Termination w.r.t. Q proof of /tmp/tmp8z3b7d/Ex5_DLMMU04

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 QDPOrderProof (⇔)

    ↳9 QDP

    ↳10 PisEmptyProof (⇔)

    ↳11 TRUE

  ↳12 QDP

    ↳13 QDPOrderProof (⇔)

    ↳14 QDP

    ↳15 QDPOrderProof (⇔)

    ↳16 QDP

    ↳17 PisEmptyProof (⇔)

    ↳18 TRUE

  ↳19 QDP

    ↳20 QDPOrderProof (⇔)

    ↳21 QDP

    ↳22 QDPOrderProof (⇔)

    ↳23 QDP

    ↳24 QDPOrderProof (⇔)

    ↳25 QDP

    ↳26 PisEmptyProof (⇔)

    ↳27 TRUE

  ↳28 QDP

    ↳29 QDPOrderProof (⇔)

    ↳30 QDP

    ↳31 QDPOrderProof (⇔)

    ↳32 QDP

    ↳33 QDPOrderProof (⇔)

    ↳34 QDP

    ↳35 PisEmptyProof (⇔)

    ↳36 TRUE

  ↳37 QDP

    ↳38 QDPOrderProof (⇔)

    ↳39 QDP

    ↳40 QDPOrderProof (⇔)

    ↳41 QDP

    ↳42 QDPOrderProof (⇔)

    ↳43 QDP

    ↳44 PisEmptyProof (⇔)

    ↳45 TRUE

  ↳46 QDP

    ↳47 QDPOrderProof (⇔)

    ↳48 QDP

    ↳49 QDPOrderProof (⇔)

    ↳50 QDP

    ↳51 PisEmptyProof (⇔)

    ↳52 TRUE

  ↳53 QDP

    ↳54 QDPOrderProof (⇔)

    ↳55 QDP

    ↳56 QDPOrderProof (⇔)

    ↳57 QDP

    ↳58 PisEmptyProof (⇔)

    ↳59 TRUE

  ↳60 QDP

    ↳61 QDPOrderProof (⇔)

    ↳62 QDP

    ↳63 QDPOrderProof (⇔)

    ↳64 QDP

    ↳65 PisEmptyProof (⇔)

    ↳66 TRUE

  ↳67 QDP

    ↳68 QDPOrderProof (⇔)

    ↳69 QDP

    ↳70 QDPOrderProof (⇔)

    ↳71 QDP

    ↳72 QDPOrderProof (⇔)

    ↳73 QDP

    ↳74 QDPOrderProof (⇔)

    ↳75 QDP

    ↳76 PisEmptyProof (⇔)

    ↳77 TRUE

  ↳78 QDP

    ↳79 QDPOrderProof (⇔)

    ↳80 QDP

    ↳81 QDPOrderProof (⇔)

    ↳82 QDP

    ↳83 QDPOrderProof (⇔)

    ↳84 QDP

    ↳85 PisEmptyProof (⇔)

    ↳86 TRUE

  ↳87 QDP

(0) Obligation:

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

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(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(pairNs) → CONS(0, incr(oddNs))
ACTIVE(pairNs) → INCR(oddNs)
ACTIVE(oddNs) → INCR(pairNs)
ACTIVE(incr(cons(X, XS))) → CONS(s(X), incr(XS))
ACTIVE(incr(cons(X, XS))) → S(X)
ACTIVE(incr(cons(X, XS))) → INCR(XS)
ACTIVE(take(s(N), cons(X, XS))) → CONS(X, take(N, XS))
ACTIVE(take(s(N), cons(X, XS))) → TAKE(N, XS)
ACTIVE(zip(cons(X, XS), cons(Y, YS))) → CONS(pair(X, Y), zip(XS, YS))
ACTIVE(zip(cons(X, XS), cons(Y, YS))) → PAIR(X, Y)
ACTIVE(zip(cons(X, XS), cons(Y, YS))) → ZIP(XS, YS)
ACTIVE(repItems(cons(X, XS))) → CONS(X, cons(X, repItems(XS)))
ACTIVE(repItems(cons(X, XS))) → CONS(X, repItems(XS))
ACTIVE(repItems(cons(X, XS))) → REPITEMS(XS)
ACTIVE(cons(X1, X2)) → CONS(active(X1), X2)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(incr(X)) → INCR(active(X))
ACTIVE(incr(X)) → ACTIVE(X)
ACTIVE(s(X)) → S(active(X))
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(take(X1, X2)) → TAKE(active(X1), X2)
ACTIVE(take(X1, X2)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → TAKE(X1, active(X2))
ACTIVE(take(X1, X2)) → ACTIVE(X2)
ACTIVE(zip(X1, X2)) → ZIP(active(X1), X2)
ACTIVE(zip(X1, X2)) → ACTIVE(X1)
ACTIVE(zip(X1, X2)) → ZIP(X1, active(X2))
ACTIVE(zip(X1, X2)) → ACTIVE(X2)
ACTIVE(pair(X1, X2)) → PAIR(active(X1), X2)
ACTIVE(pair(X1, X2)) → ACTIVE(X1)
ACTIVE(pair(X1, X2)) → PAIR(X1, active(X2))
ACTIVE(pair(X1, X2)) → ACTIVE(X2)
ACTIVE(tail(X)) → TAIL(active(X))
ACTIVE(tail(X)) → ACTIVE(X)
ACTIVE(repItems(X)) → REPITEMS(active(X))
ACTIVE(repItems(X)) → ACTIVE(X)
CONS(mark(X1), X2) → CONS(X1, X2)
INCR(mark(X)) → INCR(X)
S(mark(X)) → S(X)
TAKE(mark(X1), X2) → TAKE(X1, X2)
TAKE(X1, mark(X2)) → TAKE(X1, X2)
ZIP(mark(X1), X2) → ZIP(X1, X2)
ZIP(X1, mark(X2)) → ZIP(X1, X2)
PAIR(mark(X1), X2) → PAIR(X1, X2)
PAIR(X1, mark(X2)) → PAIR(X1, X2)
TAIL(mark(X)) → TAIL(X)
REPITEMS(mark(X)) → REPITEMS(X)
PROPER(cons(X1, X2)) → CONS(proper(X1), proper(X2))
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(incr(X)) → INCR(proper(X))
PROPER(incr(X)) → PROPER(X)
PROPER(s(X)) → S(proper(X))
PROPER(s(X)) → PROPER(X)
PROPER(take(X1, X2)) → TAKE(proper(X1), proper(X2))
PROPER(take(X1, X2)) → PROPER(X1)
PROPER(take(X1, X2)) → PROPER(X2)
PROPER(zip(X1, X2)) → ZIP(proper(X1), proper(X2))
PROPER(zip(X1, X2)) → PROPER(X1)
PROPER(zip(X1, X2)) → PROPER(X2)
PROPER(pair(X1, X2)) → PAIR(proper(X1), proper(X2))
PROPER(pair(X1, X2)) → PROPER(X1)
PROPER(pair(X1, X2)) → PROPER(X2)
PROPER(tail(X)) → TAIL(proper(X))
PROPER(tail(X)) → PROPER(X)
PROPER(repItems(X)) → REPITEMS(proper(X))
PROPER(repItems(X)) → PROPER(X)
CONS(ok(X1), ok(X2)) → CONS(X1, X2)
INCR(ok(X)) → INCR(X)
S(ok(X)) → S(X)
TAKE(ok(X1), ok(X2)) → TAKE(X1, X2)
ZIP(ok(X1), ok(X2)) → ZIP(X1, X2)
PAIR(ok(X1), ok(X2)) → PAIR(X1, X2)
TAIL(ok(X)) → TAIL(X)
REPITEMS(ok(X)) → REPITEMS(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(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(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 35 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

REPITEMS(ok(X)) → REPITEMS(X)
REPITEMS(mark(X)) → REPITEMS(X)

The TRS R consists of the following rules:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(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.

REPITEMS(mark(X)) → REPITEMS(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
REPITEMS(x1) = x1
ok(x1) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
pairNs = pairNs
cons(x1, x2) = cons(x1, x2)
0 = 0
incr(x1) = incr(x1)
oddNs = oddNs
s(x1) = s(x1)
take(x1, x2) = take(x1, x2)
nil = nil
zip(x1, x2) = zip(x1, x2)
pair(x1, x2) = pair(x1, x2)
tail(x1) = x1
repItems(x1) = repItems(x1)
proper(x1) = x1
top(x1) = top

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

[active₁, cons₂] > pairNs > 0 > [mark₁, incr₁, oddNs]
[active₁, cons₂] > [s₁, take₂] > [mark₁, incr₁, oddNs]
[active₁, cons₂] > nil > [mark₁, incr₁, oddNs]
[active₁, cons₂] > zip₂ > pair₂ > [mark₁, incr₁, oddNs]
[active₁, cons₂] > repItems₁ > [mark₁, incr₁, oddNs]
top > [mark₁, incr₁, oddNs]

Status:

mark₁: [1]
active₁: [1]
pairNs: []
cons₂: [1,2]
0: []
incr₁: [1]
oddNs: []
s₁: [1]
take₂: [2,1]
nil: []
zip₂: [2,1]
pair₂: [2,1]
repItems₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(7) Obligation:

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

REPITEMS(ok(X)) → REPITEMS(X)

The TRS R consists of the following rules:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(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) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

REPITEMS(ok(X)) → REPITEMS(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
REPITEMS(x1) = REPITEMS(x1)
ok(x1) = ok(x1)
active(x1) = x1
pairNs = pairNs
mark(x1) = mark
cons(x1, x2) = cons(x2)
0 = 0
incr(x1) = x1
oddNs = oddNs
s(x1) = x1
take(x1, x2) = x1
nil = nil
zip(x1, x2) = x2
pair(x1, x2) = x1
tail(x1) = x1
repItems(x1) = x1
proper(x1) = proper(x1)
top(x1) = top

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

[REPITEMS₁, ok₁, cons₁, oddNs, proper₁] > [pairNs, 0] > nil > mark
[REPITEMS₁, ok₁, cons₁, oddNs, proper₁] > top > mark

Status:

REPITEMS₁: [1]
ok₁: [1]
pairNs: []
mark: []
cons₁: [1]
0: []
oddNs: []
nil: []
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(9) Obligation:

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

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(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.

(10) PisEmptyProof (EQUIVALENT transformation)

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

(11) TRUE

(12) Obligation:

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

TAIL(ok(X)) → TAIL(X)
TAIL(mark(X)) → TAIL(X)

The TRS R consists of the following rules:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(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) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAIL(mark(X)) → TAIL(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TAIL(x1) = x1
ok(x1) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
pairNs = pairNs
cons(x1, x2) = cons(x1, x2)
0 = 0
incr(x1) = incr(x1)
oddNs = oddNs
s(x1) = s(x1)
take(x1, x2) = take(x1, x2)
nil = nil
zip(x1, x2) = zip(x1, x2)
pair(x1, x2) = pair(x1, x2)
tail(x1) = x1
repItems(x1) = repItems(x1)
proper(x1) = x1
top(x1) = top

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

[active₁, cons₂] > pairNs > 0 > [mark₁, incr₁, oddNs]
[active₁, cons₂] > [s₁, take₂] > [mark₁, incr₁, oddNs]
[active₁, cons₂] > nil > [mark₁, incr₁, oddNs]
[active₁, cons₂] > zip₂ > pair₂ > [mark₁, incr₁, oddNs]
[active₁, cons₂] > repItems₁ > [mark₁, incr₁, oddNs]
top > [mark₁, incr₁, oddNs]

Status:

mark₁: [1]
active₁: [1]
pairNs: []
cons₂: [1,2]
0: []
incr₁: [1]
oddNs: []
s₁: [1]
take₂: [2,1]
nil: []
zip₂: [2,1]
pair₂: [2,1]
repItems₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(14) Obligation:

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

TAIL(ok(X)) → TAIL(X)

The TRS R consists of the following rules:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(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.

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAIL(ok(X)) → TAIL(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TAIL(x1) = TAIL(x1)
ok(x1) = ok(x1)
active(x1) = x1
pairNs = pairNs
mark(x1) = mark
cons(x1, x2) = cons(x2)
0 = 0
incr(x1) = x1
oddNs = oddNs
s(x1) = x1
take(x1, x2) = x1
nil = nil
zip(x1, x2) = x2
pair(x1, x2) = x1
tail(x1) = x1
repItems(x1) = x1
proper(x1) = proper(x1)
top(x1) = top

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

[TAIL₁, ok₁, cons₁, oddNs, proper₁] > [pairNs, 0] > nil > mark
[TAIL₁, ok₁, cons₁, oddNs, proper₁] > top > mark

Status:

TAIL₁: [1]
ok₁: [1]
pairNs: []
mark: []
cons₁: [1]
0: []
oddNs: []
nil: []
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(16) Obligation:

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

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(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.

(17) PisEmptyProof (EQUIVALENT transformation)

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

(18) TRUE

(19) Obligation:

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

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

The TRS R consists of the following rules:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(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.

(20) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PAIR(ok(X1), ok(X2)) → PAIR(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PAIR(x1, x2) = PAIR(x1, x2)
mark(x1) = x1
ok(x1) = ok(x1)
active(x1) = x1
pairNs = pairNs
cons(x1, x2) = x2
0 = 0
incr(x1) = x1
oddNs = oddNs
s(x1) = s(x1)
take(x1, x2) = x1
nil = nil
zip(x1, x2) = zip(x1)
pair(x1, x2) = x1
tail(x1) = x1
repItems(x1) = repItems(x1)
proper(x1) = proper(x1)
top(x1) = top

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

[repItems₁, proper₁] > [pairNs, oddNs] > [0, nil] > ok₁
[repItems₁, proper₁] > s₁ > ok₁
[repItems₁, proper₁] > zip₁ > [0, nil] > ok₁

Status:

PAIR₂: [2,1]
ok₁: [1]
pairNs: []
0: []
oddNs: []
s₁: [1]
nil: []
zip₁: [1]
repItems₁: [1]
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(21) Obligation:

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

PAIR(X1, mark(X2)) → PAIR(X1, X2)
PAIR(mark(X1), X2) → PAIR(X1, X2)

The TRS R consists of the following rules:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(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.

(22) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PAIR(mark(X1), X2) → PAIR(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PAIR(x1, x2) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
pairNs = pairNs
cons(x1, x2) = cons(x1, x2)
0 = 0
incr(x1) = incr(x1)
oddNs = oddNs
s(x1) = s(x1)
take(x1, x2) = take(x1, x2)
nil = nil
zip(x1, x2) = zip(x1, x2)
pair(x1, x2) = pair(x1, x2)
tail(x1) = tail(x1)
repItems(x1) = repItems(x1)
proper(x1) = x1
ok(x1) = x1
top(x1) = top

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

[active₁, cons₂, repItems₁] > [pairNs, incr₁] > 0 > mark₁ > top
[active₁, cons₂, repItems₁] > [pairNs, incr₁] > oddNs > mark₁ > top
[active₁, cons₂, repItems₁] > s₁ > take₂ > mark₁ > top
[active₁, cons₂, repItems₁] > nil > mark₁ > top
[active₁, cons₂, repItems₁] > [zip₂, pair₂] > mark₁ > top
[active₁, cons₂, repItems₁] > tail₁ > mark₁ > top

Status:

mark₁: [1]
active₁: [1]
pairNs: []
cons₂: [1,2]
0: []
incr₁: [1]
oddNs: []
s₁: [1]
take₂: [1,2]
nil: []
zip₂: [1,2]
pair₂: [1,2]
tail₁: [1]
repItems₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(23) Obligation:

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

PAIR(X1, mark(X2)) → PAIR(X1, X2)

The TRS R consists of the following rules:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(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.

(24) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PAIR(X1, mark(X2)) → PAIR(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PAIR(x1, x2) = PAIR(x2)
mark(x1) = mark(x1)
active(x1) = active(x1)
pairNs = pairNs
cons(x1, x2) = cons(x1, x2)
0 = 0
incr(x1) = incr(x1)
oddNs = oddNs
s(x1) = s(x1)
take(x1, x2) = take(x1, x2)
nil = nil
zip(x1, x2) = zip(x1, x2)
pair(x1, x2) = pair(x1, x2)
tail(x1) = x1
repItems(x1) = x1
proper(x1) = x1
ok(x1) = x1
top(x1) = top

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

[active₁, take₂, pair₂] > oddNs > pairNs > incr₁ > [PAIR₁, mark₁] > top
[active₁, take₂, pair₂] > s₁ > cons₂ > incr₁ > [PAIR₁, mark₁] > top
[active₁, take₂, pair₂] > zip₂ > cons₂ > incr₁ > [PAIR₁, mark₁] > top
[active₁, take₂, pair₂] > zip₂ > [0, nil] > [PAIR₁, mark₁] > top

Status:

PAIR₁: [1]
mark₁: [1]
active₁: [1]
pairNs: []
cons₂: [2,1]
0: []
incr₁: [1]
oddNs: []
s₁: [1]
take₂: [1,2]
nil: []
zip₂: [2,1]
pair₂: [1,2]
top: []

The following usable rules [FROCOS05] were oriented:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(25) Obligation:

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

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(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.

(26) PisEmptyProof (EQUIVALENT transformation)

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

(27) TRUE

(28) Obligation:

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

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

The TRS R consists of the following rules:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(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.

(29) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ZIP(ok(X1), ok(X2)) → ZIP(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ZIP(x1, x2) = ZIP(x1, x2)
mark(x1) = x1
ok(x1) = ok(x1)
active(x1) = x1
pairNs = pairNs
cons(x1, x2) = x2
0 = 0
incr(x1) = x1
oddNs = oddNs
s(x1) = s(x1)
take(x1, x2) = x1
nil = nil
zip(x1, x2) = zip(x1)
pair(x1, x2) = x1
tail(x1) = x1
repItems(x1) = repItems(x1)
proper(x1) = proper(x1)
top(x1) = top

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

[repItems₁, proper₁] > [pairNs, oddNs] > [0, nil] > ok₁
[repItems₁, proper₁] > s₁ > ok₁
[repItems₁, proper₁] > zip₁ > [0, nil] > ok₁

Status:

ZIP₂: [2,1]
ok₁: [1]
pairNs: []
0: []
oddNs: []
s₁: [1]
nil: []
zip₁: [1]
repItems₁: [1]
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(30) Obligation:

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

ZIP(X1, mark(X2)) → ZIP(X1, X2)
ZIP(mark(X1), X2) → ZIP(X1, X2)

The TRS R consists of the following rules:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(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.

(31) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ZIP(mark(X1), X2) → ZIP(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ZIP(x1, x2) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
pairNs = pairNs
cons(x1, x2) = cons(x1, x2)
0 = 0
incr(x1) = incr(x1)
oddNs = oddNs
s(x1) = s(x1)
take(x1, x2) = take(x1, x2)
nil = nil
zip(x1, x2) = zip(x1, x2)
pair(x1, x2) = pair(x1, x2)
tail(x1) = tail(x1)
repItems(x1) = x1
proper(x1) = x1
ok(x1) = ok
top(x1) = top

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

[active₁, nil] > 0 > [pairNs, s₁, ok, top] > [cons₂, zip₂, pair₂] > mark₁
[active₁, nil] > oddNs > incr₁ > [pairNs, s₁, ok, top] > [cons₂, zip₂, pair₂] > mark₁
[active₁, nil] > take₂ > [pairNs, s₁, ok, top] > [cons₂, zip₂, pair₂] > mark₁
[active₁, nil] > tail₁ > [pairNs, s₁, ok, top] > [cons₂, zip₂, pair₂] > mark₁

Status:

mark₁: [1]
active₁: [1]
pairNs: []
cons₂: [1,2]
0: []
incr₁: [1]
oddNs: []
s₁: [1]
take₂: [1,2]
nil: []
zip₂: [1,2]
pair₂: [1,2]
tail₁: [1]
ok: []
top: []

The following usable rules [FROCOS05] were oriented:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(32) Obligation:

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

ZIP(X1, mark(X2)) → ZIP(X1, X2)

The TRS R consists of the following rules:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(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.

(33) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ZIP(X1, mark(X2)) → ZIP(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ZIP(x1, x2) = ZIP(x2)
mark(x1) = mark(x1)
active(x1) = active(x1)
pairNs = pairNs
cons(x1, x2) = cons(x1, x2)
0 = 0
incr(x1) = incr(x1)
oddNs = oddNs
s(x1) = s(x1)
take(x1, x2) = take(x1, x2)
nil = nil
zip(x1, x2) = zip(x1, x2)
pair(x1, x2) = pair(x1, x2)
tail(x1) = tail(x1)
repItems(x1) = repItems(x1)
proper(x1) = x1
ok(x1) = x1
top(x1) = top

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

[active₁, take₂, pair₂] > oddNs > pairNs > incr₁ > [ZIP₁, mark₁, tail₁] > top
[active₁, take₂, pair₂] > s₁ > cons₂ > incr₁ > [ZIP₁, mark₁, tail₁] > top
[active₁, take₂, pair₂] > zip₂ > cons₂ > incr₁ > [ZIP₁, mark₁, tail₁] > top
[active₁, take₂, pair₂] > zip₂ > [0, nil] > [ZIP₁, mark₁, tail₁] > top
[active₁, take₂, pair₂] > repItems₁ > cons₂ > incr₁ > [ZIP₁, mark₁, tail₁] > top
[active₁, take₂, pair₂] > repItems₁ > [0, nil] > [ZIP₁, mark₁, tail₁] > top

Status:

ZIP₁: [1]
mark₁: [1]
active₁: [1]
pairNs: []
cons₂: [2,1]
0: []
incr₁: [1]
oddNs: []
s₁: [1]
take₂: [1,2]
nil: []
zip₂: [2,1]
pair₂: [1,2]
tail₁: [1]
repItems₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(34) Obligation:

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

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(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) PisEmptyProof (EQUIVALENT transformation)

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

(36) TRUE

(37) Obligation:

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

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

The TRS R consists of the following rules:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(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.

(38) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAKE(ok(X1), ok(X2)) → TAKE(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TAKE(x1, x2) = TAKE(x1, x2)
mark(x1) = x1
ok(x1) = ok(x1)
active(x1) = x1
pairNs = pairNs
cons(x1, x2) = x2
0 = 0
incr(x1) = x1
oddNs = oddNs
s(x1) = s(x1)
take(x1, x2) = x1
nil = nil
zip(x1, x2) = zip(x1)
pair(x1, x2) = x1
tail(x1) = x1
repItems(x1) = repItems(x1)
proper(x1) = proper(x1)
top(x1) = top

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

[repItems₁, proper₁] > [pairNs, oddNs] > [0, nil] > ok₁
[repItems₁, proper₁] > s₁ > ok₁
[repItems₁, proper₁] > zip₁ > [0, nil] > ok₁

Status:

TAKE₂: [2,1]
ok₁: [1]
pairNs: []
0: []
oddNs: []
s₁: [1]
nil: []
zip₁: [1]
repItems₁: [1]
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(39) Obligation:

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

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

The TRS R consists of the following rules:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(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.

(40) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAKE(mark(X1), X2) → TAKE(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TAKE(x1, x2) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
pairNs = pairNs
cons(x1, x2) = cons(x1, x2)
0 = 0
incr(x1) = incr(x1)
oddNs = oddNs
s(x1) = x1
take(x1, x2) = take(x1, x2)
nil = nil
zip(x1, x2) = zip(x1, x2)
pair(x1, x2) = pair(x1, x2)
tail(x1) = x1
repItems(x1) = x1
proper(x1) = proper(x1)
ok(x1) = ok
top(x1) = top

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

active₁ > [cons₂, proper₁] > [incr₁, take₂, nil, zip₂, pair₂, ok, top] > [mark₁, pairNs, 0]
active₁ > oddNs > [incr₁, take₂, nil, zip₂, pair₂, ok, top] > [mark₁, pairNs, 0]

Status:

mark₁: [1]
active₁: [1]
pairNs: []
cons₂: [1,2]
0: []
incr₁: [1]
oddNs: []
take₂: [1,2]
nil: []
zip₂: [2,1]
pair₂: [2,1]
proper₁: [1]
ok: []
top: []

The following usable rules [FROCOS05] were oriented:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(41) Obligation:

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

TAKE(X1, mark(X2)) → TAKE(X1, X2)

The TRS R consists of the following rules:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(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.

TAKE(X1, mark(X2)) → TAKE(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TAKE(x1, x2) = TAKE(x2)
mark(x1) = mark(x1)
active(x1) = active(x1)
pairNs = pairNs
cons(x1, x2) = cons(x1, x2)
0 = 0
incr(x1) = x1
oddNs = oddNs
s(x1) = x1
take(x1, x2) = take(x1, x2)
nil = nil
zip(x1, x2) = zip(x1, x2)
pair(x1, x2) = pair(x1, x2)
tail(x1) = x1
repItems(x1) = x1
proper(x1) = proper(x1)
ok(x1) = ok
top(x1) = top

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

active₁ > [pairNs, 0] > oddNs
active₁ > [pairNs, 0] > nil
active₁ > take₂ > cons₂ > [TAKE₁, mark₁] > [ok, top]
active₁ > take₂ > nil
active₁ > zip₂ > cons₂ > [TAKE₁, mark₁] > [ok, top]
active₁ > zip₂ > nil
active₁ > pair₂ > [TAKE₁, mark₁] > [ok, top]
proper₁ > [pairNs, 0] > oddNs
proper₁ > [pairNs, 0] > nil
proper₁ > take₂ > cons₂ > [TAKE₁, mark₁] > [ok, top]
proper₁ > take₂ > nil
proper₁ > zip₂ > cons₂ > [TAKE₁, mark₁] > [ok, top]
proper₁ > zip₂ > nil
proper₁ > pair₂ > [TAKE₁, mark₁] > [ok, top]

Status:

TAKE₁: [1]
mark₁: [1]
active₁: [1]
pairNs: []
cons₂: [1,2]
0: []
oddNs: []
take₂: [1,2]
nil: []
zip₂: [2,1]
pair₂: [2,1]
proper₁: [1]
ok: []
top: []

The following usable rules [FROCOS05] were oriented:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(43) Obligation:

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

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(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) PisEmptyProof (EQUIVALENT transformation)

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

(45) TRUE

(46) 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(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(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.

(47) 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) = x1
ok(x1) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
pairNs = pairNs
cons(x1, x2) = cons(x1, x2)
0 = 0
incr(x1) = incr(x1)
oddNs = oddNs
s(x1) = s(x1)
take(x1, x2) = take(x1, x2)
nil = nil
zip(x1, x2) = zip(x1, x2)
pair(x1, x2) = pair(x1, x2)
tail(x1) = x1
repItems(x1) = repItems(x1)
proper(x1) = x1
top(x1) = top

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

[active₁, cons₂] > pairNs > 0 > [mark₁, incr₁, oddNs]
[active₁, cons₂] > [s₁, take₂] > [mark₁, incr₁, oddNs]
[active₁, cons₂] > nil > [mark₁, incr₁, oddNs]
[active₁, cons₂] > zip₂ > pair₂ > [mark₁, incr₁, oddNs]
[active₁, cons₂] > repItems₁ > [mark₁, incr₁, oddNs]
top > [mark₁, incr₁, oddNs]

Status:

mark₁: [1]
active₁: [1]
pairNs: []
cons₂: [1,2]
0: []
incr₁: [1]
oddNs: []
s₁: [1]
take₂: [2,1]
nil: []
zip₂: [2,1]
pair₂: [2,1]
repItems₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(48) 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(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(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.

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) = S(x1)
ok(x1) = ok(x1)
active(x1) = x1
pairNs = pairNs
mark(x1) = mark
cons(x1, x2) = cons(x2)
0 = 0
incr(x1) = x1
oddNs = oddNs
s(x1) = x1
take(x1, x2) = x1
nil = nil
zip(x1, x2) = x2
pair(x1, x2) = x1
tail(x1) = x1
repItems(x1) = x1
proper(x1) = proper(x1)
top(x1) = top

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

[S₁, ok₁, cons₁, oddNs, proper₁] > [pairNs, 0] > nil > mark
[S₁, ok₁, cons₁, oddNs, proper₁] > top > mark

Status:

S₁: [1]
ok₁: [1]
pairNs: []
mark: []
cons₁: [1]
0: []
oddNs: []
nil: []
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(50) Obligation:

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

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(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) PisEmptyProof (EQUIVALENT transformation)

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

(52) TRUE

(53) Obligation:

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

INCR(ok(X)) → INCR(X)
INCR(mark(X)) → INCR(X)

The TRS R consists of the following rules:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(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.

(54) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

INCR(mark(X)) → INCR(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
INCR(x1) = x1
ok(x1) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
pairNs = pairNs
cons(x1, x2) = cons(x1, x2)
0 = 0
incr(x1) = incr(x1)
oddNs = oddNs
s(x1) = s(x1)
take(x1, x2) = take(x1, x2)
nil = nil
zip(x1, x2) = zip(x1, x2)
pair(x1, x2) = pair(x1, x2)
tail(x1) = x1
repItems(x1) = repItems(x1)
proper(x1) = x1
top(x1) = top

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

[active₁, cons₂] > pairNs > 0 > [mark₁, incr₁, oddNs]
[active₁, cons₂] > [s₁, take₂] > [mark₁, incr₁, oddNs]
[active₁, cons₂] > nil > [mark₁, incr₁, oddNs]
[active₁, cons₂] > zip₂ > pair₂ > [mark₁, incr₁, oddNs]
[active₁, cons₂] > repItems₁ > [mark₁, incr₁, oddNs]
top > [mark₁, incr₁, oddNs]

Status:

mark₁: [1]
active₁: [1]
pairNs: []
cons₂: [1,2]
0: []
incr₁: [1]
oddNs: []
s₁: [1]
take₂: [2,1]
nil: []
zip₂: [2,1]
pair₂: [2,1]
repItems₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(55) Obligation:

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

INCR(ok(X)) → INCR(X)

The TRS R consists of the following rules:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(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.

INCR(ok(X)) → INCR(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
INCR(x1) = INCR(x1)
ok(x1) = ok(x1)
active(x1) = x1
pairNs = pairNs
mark(x1) = mark
cons(x1, x2) = cons(x2)
0 = 0
incr(x1) = x1
oddNs = oddNs
s(x1) = x1
take(x1, x2) = x1
nil = nil
zip(x1, x2) = x2
pair(x1, x2) = x1
tail(x1) = x1
repItems(x1) = x1
proper(x1) = proper(x1)
top(x1) = top

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

[INCR₁, ok₁, cons₁, oddNs, proper₁] > [pairNs, 0] > nil > mark
[INCR₁, ok₁, cons₁, oddNs, proper₁] > top > mark

Status:

INCR₁: [1]
ok₁: [1]
pairNs: []
mark: []
cons₁: [1]
0: []
oddNs: []
nil: []
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(57) Obligation:

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

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(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) PisEmptyProof (EQUIVALENT transformation)

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

(59) TRUE

(60) 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(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(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.

(61) 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(x2)
ok(x1) = ok(x1)
mark(x1) = x1
active(x1) = active(x1)
pairNs = pairNs
cons(x1, x2) = cons(x1, x2)
0 = 0
incr(x1) = x1
oddNs = oddNs
s(x1) = x1
take(x1, x2) = take(x2)
nil = nil
zip(x1, x2) = zip(x2)
pair(x1, x2) = x2
tail(x1) = x1
repItems(x1) = x1
proper(x1) = proper(x1)
top(x1) = top

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

proper₁ > [active₁, take₁] > cons₂ > zip₁ > [ok₁, pairNs] > CONS₁
proper₁ > [active₁, take₁] > cons₂ > zip₁ > [ok₁, pairNs] > 0 > nil
proper₁ > [active₁, take₁] > cons₂ > zip₁ > [ok₁, pairNs] > oddNs
top > [active₁, take₁] > cons₂ > zip₁ > [ok₁, pairNs] > CONS₁
top > [active₁, take₁] > cons₂ > zip₁ > [ok₁, pairNs] > 0 > nil
top > [active₁, take₁] > cons₂ > zip₁ > [ok₁, pairNs] > oddNs

Status:

CONS₁: [1]
ok₁: [1]
active₁: [1]
pairNs: []
cons₂: [1,2]
0: []
oddNs: []
take₁: [1]
nil: []
zip₁: [1]
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(62) 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(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(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.

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) = CONS(x1)
mark(x1) = mark(x1)
active(x1) = active(x1)
pairNs = pairNs
cons(x1, x2) = cons(x1, x2)
0 = 0
incr(x1) = incr(x1)
oddNs = oddNs
s(x1) = s(x1)
take(x1, x2) = take(x1, x2)
nil = nil
zip(x1, x2) = zip(x1, x2)
pair(x1, x2) = pair(x1, x2)
tail(x1) = tail(x1)
repItems(x1) = x1
proper(x1) = x1
ok(x1) = x1
top(x1) = top

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

top > [active₁, zip₂] > [pairNs, oddNs] > cons₂ > [CONS₁, mark₁, incr₁, s₁]
top > [active₁, zip₂] > [pairNs, oddNs] > 0
top > [active₁, zip₂] > take₂ > cons₂ > [CONS₁, mark₁, incr₁, s₁]
top > [active₁, zip₂] > take₂ > nil
top > [active₁, zip₂] > pair₂ > [CONS₁, mark₁, incr₁, s₁]
top > [active₁, zip₂] > tail₁ > [CONS₁, mark₁, incr₁, s₁]

Status:

CONS₁: [1]
mark₁: [1]
active₁: [1]
pairNs: []
cons₂: [1,2]
0: []
incr₁: [1]
oddNs: []
s₁: [1]
take₂: [1,2]
nil: []
zip₂: [2,1]
pair₂: [2,1]
tail₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(64) Obligation:

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

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(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) PisEmptyProof (EQUIVALENT transformation)

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

(66) TRUE

(67) 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(incr(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(take(X1, X2)) → PROPER(X1)
PROPER(take(X1, X2)) → PROPER(X2)
PROPER(zip(X1, X2)) → PROPER(X1)
PROPER(zip(X1, X2)) → PROPER(X2)
PROPER(pair(X1, X2)) → PROPER(X1)
PROPER(pair(X1, X2)) → PROPER(X2)
PROPER(tail(X)) → PROPER(X)
PROPER(repItems(X)) → PROPER(X)

The TRS R consists of the following rules:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(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.

(68) 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(take(X1, X2)) → PROPER(X1)
PROPER(take(X1, X2)) → PROPER(X2)
PROPER(zip(X1, X2)) → PROPER(X1)
PROPER(zip(X1, X2)) → PROPER(X2)
PROPER(pair(X1, X2)) → PROPER(X1)
PROPER(pair(X1, X2)) → PROPER(X2)

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)
incr(x1) = x1
s(x1) = x1
take(x1, x2) = take(x1, x2)
zip(x1, x2) = zip(x1, x2)
pair(x1, x2) = pair(x1, x2)
tail(x1) = x1
repItems(x1) = x1
active(x1) = x1
pairNs = pairNs
mark(x1) = mark
0 = 0
oddNs = oddNs
nil = nil
proper(x1) = x1
ok(x1) = ok
top(x1) = top

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

[pairNs, oddNs] > [ok, top] > zip₂ > pair₂ > [cons₂, take₂, mark]
0 > [ok, top] > zip₂ > pair₂ > [cons₂, take₂, mark]
nil > [ok, top] > zip₂ > pair₂ > [cons₂, take₂, mark]

Status:

PROPER₁: [1]
cons₂: [1,2]
take₂: [1,2]
zip₂: [2,1]
pair₂: [1,2]
pairNs: []
mark: []
0: []
oddNs: []
nil: []
ok: []
top: []

The following usable rules [FROCOS05] were oriented:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(69) Obligation:

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

PROPER(incr(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(tail(X)) → PROPER(X)
PROPER(repItems(X)) → PROPER(X)

The TRS R consists of the following rules:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(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.

(70) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(tail(X)) → PROPER(X)
PROPER(repItems(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1) = x1
incr(x1) = x1
s(x1) = x1
tail(x1) = tail(x1)
repItems(x1) = repItems(x1)
active(x1) = x1
pairNs = pairNs
mark(x1) = mark
cons(x1, x2) = x2
0 = 0
oddNs = oddNs
take(x1, x2) = x1
nil = nil
zip(x1, x2) = x2
pair(x1, x2) = x2
proper(x1) = proper(x1)
ok(x1) = ok
top(x1) = top

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

oddNs > pairNs > [mark, 0]
oddNs > ok > top > [mark, 0]
proper₁ > tail₁ > ok > top > [mark, 0]
proper₁ > repItems₁ > nil > ok > top > [mark, 0]
proper₁ > pairNs > [mark, 0]

Status:

tail₁: [1]
repItems₁: [1]
pairNs: []
mark: []
0: []
oddNs: []
nil: []
proper₁: [1]
ok: []
top: []

The following usable rules [FROCOS05] were oriented:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(71) Obligation:

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

PROPER(incr(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)

The TRS R consists of the following rules:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(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.

(72) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(s(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)
incr(x1) = x1
s(x1) = s(x1)
active(x1) = active(x1)
pairNs = pairNs
mark(x1) = mark
cons(x1, x2) = cons(x1, x2)
0 = 0
oddNs = oddNs
take(x1, x2) = take(x1, x2)
nil = nil
zip(x1, x2) = zip(x1, x2)
pair(x1, x2) = pair
tail(x1) = x1
repItems(x1) = repItems(x1)
proper(x1) = x1
ok(x1) = x1
top(x1) = top

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

pair > [active₁, repItems₁] > [mark, cons₂, oddNs] > pairNs > [PROPER₁, s₁, take₂, zip₂, top]
pair > [active₁, repItems₁] > 0 > nil > [PROPER₁, s₁, take₂, zip₂, top]

Status:

PROPER₁: [1]
s₁: [1]
active₁: [1]
pairNs: []
mark: []
cons₂: [2,1]
0: []
oddNs: []
take₂: [1,2]
nil: []
zip₂: [1,2]
pair: []
repItems₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(73) Obligation:

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

PROPER(incr(X)) → PROPER(X)

The TRS R consists of the following rules:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(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.

(74) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(incr(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)
incr(x1) = incr(x1)
active(x1) = x1
pairNs = pairNs
mark(x1) = mark
cons(x1, x2) = cons(x1)
0 = 0
oddNs = oddNs
s(x1) = x1
take(x1, x2) = x1
nil = nil
zip(x1, x2) = x2
pair(x1, x2) = pair(x1)
tail(x1) = tail(x1)
repItems(x1) = repItems
proper(x1) = proper(x1)
ok(x1) = ok
top(x1) = top

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

0 > [nil, proper₁] > [pairNs, oddNs] > [mark, cons₁, tail₁]
0 > [nil, proper₁] > [ok, top] > incr₁ > PROPER₁ > [mark, cons₁, tail₁]
0 > [nil, proper₁] > [ok, top] > pair₁ > [mark, cons₁, tail₁]
repItems > [nil, proper₁] > [pairNs, oddNs] > [mark, cons₁, tail₁]
repItems > [nil, proper₁] > [ok, top] > incr₁ > PROPER₁ > [mark, cons₁, tail₁]
repItems > [nil, proper₁] > [ok, top] > pair₁ > [mark, cons₁, tail₁]

Status:

PROPER₁: [1]
incr₁: [1]
pairNs: []
mark: []
cons₁: [1]
0: []
oddNs: []
nil: []
pair₁: [1]
tail₁: [1]
repItems: []
proper₁: [1]
ok: []
top: []

The following usable rules [FROCOS05] were oriented:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(75) Obligation:

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

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(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.

(76) PisEmptyProof (EQUIVALENT transformation)

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

(77) TRUE

(78) Obligation:

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

ACTIVE(incr(X)) → ACTIVE(X)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(take(X1, X2)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → ACTIVE(X2)
ACTIVE(zip(X1, X2)) → ACTIVE(X1)
ACTIVE(zip(X1, X2)) → ACTIVE(X2)
ACTIVE(pair(X1, X2)) → ACTIVE(X1)
ACTIVE(pair(X1, X2)) → ACTIVE(X2)
ACTIVE(tail(X)) → ACTIVE(X)
ACTIVE(repItems(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(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.

(79) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(take(X1, X2)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → ACTIVE(X2)
ACTIVE(zip(X1, X2)) → ACTIVE(X1)
ACTIVE(zip(X1, X2)) → ACTIVE(X2)
ACTIVE(pair(X1, X2)) → ACTIVE(X1)
ACTIVE(pair(X1, X2)) → ACTIVE(X2)
ACTIVE(tail(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1) = ACTIVE(x1)
incr(x1) = x1
cons(x1, x2) = x1
s(x1) = x1
take(x1, x2) = take(x1, x2)
zip(x1, x2) = zip(x1, x2)
pair(x1, x2) = pair(x1, x2)
tail(x1) = tail(x1)
repItems(x1) = x1
active(x1) = x1
pairNs = pairNs
mark(x1) = mark
0 = 0
oddNs = oddNs
nil = nil
proper(x1) = x1
ok(x1) = x1
top(x1) = top

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

take₂ > ACTIVE₁ > [mark, oddNs, top]
take₂ > nil > [mark, oddNs, top]
[zip₂, pair₂] > ACTIVE₁ > [mark, oddNs, top]
[zip₂, pair₂] > nil > [mark, oddNs, top]
tail₁ > ACTIVE₁ > [mark, oddNs, top]
pairNs > [mark, oddNs, top]
0 > nil > [mark, oddNs, top]

Status:

ACTIVE₁: [1]
take₂: [1,2]
zip₂: [1,2]
pair₂: [2,1]
tail₁: [1]
pairNs: []
mark: []
0: []
oddNs: []
nil: []
top: []

The following usable rules [FROCOS05] were oriented:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(80) Obligation:

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

ACTIVE(incr(X)) → ACTIVE(X)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(repItems(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(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.

(81) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(incr(X)) → ACTIVE(X)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(repItems(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
incr(x1) = incr(x1)
cons(x1, x2) = x1
s(x1) = s(x1)
repItems(x1) = repItems(x1)
active(x1) = x1
pairNs = pairNs
mark(x1) = mark
0 = 0
oddNs = oddNs
take(x1, x2) = x2
nil = nil
zip(x1, x2) = x2
pair(x1, x2) = x2
tail(x1) = x1
proper(x1) = x1
ok(x1) = ok
top(x1) = top

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

incr₁ > s₁ > [mark, ok, top]
repItems₁ > [0, nil] > [mark, ok, top]
oddNs > pairNs > [mark, ok, top]

Status:

incr₁: [1]
s₁: [1]
repItems₁: [1]
pairNs: []
mark: []
0: []
oddNs: []
nil: []
ok: []
top: []

The following usable rules [FROCOS05] were oriented:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(82) Obligation:

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

ACTIVE(cons(X1, X2)) → ACTIVE(X1)

The TRS R consists of the following rules:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(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.

(83) 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)

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)
active(x1) = active(x1)
pairNs = pairNs
mark(x1) = mark
0 = 0
incr(x1) = x1
oddNs = oddNs
s(x1) = x1
take(x1, x2) = take(x1)
nil = nil
zip(x1, x2) = zip
pair(x1, x2) = pair
tail(x1) = x1
repItems(x1) = repItems
proper(x1) = proper(x1)
ok(x1) = ok
top(x1) = top

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

[zip, pair] > [proper₁, ok, top] > oddNs > [cons₁, active₁, mark, take₁] > [pairNs, 0]
[zip, pair] > [proper₁, ok, top] > nil > [cons₁, active₁, mark, take₁] > [pairNs, 0]
repItems > [proper₁, ok, top] > oddNs > [cons₁, active₁, mark, take₁] > [pairNs, 0]
repItems > [proper₁, ok, top] > nil > [cons₁, active₁, mark, take₁] > [pairNs, 0]

Status:

ACTIVE₁: [1]
cons₁: [1]
active₁: [1]
pairNs: []
mark: []
0: []
oddNs: []
take₁: [1]
nil: []
zip: []
pair: []
repItems: []
proper₁: [1]
ok: []
top: []

The following usable rules [FROCOS05] were oriented:

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(84) Obligation:

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

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(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.

(85) PisEmptyProof (EQUIVALENT transformation)

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

(86) TRUE

(87) 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(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(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.