACTIVE(uTake1(mark(X)))

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

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = 1
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = 1
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 1
POL(isNatIList(x₁)) = 1
POL(isNatList(x₁)) = 1
POL(length(x₁)) = 1
POL(mark(x₁)) = 0
POL(nil) = 0
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 1
POL(tt) = 0
POL(uLength(x₁, x₂)) = 1
POL(uTake1(x₁)) = 0
POL(uTake2(x₁, x₂, x₃, x₄)) = 1
POL(zeros) = 1

The following usable rules [FROCOS05] were oriented:

uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)

(62) Obligation:

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

MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(and(tt, T)) → MARK(T)
MARK(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X2)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNat(s(N))) → MARK(isNat(N))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(isNat(length(L))) → MARK(isNatList(L))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(s(X)) → MARK(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(length(X)) → MARK(X)
MARK(zeros) → ACTIVE(zeros)
ACTIVE(zeros) → MARK(cons(0, zeros))
ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
ACTIVE(take(0, IL)) → MARK(uTake1(isNatIList(IL)))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
ACTIVE(take(s(M), cons(N, IL))) → MARK(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
MARK(uTake1(X)) → MARK(X)
MARK(uTake2(X1, X2, X3, X4)) → ACTIVE(uTake2(mark(X1), X2, X3, X4))
ACTIVE(uTake2(tt, M, N, IL)) → MARK(cons(N, take(M, IL)))
MARK(uTake2(X1, X2, X3, X4)) → MARK(X1)
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
MARK(uLength(X1, X2)) → MARK(X1)
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

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

(63) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(take(0, IL)) → MARK(uTake1(isNatIList(IL)))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
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(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(active(x₁)) = x₁
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:

active(uTake1(tt)) → mark(nil)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
active(isNat(0)) → mark(tt)
active(isNatIList(zeros)) → mark(tt)
active(isNatList(nil)) → mark(tt)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
mark(0) → active(0)
mark(tt) → active(tt)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
mark(nil) → active(nil)

(64) Obligation:

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

MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(and(tt, T)) → MARK(T)
MARK(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X2)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNat(s(N))) → MARK(isNat(N))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(isNat(length(L))) → MARK(isNatList(L))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(s(X)) → MARK(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(length(X)) → MARK(X)
MARK(zeros) → ACTIVE(zeros)
ACTIVE(zeros) → MARK(cons(0, zeros))
ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
ACTIVE(take(s(M), cons(N, IL))) → MARK(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
MARK(uTake1(X)) → MARK(X)
MARK(uTake2(X1, X2, X3, X4)) → ACTIVE(uTake2(mark(X1), X2, X3, X4))
ACTIVE(uTake2(tt, M, N, IL)) → MARK(cons(N, take(M, IL)))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
MARK(uLength(X1, X2)) → MARK(X1)
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

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

(65) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(length(X)) → MARK(X)
MARK(uLength(X1, X2)) → MARK(X1)

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

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(active(x₁)) = x₁
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:

active(uTake1(tt)) → mark(nil)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
active(isNat(0)) → mark(tt)
active(isNatIList(zeros)) → mark(tt)
active(isNatList(nil)) → mark(tt)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
mark(0) → active(0)
mark(tt) → active(tt)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
mark(nil) → active(nil)

(66) Obligation:

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

MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(and(tt, T)) → MARK(T)
MARK(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X2)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNat(s(N))) → MARK(isNat(N))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(isNat(length(L))) → MARK(isNatList(L))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(s(X)) → MARK(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(zeros) → ACTIVE(zeros)
ACTIVE(zeros) → MARK(cons(0, zeros))
ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
ACTIVE(take(s(M), cons(N, IL))) → MARK(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
MARK(uTake1(X)) → MARK(X)
MARK(uTake2(X1, X2, X3, X4)) → ACTIVE(uTake2(mark(X1), X2, X3, X4))
ACTIVE(uTake2(tt, M, N, IL)) → MARK(cons(N, take(M, IL)))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

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

(67) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(zeros) → MARK(cons(0, zeros))

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

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = 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)) = x₂
POL(tt) = 0
POL(uLength(x₁, x₂)) = 0
POL(uTake1(x₁)) = x₁
POL(uTake2(x₁, x₂, x₃, x₄)) = x₃
POL(zeros) = 1

The following usable rules [FROCOS05] were oriented:

active(uTake1(tt)) → mark(nil)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
active(isNat(0)) → mark(tt)
active(isNatIList(zeros)) → mark(tt)
active(isNatList(nil)) → mark(tt)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
mark(0) → active(0)
mark(tt) → active(tt)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
mark(nil) → active(nil)

(68) Obligation:

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

MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(and(tt, T)) → MARK(T)
MARK(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X2)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNat(s(N))) → MARK(isNat(N))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(isNat(length(L))) → MARK(isNatList(L))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(s(X)) → MARK(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(zeros) → ACTIVE(zeros)
ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
ACTIVE(take(s(M), cons(N, IL))) → MARK(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
MARK(uTake1(X)) → MARK(X)
MARK(uTake2(X1, X2, X3, X4)) → ACTIVE(uTake2(mark(X1), X2, X3, X4))
ACTIVE(uTake2(tt, M, N, IL)) → MARK(cons(N, take(M, IL)))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

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

(69) DependencyGraphProof (EQUIVALENT transformation)

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

(70) Obligation:

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

ACTIVE(and(tt, T)) → MARK(T)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X2)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNat(s(N))) → MARK(isNat(N))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNat(length(L))) → MARK(isNatList(L))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(s(X)) → MARK(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(uTake1(X)) → MARK(X)
MARK(uTake2(X1, X2, X3, X4)) → ACTIVE(uTake2(mark(X1), X2, X3, X4))
ACTIVE(take(s(M), cons(N, IL))) → MARK(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(uTake2(tt, M, N, IL)) → MARK(cons(N, take(M, IL)))
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

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

(71) 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) = 1
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = 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)) = 1 + x₁
POL(uTake2(x₁, x₂, x₃, x₄)) = x₃
POL(zeros) = 1

The following usable rules [FROCOS05] were oriented:

active(uTake1(tt)) → mark(nil)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
active(isNat(0)) → mark(tt)
active(isNatIList(zeros)) → mark(tt)
active(isNatList(nil)) → mark(tt)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
mark(0) → active(0)
mark(tt) → active(tt)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
mark(nil) → active(nil)

(72) Obligation:

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

ACTIVE(and(tt, T)) → MARK(T)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X2)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNat(s(N))) → MARK(isNat(N))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNat(length(L))) → MARK(isNatList(L))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(s(X)) → MARK(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(uTake2(X1, X2, X3, X4)) → ACTIVE(uTake2(mark(X1), X2, X3, X4))
ACTIVE(take(s(M), cons(N, IL))) → MARK(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(uTake2(tt, M, N, IL)) → MARK(cons(N, take(M, IL)))
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

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

(73) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(take(s(M), cons(N, IL))) → MARK(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))

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

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = 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)) = 1 + x₂
POL(tt) = 0
POL(uLength(x₁, x₂)) = 0
POL(uTake1(x₁)) = 0
POL(uTake2(x₁, x₂, x₃, x₄)) = x₃
POL(zeros) = 1

The following usable rules [FROCOS05] were oriented:

active(uTake1(tt)) → mark(nil)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
active(isNat(0)) → mark(tt)
active(isNatIList(zeros)) → mark(tt)
active(isNatList(nil)) → mark(tt)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
mark(0) → active(0)
mark(tt) → active(tt)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
mark(nil) → active(nil)

(74) Obligation:

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

ACTIVE(and(tt, T)) → MARK(T)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X2)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNat(s(N))) → MARK(isNat(N))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNat(length(L))) → MARK(isNatList(L))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(s(X)) → MARK(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(uTake2(X1, X2, X3, X4)) → ACTIVE(uTake2(mark(X1), X2, X3, X4))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(uTake2(tt, M, N, IL)) → MARK(cons(N, take(M, IL)))
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

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

(75) 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(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(active(x₁)) = x₁
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)) = 0
POL(uTake2(x₁, x₂, x₃, x₄)) = 1 + x₃
POL(zeros) = 1

The following usable rules [FROCOS05] were oriented:

isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
active(uTake1(tt)) → mark(nil)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
active(isNat(0)) → mark(tt)
length(active(X)) → length(X)
length(mark(X)) → length(X)
active(isNatIList(zeros)) → mark(tt)
s(active(X)) → s(X)
s(mark(X)) → s(X)
active(isNatList(nil)) → mark(tt)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
mark(0) → active(0)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
mark(tt) → active(tt)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
mark(nil) → active(nil)

(76) Obligation:

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

ACTIVE(and(tt, T)) → MARK(T)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X2)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNat(s(N))) → MARK(isNat(N))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNat(length(L))) → MARK(isNatList(L))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(s(X)) → MARK(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(uTake2(X1, X2, X3, X4)) → ACTIVE(uTake2(mark(X1), X2, X3, X4))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(uTake2(tt, M, N, IL)) → MARK(cons(N, take(M, IL)))
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

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

(77) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(uTake2(tt, M, N, IL)) → MARK(cons(N, take(M, IL)))

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

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(active(x₁)) = x₁
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
POL(tt) = 0
POL(uLength(x₁, x₂)) = 0
POL(uTake1(x₁)) = 0
POL(uTake2(x₁, x₂, x₃, x₄)) = 1
POL(zeros) = 1

The following usable rules [FROCOS05] were oriented:

isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
active(uTake1(tt)) → mark(nil)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
active(isNat(0)) → mark(tt)
length(active(X)) → length(X)
length(mark(X)) → length(X)
active(isNatIList(zeros)) → mark(tt)
s(active(X)) → s(X)
s(mark(X)) → s(X)
active(isNatList(nil)) → mark(tt)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
mark(0) → active(0)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
mark(tt) → active(tt)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
mark(nil) → active(nil)

(78) Obligation:

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

ACTIVE(and(tt, T)) → MARK(T)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X2)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNat(s(N))) → MARK(isNat(N))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNat(length(L))) → MARK(isNatList(L))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(s(X)) → MARK(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(uTake2(X1, X2, X3, X4)) → ACTIVE(uTake2(mark(X1), X2, X3, X4))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

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

(79) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))

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

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(active(x₁)) = x₁
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) = 1
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:

isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
active(uTake1(tt)) → mark(nil)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
active(isNat(0)) → mark(tt)
length(active(X)) → length(X)
length(mark(X)) → length(X)
active(isNatIList(zeros)) → mark(tt)
s(active(X)) → s(X)
s(mark(X)) → s(X)
active(isNatList(nil)) → mark(tt)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
mark(0) → active(0)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
mark(tt) → active(tt)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
mark(nil) → active(nil)

(80) Obligation:

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

ACTIVE(and(tt, T)) → MARK(T)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X2)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNat(s(N))) → MARK(isNat(N))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNat(length(L))) → MARK(isNatList(L))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(s(X)) → MARK(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(uTake2(X1, X2, X3, X4)) → ACTIVE(uTake2(mark(X1), X2, X3, X4))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

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

(81) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))

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

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(active(x₁)) = x₁
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)) = 0
POL(uTake2(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₃ + x₄
POL(zeros) = 0

The following usable rules [FROCOS05] were oriented:

isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
active(uTake1(tt)) → mark(nil)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
active(isNat(0)) → mark(tt)
length(active(X)) → length(X)
length(mark(X)) → length(X)
active(isNatIList(zeros)) → mark(tt)
s(active(X)) → s(X)
s(mark(X)) → s(X)
active(isNatList(nil)) → mark(tt)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
mark(0) → active(0)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
mark(tt) → active(tt)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
mark(nil) → active(nil)

(82) Obligation:

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

ACTIVE(and(tt, T)) → MARK(T)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X2)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNat(s(N))) → MARK(isNat(N))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNat(length(L))) → MARK(isNatList(L))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(s(X)) → MARK(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(uTake2(X1, X2, X3, X4)) → ACTIVE(uTake2(mark(X1), X2, X3, X4))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

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

(83) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(isNat(length(L))) → MARK(isNatList(L))

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

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(active(x₁)) = x₁
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)) = 0
POL(uTake2(x₁, x₂, x₃, x₄)) = x₂ + x₃ + x₄
POL(zeros) = 0

The following usable rules [FROCOS05] were oriented:

isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
active(uTake1(tt)) → mark(nil)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
active(isNat(0)) → mark(tt)
length(active(X)) → length(X)
length(mark(X)) → length(X)
active(isNatIList(zeros)) → mark(tt)
s(active(X)) → s(X)
s(mark(X)) → s(X)
active(isNatList(nil)) → mark(tt)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
mark(0) → active(0)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
mark(tt) → active(tt)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
mark(nil) → active(nil)

(84) Obligation:

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

ACTIVE(and(tt, T)) → MARK(T)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X2)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNat(s(N))) → MARK(isNat(N))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(s(X)) → MARK(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(uTake2(X1, X2, X3, X4)) → ACTIVE(uTake2(mark(X1), X2, X3, X4))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

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

(85) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(uTake2(X1, X2, X3, X4)) → ACTIVE(uTake2(mark(X1), X2, X3, X4))

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

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = 1
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = 1
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 1
POL(isNatIList(x₁)) = 1
POL(isNatList(x₁)) = 1
POL(length(x₁)) = 1
POL(mark(x₁)) = 0
POL(nil) = 0
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 0
POL(tt) = 0
POL(uLength(x₁, x₂)) = 1
POL(uTake1(x₁)) = 0
POL(uTake2(x₁, x₂, x₃, x₄)) = 0
POL(zeros) = 0

The following usable rules [FROCOS05] were oriented:

isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)

(86) Obligation:

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

ACTIVE(and(tt, T)) → MARK(T)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X2)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNat(s(N))) → MARK(isNat(N))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(s(X)) → MARK(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

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

(87) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(isNat(s(N))) → MARK(isNat(N))

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

POL(ACTIVE(x₁)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 0 0 /

· x₁

POL(and(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 0 0 /

· x₁ +
/ 1 0 \

\ 0 1 /

· x₂

POL(tt) =
/ 0 \

\ 1 /

POL(MARK(x₁)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 0 0 /

· x₁

POL(mark(x₁)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 0 1 /

· x₁

POL(isNatIList(x₁)) =
/ 0 \

\ 1 /

+
/ 0 1 \

\ 1 0 /

· x₁

POL(isNat(x₁)) =
/ 0 \

\ 0 /

+
/ 0 1 \

\ 1 1 /

· x₁

POL(s(x₁)) =
/ 0 \

\ 1 /

+
/ 1 0 \

\ 0 1 /

· x₁

POL(isNatList(x₁)) =
/ 0 \

\ 0 /

+
/ 0 1 \

\ 1 0 /

· x₁

POL(cons(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 1 /

· x₁ +
/ 1 0 \

\ 1 1 /

· x₂

POL(length(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 1 1 /

· x₁

POL(uLength(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 1 /

· x₁ +
/ 0 0 \

\ 1 1 /

· x₂

POL(active(x₁)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 0 1 /

· x₁

POL(uTake1(x₁)) =
/ 0 \

\ 0 /

+
/ 0 1 \

\ 0 0 /

· x₁

POL(nil) =
/ 1 \

\ 0 /

POL(0) =
/ 1 \

\ 0 /

POL(zeros) =
/ 0 \

\ 0 /

POL(take(x₁, x₂)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 1 /

· x₁ +
/ 1 0 \

\ 0 1 /

· x₂

POL(uTake2(x₁, x₂, x₃, x₄)) =
/ 1 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 1 /

· x₂ +
/ 0 0 \

\ 0 1 /

· x₃ +
/ 1 0 \

\ 1 1 /

· x₄

The following usable rules [FROCOS05] were oriented:

isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
active(uTake1(tt)) → mark(nil)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
active(isNat(0)) → mark(tt)
length(active(X)) → length(X)
length(mark(X)) → length(X)
active(isNatIList(zeros)) → mark(tt)
s(active(X)) → s(X)
s(mark(X)) → s(X)
active(isNatList(nil)) → mark(tt)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
mark(0) → active(0)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
mark(tt) → active(tt)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
mark(nil) → active(nil)

(88) Obligation:

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

ACTIVE(and(tt, T)) → MARK(T)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X2)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(s(X)) → MARK(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

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

(89) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(isNat(X)) → ACTIVE(isNat(X))

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

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = 1
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = 1
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 1
POL(isNatList(x₁)) = 1
POL(length(x₁)) = 1
POL(mark(x₁)) = 0
POL(nil) = 0
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 0
POL(tt) = 0
POL(uLength(x₁, x₂)) = 1
POL(uTake1(x₁)) = 0
POL(uTake2(x₁, x₂, x₃, x₄)) = 0
POL(zeros) = 0

The following usable rules [FROCOS05] were oriented:

isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)

(90) Obligation:

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

ACTIVE(and(tt, T)) → MARK(T)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X2)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(s(X)) → MARK(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

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

(91) 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(ACTIVE(x₁)) =
/ 0 \

\ 0 /

+
/ 0 1 \

\ 0 0 /

· x₁

POL(and(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 1 /

· x₁ +
/ 1 0 \

\ 0 1 /

· x₂

POL(tt) =
/ 1 \

\ 0 /

POL(MARK(x₁)) =
/ 0 \

\ 0 /

+
/ 0 1 \

\ 0 0 /

· x₁

POL(mark(x₁)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 0 1 /

· x₁

POL(isNatIList(x₁)) =
/ 1 \

\ 0 /

+
/ 1 0 \

\ 0 0 /

· x₁

POL(isNatList(x₁)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 0 0 /

· x₁

POL(cons(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 1 0 \

\ 1 1 /

· x₂

POL(isNat(x₁)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 0 0 /

· x₁

POL(s(x₁)) =
/ 0 \

\ 1 /

+
/ 1 0 \

\ 0 1 /

· x₁

POL(length(x₁)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 0 1 /

· x₁

POL(uLength(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 1 0 /

· x₁ +
/ 1 0 \

\ 0 1 /

· x₂

POL(active(x₁)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 0 1 /

· x₁

POL(uTake1(x₁)) =
/ 0 \

\ 0 /

+
/ 1 1 \

\ 0 0 /

· x₁

POL(nil) =
/ 1 \

\ 0 /

POL(0) =
/ 1 \

\ 0 /

POL(zeros) =
/ 0 \

\ 0 /

POL(take(x₁, x₂)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 1 /

· x₁ +
/ 1 0 \

\ 0 1 /

· x₂

POL(uTake2(x₁, x₂, x₃, x₄)) =
/ 1 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 1 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 1 0 \

\ 1 1 /

· x₄

The following usable rules [FROCOS05] were oriented:

isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
active(uTake1(tt)) → mark(nil)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
active(isNat(0)) → mark(tt)
length(active(X)) → length(X)
length(mark(X)) → length(X)
active(isNatIList(zeros)) → mark(tt)
s(active(X)) → s(X)
s(mark(X)) → s(X)
active(isNatList(nil)) → mark(tt)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
mark(0) → active(0)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
mark(tt) → active(tt)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
mark(nil) → active(nil)

(92) Obligation:

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

ACTIVE(and(tt, T)) → MARK(T)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X2)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

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

(93) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))

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

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = 1
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)) = 0
POL(take(x₁, x₂)) = 1 + x₁
POL(tt) = 0
POL(uLength(x₁, x₂)) = 0
POL(uTake1(x₁)) = 0
POL(uTake2(x₁, x₂, x₃, x₄)) = 1 + x₁
POL(zeros) = 1

The following usable rules [FROCOS05] were oriented:

isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
active(uTake1(tt)) → mark(nil)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
active(isNat(0)) → mark(tt)
length(active(X)) → length(X)
length(mark(X)) → length(X)
active(isNatIList(zeros)) → mark(tt)
s(active(X)) → s(X)
s(mark(X)) → s(X)
active(isNatList(nil)) → mark(tt)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
mark(0) → active(0)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
mark(tt) → active(tt)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
mark(nil) → active(nil)

(94) Obligation:

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

ACTIVE(and(tt, T)) → MARK(T)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X2)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

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

(95) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(uLength(tt, L)) → MARK(s(length(L)))

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

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = 1
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)) = 0
POL(take(x₁, x₂)) = 1 + x₁
POL(tt) = 0
POL(uLength(x₁, x₂)) = 1
POL(uTake1(x₁)) = x₁
POL(uTake2(x₁, x₂, x₃, x₄)) = 1 + x₁
POL(zeros) = 1

The following usable rules [FROCOS05] were oriented:

isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
active(uTake1(tt)) → mark(nil)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
active(isNat(0)) → mark(tt)
length(active(X)) → length(X)
length(mark(X)) → length(X)
active(isNatIList(zeros)) → mark(tt)
s(active(X)) → s(X)
s(mark(X)) → s(X)
active(isNatList(nil)) → mark(tt)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
mark(0) → active(0)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
mark(tt) → active(tt)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
mark(nil) → active(nil)

(96) Obligation:

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

ACTIVE(and(tt, T)) → MARK(T)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X2)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

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

(97) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))

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

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = 1
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = 1
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 1
POL(isNatList(x₁)) = 1
POL(length(x₁)) = 0
POL(mark(x₁)) = 0
POL(nil) = 0
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 0
POL(tt) = 0
POL(uLength(x₁, x₂)) = 0
POL(uTake1(x₁)) = 0
POL(uTake2(x₁, x₂, x₃, x₄)) = 0
POL(zeros) = 0

The following usable rules [FROCOS05] were oriented:

isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)

(98) Obligation:

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

ACTIVE(and(tt, T)) → MARK(T)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X2)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

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

(99) 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(ACTIVE(x₁)) = -I + 0A · x₁

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

POL(tt) = 0A

POL(MARK(x₁)) = -I + 0A · x₁

POL(mark(x₁)) = -I + 0A · x₁

POL(isNatIList(x₁)) = -I + 0A · x₁

POL(isNatList(x₁)) = -I + 0A · x₁

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

POL(isNat(x₁)) = -I + 0A · x₁

POL(active(x₁)) = -I + 0A · x₁

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

POL(uTake1(x₁)) = 1A + -I · x₁

POL(nil) = 0A

POL(length(x₁)) = -I + 0A · x₁

POL(0) = 0A

POL(zeros) = 1A

POL(s(x₁)) = -I + 0A · x₁

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

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

The following usable rules [FROCOS05] were oriented:

isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
active(uTake1(tt)) → mark(nil)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
active(isNat(0)) → mark(tt)
active(isNatIList(zeros)) → mark(tt)
s(active(X)) → s(X)
s(mark(X)) → s(X)
active(isNatList(nil)) → mark(tt)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
mark(0) → active(0)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
mark(tt) → active(tt)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
mark(nil) → active(nil)

(100) Obligation:

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

ACTIVE(and(tt, T)) → MARK(T)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X2)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

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

(101) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))

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

POL(ACTIVE(x₁)) = -I + 0A · x₁

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

POL(tt) = 0A

POL(MARK(x₁)) = -I + 0A · x₁

POL(mark(x₁)) = 1A + 0A · x₁

POL(isNatIList(x₁)) = -I + 1A · x₁

POL(isNatList(x₁)) = 0A + 1A · x₁

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

POL(isNat(x₁)) = -I + 1A · x₁

POL(active(x₁)) = 1A + 0A · x₁

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

POL(uTake1(x₁)) = 0A + -I · x₁

POL(nil) = 0A

POL(length(x₁)) = -I + 0A · x₁

POL(0) = 0A

POL(zeros) = 0A

POL(s(x₁)) = -I + 0A · x₁

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

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

The following usable rules [FROCOS05] were oriented:

isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
active(uTake1(tt)) → mark(nil)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
active(isNat(0)) → mark(tt)
active(isNatIList(zeros)) → mark(tt)
s(active(X)) → s(X)
s(mark(X)) → s(X)
active(isNatList(nil)) → mark(tt)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
mark(0) → active(0)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
mark(tt) → active(tt)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
mark(nil) → active(nil)

(102) Obligation:

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

ACTIVE(and(tt, T)) → MARK(T)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
MARK(and(X1, X2)) → MARK(X2)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(isNatList(X)) → ACTIVE(isNatList(X))

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

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

(103) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(isNatList(X)) → ACTIVE(isNatList(X))

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

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = 1
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = 1
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = 0
POL(nil) = 0
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 0
POL(tt) = 0
POL(uLength(x₁, x₂)) = 0
POL(uTake1(x₁)) = 0
POL(uTake2(x₁, x₂, x₃, x₄)) = 0
POL(zeros) = 0

The following usable rules [FROCOS05] were oriented:

isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)

(104) Obligation:

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

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

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

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

(105) 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] with arctic natural numbers [ARCTIC]:

POL(ACTIVE(x₁)) = 4A + 1A · x₁

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

POL(tt) = 3A

POL(MARK(x₁)) = -I + 2A · x₁

POL(mark(x₁)) = 1A + 0A · x₁

POL(isNatList(x₁)) = -I + 1A · x₁

POL(active(x₁)) = 1A + 0A · x₁

POL(isNatIList(x₁)) = 0A + 3A · x₁

POL(uLength(x₁, x₂)) = -I + 0A · x₁ + 2A · x₂

POL(uTake1(x₁)) = 4A + 0A · x₁

POL(nil) = 2A

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

POL(length(x₁)) = 2A + 1A · x₁

POL(isNat(x₁)) = -I + 2A · x₁

POL(0) = 1A

POL(zeros) = 0A

POL(s(x₁)) = 3A + 1A · x₁

POL(take(x₁, x₂)) = 4A + 2A · x₁ + 3A · x₂

POL(uTake2(x₁, x₂, x₃, x₄)) = 5A + -I · x₁ + 3A · x₂ + 1A · x₃ + 4A · x₄

The following usable rules [FROCOS05] were oriented:

isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
active(uTake1(tt)) → mark(nil)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
active(isNat(0)) → mark(tt)
active(isNatIList(zeros)) → mark(tt)
s(active(X)) → s(X)
s(mark(X)) → s(X)
active(isNatList(nil)) → mark(tt)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
mark(0) → active(0)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
mark(tt) → active(tt)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
mark(nil) → active(nil)

(106) Obligation:

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

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

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

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

(107) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(and(tt, T)) → MARK(T)

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

POL(ACTIVE(x₁)) = -I + 3A · x₁

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

POL(tt) = 3A

POL(MARK(x₁)) = 2A + 3A · x₁

POL(mark(x₁)) = 1A + 0A · x₁

POL(isNatList(x₁)) = -I + 1A · x₁

POL(active(x₁)) = 1A + 0A · x₁

POL(isNatIList(x₁)) = -I + 3A · x₁

POL(uLength(x₁, x₂)) = -I + 0A · x₁ + 2A · x₂

POL(uTake1(x₁)) = 4A + 0A · x₁

POL(nil) = 2A

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

POL(length(x₁)) = 2A + 1A · x₁

POL(isNat(x₁)) = 3A + 0A · x₁

POL(0) = 0A

POL(zeros) = 0A

POL(s(x₁)) = 3A + 1A · x₁

POL(take(x₁, x₂)) = 4A + 2A · x₁ + 3A · x₂

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

The following usable rules [FROCOS05] were oriented:

isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
active(uTake1(tt)) → mark(nil)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
active(isNat(0)) → mark(tt)
active(isNatIList(zeros)) → mark(tt)
s(active(X)) → s(X)
s(mark(X)) → s(X)
active(isNatList(nil)) → mark(tt)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
mark(0) → active(0)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)
mark(tt) → active(tt)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
mark(zeros) → active(zeros)
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(zeros) → mark(cons(0, zeros))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(isNatList(X)) → active(isNatList(X))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(uTake1(X)) → active(uTake1(mark(X)))
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(length(L))) → mark(isNatList(L))
active(and(tt, T)) → mark(T)
mark(length(X)) → active(length(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(N))) → mark(isNat(N))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
mark(nil) → active(nil)

(108) Obligation:

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

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

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

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

(109) DependencyGraphProof (EQUIVALENT transformation)

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