Termination w.r.t. Q proof of /tmp/tmp4OUUdC/LengthOfFiniteLists_nokinds

Termination w.r.t. Q of the given QTRS could not be shown:

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 AND

  ↳5 QDP

    ↳6 QDPOrderProof (⇔)

    ↳7 QDP

    ↳8 PisEmptyProof (⇔)

    ↳9 TRUE

  ↳10 QDP

    ↳11 QDPOrderProof (⇔)

    ↳12 QDP

    ↳13 PisEmptyProof (⇔)

    ↳14 TRUE

  ↳15 QDP

    ↳16 QDPOrderProof (⇔)

    ↳17 QDP

    ↳18 PisEmptyProof (⇔)

    ↳19 TRUE

  ↳20 QDP

    ↳21 QDPOrderProof (⇔)

    ↳22 QDP

    ↳23 QDPOrderProof (⇔)

    ↳24 QDP

    ↳25 PisEmptyProof (⇔)

    ↳26 TRUE

  ↳27 QDP

    ↳28 QDPOrderProof (⇔)

    ↳29 QDP

    ↳30 QDPOrderProof (⇔)

    ↳31 QDP

    ↳32 PisEmptyProof (⇔)

    ↳33 TRUE

  ↳34 QDP

    ↳35 QDPOrderProof (⇔)

    ↳36 QDP

    ↳37 QDPOrderProof (⇔)

    ↳38 QDP

    ↳39 PisEmptyProof (⇔)

    ↳40 TRUE

  ↳41 QDP

    ↳42 QDPOrderProof (⇔)

    ↳43 QDP

    ↳44 QDPOrderProof (⇔)

    ↳45 QDP

    ↳46 PisEmptyProof (⇔)

    ↳47 TRUE

  ↳48 QDP

    ↳49 QDPOrderProof (⇔)

    ↳50 QDP

    ↳51 QDPOrderProof (⇔)

    ↳52 QDP

    ↳53 PisEmptyProof (⇔)

    ↳54 TRUE

  ↳55 QDP

    ↳56 QDPOrderProof (⇔)

    ↳57 QDP

    ↳58 QDPOrderProof (⇔)

    ↳59 QDP

    ↳60 PisEmptyProof (⇔)

    ↳61 TRUE

  ↳62 QDP

    ↳63 QDPOrderProof (⇔)

    ↳64 QDP

    ↳65 QDPOrderProof (⇔)

    ↳66 QDP

    ↳67 QDPOrderProof (⇔)

    ↳68 QDP

    ↳69 PisEmptyProof (⇔)

    ↳70 TRUE

  ↳71 QDP

(0) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

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

(2) Obligation:

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

ACTIVE(zeros) → CONS(0, zeros)
ACTIVE(U11(tt, L)) → S(length(L))
ACTIVE(U11(tt, L)) → LENGTH(L)
ACTIVE(isNat(length(V1))) → ISNATLIST(V1)
ACTIVE(isNat(s(V1))) → ISNAT(V1)
ACTIVE(isNatIList(V)) → ISNATLIST(V)
ACTIVE(isNatIList(cons(V1, V2))) → AND(isNat(V1), isNatIList(V2))
ACTIVE(isNatIList(cons(V1, V2))) → ISNAT(V1)
ACTIVE(isNatIList(cons(V1, V2))) → ISNATILIST(V2)
ACTIVE(isNatList(cons(V1, V2))) → AND(isNat(V1), isNatList(V2))
ACTIVE(isNatList(cons(V1, V2))) → ISNAT(V1)
ACTIVE(isNatList(cons(V1, V2))) → ISNATLIST(V2)
ACTIVE(length(cons(N, L))) → U11¹(and(isNatList(L), isNat(N)), L)
ACTIVE(length(cons(N, L))) → AND(isNatList(L), isNat(N))
ACTIVE(length(cons(N, L))) → ISNATLIST(L)
ACTIVE(length(cons(N, L))) → ISNAT(N)
ACTIVE(cons(X1, X2)) → CONS(active(X1), X2)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(U11(X1, X2)) → U11¹(active(X1), X2)
ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(s(X)) → S(active(X))
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(length(X)) → LENGTH(active(X))
ACTIVE(length(X)) → ACTIVE(X)
ACTIVE(and(X1, X2)) → AND(active(X1), X2)
ACTIVE(and(X1, X2)) → ACTIVE(X1)
CONS(mark(X1), X2) → CONS(X1, X2)
U11¹(mark(X1), X2) → U11¹(X1, X2)
S(mark(X)) → S(X)
LENGTH(mark(X)) → LENGTH(X)
AND(mark(X1), X2) → AND(X1, X2)
PROPER(cons(X1, X2)) → CONS(proper(X1), proper(X2))
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(U11(X1, X2)) → U11¹(proper(X1), proper(X2))
PROPER(U11(X1, X2)) → PROPER(X1)
PROPER(U11(X1, X2)) → PROPER(X2)
PROPER(s(X)) → S(proper(X))
PROPER(s(X)) → PROPER(X)
PROPER(length(X)) → LENGTH(proper(X))
PROPER(length(X)) → PROPER(X)
PROPER(and(X1, X2)) → AND(proper(X1), proper(X2))
PROPER(and(X1, X2)) → PROPER(X1)
PROPER(and(X1, X2)) → PROPER(X2)
PROPER(isNat(X)) → ISNAT(proper(X))
PROPER(isNat(X)) → PROPER(X)
PROPER(isNatList(X)) → ISNATLIST(proper(X))
PROPER(isNatList(X)) → PROPER(X)
PROPER(isNatIList(X)) → ISNATILIST(proper(X))
PROPER(isNatIList(X)) → PROPER(X)
CONS(ok(X1), ok(X2)) → CONS(X1, X2)
U11¹(ok(X1), ok(X2)) → U11¹(X1, X2)
S(ok(X)) → S(X)
LENGTH(ok(X)) → LENGTH(X)
AND(ok(X1), ok(X2)) → AND(X1, X2)
ISNAT(ok(X)) → ISNAT(X)
ISNATLIST(ok(X)) → ISNATLIST(X)
ISNATILIST(ok(X)) → ISNATILIST(X)
TOP(mark(X)) → TOP(proper(X))
TOP(mark(X)) → PROPER(X)
TOP(ok(X)) → TOP(active(X))
TOP(ok(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 11 SCCs with 31 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

ISNATILIST(ok(X)) → ISNATILIST(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATILIST(ok(X)) → ISNATILIST(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNATILIST(x1) = x1
ok(x1) = ok(x1)
active(x1) = x1
zeros = zeros
mark(x1) = x1
cons(x1, x2) = x2
0 = 0
U11(x1, x2) = x2
tt = tt
s(x1) = x1
length(x1) = x1
and(x1, x2) = x2
isNat(x1) = x1
isNatList(x1) = x1
isNatIList(x1) = x1
nil = nil
proper(x1) = proper(x1)
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

nil > [ok₁, 0, proper₁] > [zeros, tt]
nil > [ok₁, 0, proper₁] > top

Status:

ok₁: [1]
zeros: []
0: []
tt: []
nil: []
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(7) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(8) PisEmptyProof (EQUIVALENT transformation)

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

(9) TRUE

(10) Obligation:

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

ISNATLIST(ok(X)) → ISNATLIST(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATLIST(ok(X)) → ISNATLIST(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNATLIST(x1) = x1
ok(x1) = ok(x1)
active(x1) = x1
zeros = zeros
mark(x1) = x1
cons(x1, x2) = x2
0 = 0
U11(x1, x2) = x2
tt = tt
s(x1) = x1
length(x1) = x1
and(x1, x2) = x2
isNat(x1) = x1
isNatList(x1) = x1
isNatIList(x1) = x1
nil = nil
proper(x1) = proper(x1)
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

nil > [ok₁, 0, proper₁] > [zeros, tt]
nil > [ok₁, 0, proper₁] > top

Status:

ok₁: [1]
zeros: []
0: []
tt: []
nil: []
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(12) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(13) PisEmptyProof (EQUIVALENT transformation)

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

(14) TRUE

(15) Obligation:

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

ISNAT(ok(X)) → ISNAT(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(16) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNAT(ok(X)) → ISNAT(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNAT(x1) = x1
ok(x1) = ok(x1)
active(x1) = x1
zeros = zeros
mark(x1) = x1
cons(x1, x2) = x2
0 = 0
U11(x1, x2) = x2
tt = tt
s(x1) = x1
length(x1) = x1
and(x1, x2) = x2
isNat(x1) = x1
isNatList(x1) = x1
isNatIList(x1) = x1
nil = nil
proper(x1) = proper(x1)
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

nil > [ok₁, 0, proper₁] > [zeros, tt]
nil > [ok₁, 0, proper₁] > top

Status:

ok₁: [1]
zeros: []
0: []
tt: []
nil: []
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(17) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(18) PisEmptyProof (EQUIVALENT transformation)

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

(19) TRUE

(20) Obligation:

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

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(21) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

AND(ok(X1), ok(X2)) → AND(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
AND(x1, x2) = AND(x1)
ok(x1) = ok(x1)
mark(x1) = x1
active(x1) = active(x1)
zeros = zeros
cons(x1, x2) = cons(x1, x2)
0 = 0
U11(x1, x2) = U11(x1, x2)
tt = tt
s(x1) = s(x1)
length(x1) = length(x1)
and(x1, x2) = x2
isNat(x1) = x1
isNatList(x1) = x1
isNatIList(x1) = isNatIList(x1)
nil = nil
proper(x1) = proper(x1)
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

active₁ > [cons₂, proper₁] > zeros > [ok₁, s₁]
active₁ > [cons₂, proper₁] > zeros > [tt, nil]
active₁ > [cons₂, proper₁] > U11₂ > length₁ > [ok₁, s₁]
active₁ > [cons₂, proper₁] > isNatIList₁ > [ok₁, s₁]
active₁ > [cons₂, proper₁] > isNatIList₁ > [tt, nil]
active₁ > 0 > [ok₁, s₁]
active₁ > 0 > [tt, nil]

Status:

AND₁: [1]
ok₁: [1]
active₁: [1]
zeros: []
cons₂: [1,2]
0: []
U11₂: [2,1]
tt: []
s₁: [1]
length₁: [1]
isNatIList₁: [1]
nil: []
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(22) Obligation:

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

AND(mark(X1), X2) → AND(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(23) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

AND(mark(X1), X2) → AND(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
AND(x1, x2) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
zeros = zeros
cons(x1, x2) = cons(x1, x2)
0 = 0
U11(x1, x2) = U11(x1, x2)
tt = tt
s(x1) = x1
length(x1) = length(x1)
and(x1, x2) = and(x1, x2)
isNat(x1) = x1
isNatList(x1) = x1
isNatIList(x1) = isNatIList(x1)
nil = nil
proper(x1) = x1
ok(x1) = x1
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

zeros > cons₂ > mark₁
zeros > [0, tt] > mark₁
isNatIList₁ > [active₁, U11₂, length₁, and₂] > cons₂ > mark₁
isNatIList₁ > [active₁, U11₂, length₁, and₂] > [0, tt] > mark₁
nil > [0, tt] > mark₁
top > [active₁, U11₂, length₁, and₂] > cons₂ > mark₁
top > [active₁, U11₂, length₁, and₂] > [0, tt] > mark₁

Status:

mark₁: [1]
active₁: [1]
zeros: []
cons₂: [2,1]
0: []
U11₂: [1,2]
tt: []
length₁: [1]
and₂: [2,1]
isNatIList₁: [1]
nil: []
top: []

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(24) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(25) PisEmptyProof (EQUIVALENT transformation)

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

(26) TRUE

(27) Obligation:

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

LENGTH(ok(X)) → LENGTH(X)
LENGTH(mark(X)) → LENGTH(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(28) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

LENGTH(mark(X)) → LENGTH(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
LENGTH(x1) = LENGTH(x1)
ok(x1) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
zeros = zeros
cons(x1, x2) = cons(x1, x2)
0 = 0
U11(x1, x2) = U11(x1, x2)
tt = tt
s(x1) = s(x1)
length(x1) = x1
and(x1, x2) = and(x1, x2)
isNat(x1) = isNat
isNatList(x1) = isNatList
isNatIList(x1) = isNatIList
nil = nil
proper(x1) = x1
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

LENGTH₁ > [zeros, tt, isNat, isNatList]
active₁ > [cons₂, U11₂, and₂] > mark₁ > [zeros, tt, isNat, isNatList]
active₁ > [cons₂, U11₂, and₂] > isNatIList > [zeros, tt, isNat, isNatList]
active₁ > 0 > [zeros, tt, isNat, isNatList]
active₁ > s₁ > mark₁ > [zeros, tt, isNat, isNatList]
nil > mark₁ > [zeros, tt, isNat, isNatList]
nil > 0 > [zeros, tt, isNat, isNatList]
top > [zeros, tt, isNat, isNatList]

Status:

LENGTH₁: [1]
mark₁: [1]
active₁: [1]
zeros: []
cons₂: [1,2]
0: []
U11₂: [2,1]
tt: []
s₁: [1]
and₂: [1,2]
isNat: []
isNatList: []
isNatIList: []
nil: []
top: []

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(29) Obligation:

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

LENGTH(ok(X)) → LENGTH(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(30) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

LENGTH(ok(X)) → LENGTH(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
LENGTH(x1) = x1
ok(x1) = ok(x1)
active(x1) = x1
zeros = zeros
mark(x1) = x1
cons(x1, x2) = x2
0 = 0
U11(x1, x2) = x2
tt = tt
s(x1) = x1
length(x1) = x1
and(x1, x2) = x2
isNat(x1) = x1
isNatList(x1) = x1
isNatIList(x1) = x1
nil = nil
proper(x1) = proper(x1)
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

nil > [ok₁, 0, proper₁] > [zeros, tt]
nil > [ok₁, 0, proper₁] > top

Status:

ok₁: [1]
zeros: []
0: []
tt: []
nil: []
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(31) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(32) PisEmptyProof (EQUIVALENT transformation)

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

(33) TRUE

(34) Obligation:

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

S(ok(X)) → S(X)
S(mark(X)) → S(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(35) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(mark(X)) → S(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S(x1) = S(x1)
ok(x1) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
zeros = zeros
cons(x1, x2) = cons(x1, x2)
0 = 0
U11(x1, x2) = U11(x1, x2)
tt = tt
s(x1) = s(x1)
length(x1) = x1
and(x1, x2) = and(x1, x2)
isNat(x1) = isNat
isNatList(x1) = isNatList
isNatIList(x1) = isNatIList
nil = nil
proper(x1) = x1
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

S₁ > [zeros, tt, isNat, isNatList]
active₁ > [cons₂, U11₂, and₂] > mark₁ > [zeros, tt, isNat, isNatList]
active₁ > [cons₂, U11₂, and₂] > isNatIList > [zeros, tt, isNat, isNatList]
active₁ > 0 > [zeros, tt, isNat, isNatList]
active₁ > s₁ > mark₁ > [zeros, tt, isNat, isNatList]
nil > mark₁ > [zeros, tt, isNat, isNatList]
nil > 0 > [zeros, tt, isNat, isNatList]
top > [zeros, tt, isNat, isNatList]

Status:

S₁: [1]
mark₁: [1]
active₁: [1]
zeros: []
cons₂: [1,2]
0: []
U11₂: [2,1]
tt: []
s₁: [1]
and₂: [1,2]
isNat: []
isNatList: []
isNatIList: []
nil: []
top: []

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(36) Obligation:

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

S(ok(X)) → S(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(37) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(ok(X)) → S(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S(x1) = x1
ok(x1) = ok(x1)
active(x1) = x1
zeros = zeros
mark(x1) = x1
cons(x1, x2) = x2
0 = 0
U11(x1, x2) = x2
tt = tt
s(x1) = x1
length(x1) = x1
and(x1, x2) = x2
isNat(x1) = x1
isNatList(x1) = x1
isNatIList(x1) = x1
nil = nil
proper(x1) = proper(x1)
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

nil > [ok₁, 0, proper₁] > [zeros, tt]
nil > [ok₁, 0, proper₁] > top

Status:

ok₁: [1]
zeros: []
0: []
tt: []
nil: []
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(38) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(39) PisEmptyProof (EQUIVALENT transformation)

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

(40) TRUE

(41) Obligation:

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

U11¹(ok(X1), ok(X2)) → U11¹(X1, X2)
U11¹(mark(X1), X2) → U11¹(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(42) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(ok(X1), ok(X2)) → U11¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2) = U11¹(x1)
ok(x1) = ok(x1)
mark(x1) = x1
active(x1) = active(x1)
zeros = zeros
cons(x1, x2) = cons(x1, x2)
0 = 0
U11(x1, x2) = U11(x1, x2)
tt = tt
s(x1) = s(x1)
length(x1) = length(x1)
and(x1, x2) = x2
isNat(x1) = x1
isNatList(x1) = x1
isNatIList(x1) = isNatIList(x1)
nil = nil
proper(x1) = proper(x1)
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

active₁ > [cons₂, proper₁] > zeros > [ok₁, s₁]
active₁ > [cons₂, proper₁] > zeros > [tt, nil]
active₁ > [cons₂, proper₁] > U11₂ > length₁ > [ok₁, s₁]
active₁ > [cons₂, proper₁] > isNatIList₁ > [ok₁, s₁]
active₁ > [cons₂, proper₁] > isNatIList₁ > [tt, nil]
active₁ > 0 > [ok₁, s₁]
active₁ > 0 > [tt, nil]

Status:

U11^1₁: [1]
ok₁: [1]
active₁: [1]
zeros: []
cons₂: [1,2]
0: []
U11₂: [2,1]
tt: []
s₁: [1]
length₁: [1]
isNatIList₁: [1]
nil: []
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(43) Obligation:

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

U11¹(mark(X1), X2) → U11¹(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(44) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(mark(X1), X2) → U11¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
zeros = zeros
cons(x1, x2) = cons(x1, x2)
0 = 0
U11(x1, x2) = U11(x1, x2)
tt = tt
s(x1) = x1
length(x1) = length(x1)
and(x1, x2) = and(x1, x2)
isNat(x1) = x1
isNatList(x1) = x1
isNatIList(x1) = isNatIList(x1)
nil = nil
proper(x1) = x1
ok(x1) = x1
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

zeros > cons₂ > mark₁
zeros > [0, tt] > mark₁
isNatIList₁ > [active₁, U11₂, length₁, and₂] > cons₂ > mark₁
isNatIList₁ > [active₁, U11₂, length₁, and₂] > [0, tt] > mark₁
nil > [0, tt] > mark₁
top > [active₁, U11₂, length₁, and₂] > cons₂ > mark₁
top > [active₁, U11₂, length₁, and₂] > [0, tt] > mark₁

Status:

mark₁: [1]
active₁: [1]
zeros: []
cons₂: [2,1]
0: []
U11₂: [1,2]
tt: []
length₁: [1]
and₂: [2,1]
isNatIList₁: [1]
nil: []
top: []

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(45) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(46) PisEmptyProof (EQUIVALENT transformation)

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

(47) TRUE

(48) Obligation:

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

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(49) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(ok(X1), ok(X2)) → CONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2) = CONS(x1)
ok(x1) = ok(x1)
mark(x1) = x1
active(x1) = active(x1)
zeros = zeros
cons(x1, x2) = cons(x1, x2)
0 = 0
U11(x1, x2) = U11(x1, x2)
tt = tt
s(x1) = s(x1)
length(x1) = length(x1)
and(x1, x2) = x2
isNat(x1) = x1
isNatList(x1) = x1
isNatIList(x1) = isNatIList(x1)
nil = nil
proper(x1) = proper(x1)
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

active₁ > [cons₂, proper₁] > zeros > [ok₁, s₁]
active₁ > [cons₂, proper₁] > zeros > [tt, nil]
active₁ > [cons₂, proper₁] > U11₂ > length₁ > [ok₁, s₁]
active₁ > [cons₂, proper₁] > isNatIList₁ > [ok₁, s₁]
active₁ > [cons₂, proper₁] > isNatIList₁ > [tt, nil]
active₁ > 0 > [ok₁, s₁]
active₁ > 0 > [tt, nil]

Status:

CONS₁: [1]
ok₁: [1]
active₁: [1]
zeros: []
cons₂: [1,2]
0: []
U11₂: [2,1]
tt: []
s₁: [1]
length₁: [1]
isNatIList₁: [1]
nil: []
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(50) Obligation:

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

CONS(mark(X1), X2) → CONS(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(51) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(mark(X1), X2) → CONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
zeros = zeros
cons(x1, x2) = cons(x1, x2)
0 = 0
U11(x1, x2) = U11(x1, x2)
tt = tt
s(x1) = x1
length(x1) = length(x1)
and(x1, x2) = and(x1, x2)
isNat(x1) = x1
isNatList(x1) = x1
isNatIList(x1) = isNatIList(x1)
nil = nil
proper(x1) = x1
ok(x1) = x1
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

zeros > cons₂ > mark₁
zeros > [0, tt] > mark₁
isNatIList₁ > [active₁, U11₂, length₁, and₂] > cons₂ > mark₁
isNatIList₁ > [active₁, U11₂, length₁, and₂] > [0, tt] > mark₁
nil > [0, tt] > mark₁
top > [active₁, U11₂, length₁, and₂] > cons₂ > mark₁
top > [active₁, U11₂, length₁, and₂] > [0, tt] > mark₁

Status:

mark₁: [1]
active₁: [1]
zeros: []
cons₂: [2,1]
0: []
U11₂: [1,2]
tt: []
length₁: [1]
and₂: [2,1]
isNatIList₁: [1]
nil: []
top: []

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(52) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(53) PisEmptyProof (EQUIVALENT transformation)

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

(54) TRUE

(55) Obligation:

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

PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(U11(X1, X2)) → PROPER(X1)
PROPER(U11(X1, X2)) → PROPER(X2)
PROPER(s(X)) → PROPER(X)
PROPER(length(X)) → PROPER(X)
PROPER(and(X1, X2)) → PROPER(X1)
PROPER(and(X1, X2)) → PROPER(X2)
PROPER(isNat(X)) → PROPER(X)
PROPER(isNatList(X)) → PROPER(X)
PROPER(isNatIList(X)) → PROPER(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(56) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(U11(X1, X2)) → PROPER(X1)
PROPER(U11(X1, X2)) → PROPER(X2)
PROPER(s(X)) → PROPER(X)
PROPER(length(X)) → PROPER(X)
PROPER(and(X1, X2)) → PROPER(X1)
PROPER(and(X1, X2)) → PROPER(X2)
PROPER(isNat(X)) → PROPER(X)
PROPER(isNatIList(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1) = PROPER(x1)
cons(x1, x2) = cons(x1, x2)
U11(x1, x2) = U11(x1, x2)
s(x1) = s(x1)
length(x1) = length(x1)
and(x1, x2) = and(x1, x2)
isNat(x1) = isNat(x1)
isNatList(x1) = x1
isNatIList(x1) = isNatIList(x1)
active(x1) = active(x1)
zeros = zeros
mark(x1) = mark
0 = 0
tt = tt
nil = nil
proper(x1) = proper(x1)
ok(x1) = x1
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

PROPER₁ > s₁
[active₁, zeros] > [isNatIList₁, tt] > length₁ > [and₂, isNat₁, mark] > cons₂ > s₁
[active₁, zeros] > [isNatIList₁, tt] > length₁ > [and₂, isNat₁, mark] > U11₂ > s₁
0 > [isNatIList₁, tt] > length₁ > [and₂, isNat₁, mark] > cons₂ > s₁
0 > [isNatIList₁, tt] > length₁ > [and₂, isNat₁, mark] > U11₂ > s₁
proper₁ > [isNatIList₁, tt] > length₁ > [and₂, isNat₁, mark] > cons₂ > s₁
proper₁ > [isNatIList₁, tt] > length₁ > [and₂, isNat₁, mark] > U11₂ > s₁
proper₁ > nil > s₁
top > s₁

Status:

PROPER₁: [1]
cons₂: [1,2]
U11₂: [2,1]
s₁: [1]
length₁: [1]
and₂: [1,2]
isNat₁: [1]
isNatIList₁: [1]
active₁: [1]
zeros: []
mark: []
0: []
tt: []
nil: []
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(57) Obligation:

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

PROPER(isNatList(X)) → PROPER(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(58) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(isNatList(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1) = PROPER(x1)
isNatList(x1) = isNatList(x1)
active(x1) = active
zeros = zeros
mark(x1) = mark
cons(x1, x2) = x1
0 = 0
U11(x1, x2) = U11
tt = tt
s(x1) = s
length(x1) = x1
and(x1, x2) = and
isNat(x1) = isNat
isNatIList(x1) = x1
nil = nil
proper(x1) = proper(x1)
ok(x1) = ok
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

[PROPER₁, isNatList₁, proper₁] > [zeros, 0] > [active, mark, U11, tt, s, and, nil, ok]
[PROPER₁, isNatList₁, proper₁] > isNat > [active, mark, U11, tt, s, and, nil, ok]
top > [active, mark, U11, tt, s, and, nil, ok]

Status:

PROPER₁: [1]
isNatList₁: [1]
active: []
zeros: []
mark: []
0: []
U11: []
tt: []
s: []
and: []
isNat: []
nil: []
proper₁: [1]
ok: []
top: []

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(59) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(60) PisEmptyProof (EQUIVALENT transformation)

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

(61) TRUE

(62) Obligation:

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

ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(length(X)) → ACTIVE(X)
ACTIVE(and(X1, X2)) → ACTIVE(X1)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(length(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1) = x1
U11(x1, x2) = U11(x1, x2)
cons(x1, x2) = x1
s(x1) = x1
length(x1) = length(x1)
and(x1, x2) = x1
active(x1) = active(x1)
zeros = zeros
mark(x1) = mark
0 = 0
tt = tt
isNat(x1) = isNat
isNatList(x1) = isNatList
isNatIList(x1) = x1
nil = nil
proper(x1) = x1
ok(x1) = ok
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

active₁ > U11₂ > [isNatList, nil, ok] > length₁ > mark
active₁ > U11₂ > [isNatList, nil, ok] > top > mark
active₁ > [zeros, 0] > tt > [isNatList, nil, ok] > length₁ > mark
active₁ > [zeros, 0] > tt > [isNatList, nil, ok] > top > mark
active₁ > isNat > tt > [isNatList, nil, ok] > length₁ > mark
active₁ > isNat > tt > [isNatList, nil, ok] > top > mark

Status:

U11₂: [2,1]
length₁: [1]
active₁: [1]
zeros: []
mark: []
0: []
tt: []
isNat: []
isNatList: []
nil: []
ok: []
top: []

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(64) Obligation:

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

ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(and(X1, X2)) → ACTIVE(X1)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(and(X1, X2)) → ACTIVE(X1)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1) = ACTIVE(x1)
cons(x1, x2) = cons(x1)
s(x1) = x1
and(x1, x2) = and(x1, x2)
active(x1) = active(x1)
zeros = zeros
mark(x1) = mark
0 = 0
U11(x1, x2) = U11
tt = tt
length(x1) = x1
isNat(x1) = isNat
isNatList(x1) = x1
isNatIList(x1) = isNatIList(x1)
nil = nil
proper(x1) = proper(x1)
ok(x1) = ok
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

ACTIVE₁ > top
0 > mark > proper₁ > cons₁ > isNatIList₁ > and₂ > [tt, nil, ok] > top
0 > mark > proper₁ > zeros > top
U11 > active₁ > mark > proper₁ > cons₁ > isNatIList₁ > and₂ > [tt, nil, ok] > top
U11 > active₁ > mark > proper₁ > zeros > top
isNat > mark > proper₁ > cons₁ > isNatIList₁ > and₂ > [tt, nil, ok] > top
isNat > mark > proper₁ > zeros > top

Status:

ACTIVE₁: [1]
cons₁: [1]
and₂: [2,1]
active₁: [1]
zeros: []
mark: []
0: []
U11: []
tt: []
isNat: []
isNatIList₁: [1]
nil: []
proper₁: [1]
ok: []
top: []

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(66) Obligation:

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

ACTIVE(s(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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(s(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1) = x1
s(x1) = s(x1)
active(x1) = active(x1)
zeros = zeros
mark(x1) = mark
cons(x1, x2) = x1
0 = 0
U11(x1, x2) = U11
tt = tt
length(x1) = length
and(x1, x2) = and(x1)
isNat(x1) = isNat
isNatList(x1) = isNatList
isNatIList(x1) = isNatIList
nil = nil
proper(x1) = proper(x1)
ok(x1) = ok
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

[zeros, mark, 0, U11, tt, length, isNat, isNatList, isNatIList] > ok > top > active₁ > s₁
[zeros, mark, 0, U11, tt, length, isNat, isNatList, isNatIList] > ok > top > active₁ > and₁
[zeros, mark, 0, U11, tt, length, isNat, isNatList, isNatIList] > ok > top > proper₁ > s₁
[zeros, mark, 0, U11, tt, length, isNat, isNatList, isNatIList] > ok > top > proper₁ > and₁
nil > ok > top > active₁ > s₁
nil > ok > top > active₁ > and₁
nil > ok > top > proper₁ > s₁
nil > ok > top > proper₁ > and₁

Status:

s₁: [1]
active₁: [1]
zeros: []
mark: []
0: []
U11: []
tt: []
length: []
and₁: [1]
isNat: []
isNatList: []
isNatIList: []
nil: []
proper₁: [1]
ok: []
top: []

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(68) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(69) PisEmptyProof (EQUIVALENT transformation)

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

(70) TRUE

(71) Obligation:

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

TOP(ok(X)) → TOP(active(X))
TOP(mark(X)) → TOP(proper(X))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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