Termination w.r.t. Q proof of /tmp/tmpYyEpQD/Ex3_3_25_Bor03

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

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 AND

  ↳5 QDP

    ↳6 QDPOrderProof (⇔)

    ↳7 QDP

    ↳8 QDPOrderProof (⇔)

    ↳9 QDP

    ↳10 PisEmptyProof (⇔)

    ↳11 TRUE

  ↳12 QDP

    ↳13 QDPOrderProof (⇔)

    ↳14 QDP

    ↳15 QDPOrderProof (⇔)

    ↳16 QDP

    ↳17 QDPOrderProof (⇔)

    ↳18 QDP

    ↳19 PisEmptyProof (⇔)

    ↳20 TRUE

  ↳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 PisEmptyProof (⇔)

    ↳34 TRUE

  ↳35 QDP

    ↳36 QDPOrderProof (⇔)

    ↳37 QDP

    ↳38 QDPOrderProof (⇔)

    ↳39 QDP

    ↳40 PisEmptyProof (⇔)

    ↳41 TRUE

  ↳42 QDP

    ↳43 QDPOrderProof (⇔)

    ↳44 QDP

    ↳45 QDPOrderProof (⇔)

    ↳46 QDP

    ↳47 QDPOrderProof (⇔)

    ↳48 QDP

    ↳49 PisEmptyProof (⇔)

    ↳50 TRUE

  ↳51 QDP

    ↳52 QDPOrderProof (⇔)

    ↳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 QDPOrderProof (⇔)

    ↳66 QDP

    ↳67 PisEmptyProof (⇔)

    ↳68 TRUE

  ↳69 QDP

    ↳70 QDPOrderProof (⇔)

    ↳71 QDP

    ↳72 QDPOrderProof (⇔)

    ↳73 QDP

    ↳74 PisEmptyProof (⇔)

    ↳75 TRUE

(0) Obligation:

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

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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(app(cons(X, XS), YS)) → CONS(X, app(XS, YS))
ACTIVE(app(cons(X, XS), YS)) → APP(XS, YS)
ACTIVE(from(X)) → CONS(X, from(s(X)))
ACTIVE(from(X)) → FROM(s(X))
ACTIVE(from(X)) → S(X)
ACTIVE(zWadr(cons(X, XS), cons(Y, YS))) → CONS(app(Y, cons(X, nil)), zWadr(XS, YS))
ACTIVE(zWadr(cons(X, XS), cons(Y, YS))) → APP(Y, cons(X, nil))
ACTIVE(zWadr(cons(X, XS), cons(Y, YS))) → CONS(X, nil)
ACTIVE(zWadr(cons(X, XS), cons(Y, YS))) → ZWADR(XS, YS)
ACTIVE(prefix(L)) → CONS(nil, zWadr(L, prefix(L)))
ACTIVE(prefix(L)) → ZWADR(L, prefix(L))
ACTIVE(app(X1, X2)) → APP(active(X1), X2)
ACTIVE(app(X1, X2)) → ACTIVE(X1)
ACTIVE(app(X1, X2)) → APP(X1, active(X2))
ACTIVE(app(X1, X2)) → ACTIVE(X2)
ACTIVE(cons(X1, X2)) → CONS(active(X1), X2)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(from(X)) → FROM(active(X))
ACTIVE(from(X)) → ACTIVE(X)
ACTIVE(s(X)) → S(active(X))
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(zWadr(X1, X2)) → ZWADR(active(X1), X2)
ACTIVE(zWadr(X1, X2)) → ACTIVE(X1)
ACTIVE(zWadr(X1, X2)) → ZWADR(X1, active(X2))
ACTIVE(zWadr(X1, X2)) → ACTIVE(X2)
ACTIVE(prefix(X)) → PREFIX(active(X))
ACTIVE(prefix(X)) → ACTIVE(X)
APP(mark(X1), X2) → APP(X1, X2)
APP(X1, mark(X2)) → APP(X1, X2)
CONS(mark(X1), X2) → CONS(X1, X2)
FROM(mark(X)) → FROM(X)
S(mark(X)) → S(X)
ZWADR(mark(X1), X2) → ZWADR(X1, X2)
ZWADR(X1, mark(X2)) → ZWADR(X1, X2)
PREFIX(mark(X)) → PREFIX(X)
PROPER(app(X1, X2)) → APP(proper(X1), proper(X2))
PROPER(app(X1, X2)) → PROPER(X1)
PROPER(app(X1, X2)) → PROPER(X2)
PROPER(cons(X1, X2)) → CONS(proper(X1), proper(X2))
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(from(X)) → FROM(proper(X))
PROPER(from(X)) → PROPER(X)
PROPER(s(X)) → S(proper(X))
PROPER(s(X)) → PROPER(X)
PROPER(zWadr(X1, X2)) → ZWADR(proper(X1), proper(X2))
PROPER(zWadr(X1, X2)) → PROPER(X1)
PROPER(zWadr(X1, X2)) → PROPER(X2)
PROPER(prefix(X)) → PREFIX(proper(X))
PROPER(prefix(X)) → PROPER(X)
APP(ok(X1), ok(X2)) → APP(X1, X2)
CONS(ok(X1), ok(X2)) → CONS(X1, X2)
FROM(ok(X)) → FROM(X)
S(ok(X)) → S(X)
ZWADR(ok(X1), ok(X2)) → ZWADR(X1, X2)
PREFIX(ok(X)) → PREFIX(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(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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 9 SCCs with 27 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

PREFIX(ok(X)) → PREFIX(X)
PREFIX(mark(X)) → PREFIX(X)

The TRS R consists of the following rules:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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.

PREFIX(mark(X)) → PREFIX(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PREFIX(x1) = x1
ok(x1) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
app(x1, x2) = app(x1, x2)
nil = nil
cons(x1, x2) = x1
from(x1) = x1
s(x1) = x1
zWadr(x1, x2) = zWadr(x1, x2)
prefix(x1) = prefix(x1)
proper(x1) = proper(x1)
top(x1) = top

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

[active₁, zWadr₂, prefix₁, proper₁] > app₂ > mark₁ > [nil, top]

Status:

mark₁: [1]
active₁: [1]
app₂: [2,1]
nil: []
zWadr₂: [2,1]
prefix₁: [1]
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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:

PREFIX(ok(X)) → PREFIX(X)

The TRS R consists of the following rules:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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.

PREFIX(ok(X)) → PREFIX(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PREFIX(x1) = PREFIX(x1)
ok(x1) = ok(x1)
active(x1) = active(x1)
app(x1, x2) = app(x1)
nil = nil
mark(x1) = mark
cons(x1, x2) = cons(x1, x2)
from(x1) = from(x1)
s(x1) = x1
zWadr(x1, x2) = zWadr(x1)
prefix(x1) = x1
proper(x1) = proper(x1)
top(x1) = top

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

[active₁, cons₂, proper₁] > zWadr₁ > [ok₁, app₁, from₁] > mark > top
nil > [ok₁, app₁, from₁] > mark > top

Status:

PREFIX₁: [1]
ok₁: [1]
active₁: [1]
app₁: [1]
nil: []
mark: []
cons₂: [1,2]
from₁: [1]
zWadr₁: [1]
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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:

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

The TRS R consists of the following rules:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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.

ZWADR(ok(X1), ok(X2)) → ZWADR(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ZWADR(x1, x2) = ZWADR(x1, x2)
mark(x1) = x1
ok(x1) = ok(x1)
active(x1) = x1
app(x1, x2) = x2
nil = nil
cons(x1, x2) = x2
from(x1) = from(x1)
s(x1) = x1
zWadr(x1, x2) = x2
prefix(x1) = prefix(x1)
proper(x1) = proper(x1)
top(x1) = top

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

proper₁ > from₁ > ok₁ > ZWADR₂ > [nil, top]
proper₁ > prefix₁ > ok₁ > ZWADR₂ > [nil, top]

Status:

ZWADR₂: [2,1]
ok₁: [1]
nil: []
from₁: [1]
prefix₁: [1]
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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:

ZWADR(X1, mark(X2)) → ZWADR(X1, X2)
ZWADR(mark(X1), X2) → ZWADR(X1, X2)

The TRS R consists of the following rules:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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.

ZWADR(mark(X1), X2) → ZWADR(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ZWADR(x1, x2) = ZWADR(x1)
mark(x1) = mark(x1)
active(x1) = active(x1)
app(x1, x2) = app(x1, x2)
nil = nil
cons(x1, x2) = x1
from(x1) = x1
s(x1) = s(x1)
zWadr(x1, x2) = zWadr(x1, x2)
prefix(x1) = x1
proper(x1) = x1
ok(x1) = ok
top(x1) = top

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

[active₁, app₂, nil, zWadr₂] > [mark₁, s₁] > ok

Status:

ZWADR₁: [1]
mark₁: [1]
active₁: [1]
app₂: [1,2]
nil: []
s₁: [1]
zWadr₂: [2,1]
ok: []
top: []

The following usable rules [FROCOS05] were oriented:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(16) Obligation:

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

ZWADR(X1, mark(X2)) → ZWADR(X1, X2)

The TRS R consists of the following rules:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ZWADR(X1, mark(X2)) → ZWADR(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ZWADR(x1, x2) = ZWADR(x2)
mark(x1) = mark(x1)
active(x1) = active(x1)
app(x1, x2) = app(x1, x2)
nil = nil
cons(x1, x2) = x1
from(x1) = x1
s(x1) = s(x1)
zWadr(x1, x2) = zWadr(x1, x2)
prefix(x1) = prefix(x1)
proper(x1) = proper(x1)
ok(x1) = ok
top(x1) = top

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

active₁ > [app₂, proper₁] > zWadr₂ > [mark₁, s₁, ok, top] > [ZWADR₁, nil]
active₁ > [app₂, proper₁] > prefix₁ > [mark₁, s₁, ok, top] > [ZWADR₁, nil]

Status:

ZWADR₁: [1]
mark₁: [1]
active₁: [1]
app₂: [2,1]
nil: []
s₁: [1]
zWadr₂: [2,1]
prefix₁: [1]
proper₁: [1]
ok: []
top: []

The following usable rules [FROCOS05] were oriented:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(18) Obligation:

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

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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.

(19) PisEmptyProof (EQUIVALENT transformation)

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

(20) TRUE

(21) 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(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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.

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)
app(x1, x2) = app(x1, x2)
nil = nil
cons(x1, x2) = x1
from(x1) = x1
s(x1) = x1
zWadr(x1, x2) = zWadr(x1, x2)
prefix(x1) = prefix(x1)
proper(x1) = proper(x1)
top(x1) = top

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

[active₁, zWadr₂, prefix₁, proper₁] > app₂ > mark₁ > [nil, top]

Status:

mark₁: [1]
active₁: [1]
app₂: [2,1]
nil: []
zWadr₂: [2,1]
prefix₁: [1]
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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:

S(ok(X)) → S(X)

The TRS R consists of the following rules:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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.

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) = active(x1)
app(x1, x2) = app(x1)
nil = nil
mark(x1) = mark
cons(x1, x2) = cons(x1, x2)
from(x1) = from(x1)
s(x1) = x1
zWadr(x1, x2) = zWadr(x1)
prefix(x1) = x1
proper(x1) = proper(x1)
top(x1) = top

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

[active₁, cons₂, proper₁] > zWadr₁ > [ok₁, app₁, from₁] > mark > top
nil > [ok₁, app₁, from₁] > mark > top

Status:

S₁: [1]
ok₁: [1]
active₁: [1]
app₁: [1]
nil: []
mark: []
cons₂: [1,2]
from₁: [1]
zWadr₁: [1]
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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:

FROM(ok(X)) → FROM(X)
FROM(mark(X)) → FROM(X)

The TRS R consists of the following rules:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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.

FROM(mark(X)) → FROM(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
FROM(x1) = x1
ok(x1) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
app(x1, x2) = app(x1, x2)
nil = nil
cons(x1, x2) = x1
from(x1) = x1
s(x1) = x1
zWadr(x1, x2) = zWadr(x1, x2)
prefix(x1) = prefix(x1)
proper(x1) = proper(x1)
top(x1) = top

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

[active₁, zWadr₂, prefix₁, proper₁] > app₂ > mark₁ > [nil, top]

Status:

mark₁: [1]
active₁: [1]
app₂: [2,1]
nil: []
zWadr₂: [2,1]
prefix₁: [1]
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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:

FROM(ok(X)) → FROM(X)

The TRS R consists of the following rules:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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.

FROM(ok(X)) → FROM(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
FROM(x1) = FROM(x1)
ok(x1) = ok(x1)
active(x1) = active(x1)
app(x1, x2) = app(x1)
nil = nil
mark(x1) = mark
cons(x1, x2) = cons(x1, x2)
from(x1) = from(x1)
s(x1) = x1
zWadr(x1, x2) = zWadr(x1)
prefix(x1) = x1
proper(x1) = proper(x1)
top(x1) = top

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

[active₁, cons₂, proper₁] > zWadr₁ > [ok₁, app₁, from₁] > mark > top
nil > [ok₁, app₁, from₁] > mark > top

Status:

FROM₁: [1]
ok₁: [1]
active₁: [1]
app₁: [1]
nil: []
mark: []
cons₂: [1,2]
from₁: [1]
zWadr₁: [1]
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(32) Obligation:

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

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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) PisEmptyProof (EQUIVALENT transformation)

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

(34) TRUE

(35) 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(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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.

(36) 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) = x2
ok(x1) = ok(x1)
mark(x1) = mark
active(x1) = active(x1)
app(x1, x2) = app(x1)
nil = nil
cons(x1, x2) = cons(x2)
from(x1) = from(x1)
s(x1) = s(x1)
zWadr(x1, x2) = zWadr(x2)
prefix(x1) = x1
proper(x1) = proper(x1)
top(x1) = top

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

active₁ > [zWadr₁, proper₁] > app₁ > [ok₁, mark, cons₁, from₁, s₁]
nil > [ok₁, mark, cons₁, from₁, s₁]
top > [zWadr₁, proper₁] > app₁ > [ok₁, mark, cons₁, from₁, s₁]

Status:

ok₁: [1]
mark: []
active₁: [1]
app₁: [1]
nil: []
cons₁: [1]
from₁: [1]
s₁: [1]
zWadr₁: [1]
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(37) 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(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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.

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)
app(x1, x2) = app(x1, x2)
nil = nil
cons(x1, x2) = x1
from(x1) = x1
s(x1) = x1
zWadr(x1, x2) = zWadr(x1, x2)
prefix(x1) = prefix(x1)
proper(x1) = proper(x1)
ok(x1) = ok
top(x1) = top

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

[active₁, app₂, zWadr₂, prefix₁, proper₁] > [CONS₁, mark₁] > [nil, top]
[active₁, app₂, zWadr₂, prefix₁, proper₁] > ok > [nil, top]

Status:

CONS₁: [1]
mark₁: [1]
active₁: [1]
app₂: [1,2]
nil: []
zWadr₂: [1,2]
prefix₁: [1]
proper₁: [1]
ok: []
top: []

The following usable rules [FROCOS05] were oriented:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(39) Obligation:

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

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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) PisEmptyProof (EQUIVALENT transformation)

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

(41) TRUE

(42) Obligation:

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

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

The TRS R consists of the following rules:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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.

(43) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

APP(ok(X1), ok(X2)) → APP(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
APP(x1, x2) = APP(x1, x2)
mark(x1) = x1
ok(x1) = ok(x1)
active(x1) = x1
app(x1, x2) = x2
nil = nil
cons(x1, x2) = x2
from(x1) = from(x1)
s(x1) = x1
zWadr(x1, x2) = x2
prefix(x1) = prefix(x1)
proper(x1) = proper(x1)
top(x1) = top

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

proper₁ > from₁ > ok₁ > APP₂ > [nil, top]
proper₁ > prefix₁ > ok₁ > APP₂ > [nil, top]

Status:

APP₂: [2,1]
ok₁: [1]
nil: []
from₁: [1]
prefix₁: [1]
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(44) Obligation:

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

APP(X1, mark(X2)) → APP(X1, X2)
APP(mark(X1), X2) → APP(X1, X2)

The TRS R consists of the following rules:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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.

(45) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

APP(X1, mark(X2)) → APP(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
APP(x1, x2) = APP(x2)
mark(x1) = mark(x1)
active(x1) = active(x1)
app(x1, x2) = app(x1, x2)
nil = nil
cons(x1, x2) = x1
from(x1) = from(x1)
s(x1) = x1
zWadr(x1, x2) = zWadr(x1, x2)
prefix(x1) = x1
proper(x1) = x1
ok(x1) = x1
top(x1) = top

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

[active₁, app₂] > from₁ > mark₁ > APP₁ > nil
[active₁, app₂] > zWadr₂ > mark₁ > APP₁ > nil
top > nil

Status:

APP₁: [1]
mark₁: [1]
active₁: [1]
app₂: [1,2]
nil: []
from₁: [1]
zWadr₂: [2,1]
top: []

The following usable rules [FROCOS05] were oriented:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(46) Obligation:

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

APP(mark(X1), X2) → APP(X1, X2)

The TRS R consists of the following rules:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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.

APP(mark(X1), X2) → APP(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
APP(x1, x2) = APP(x1)
mark(x1) = mark(x1)
active(x1) = active(x1)
app(x1, x2) = app(x1, x2)
nil = nil
cons(x1, x2) = x1
from(x1) = x1
s(x1) = x1
zWadr(x1, x2) = zWadr(x1, x2)
prefix(x1) = prefix(x1)
proper(x1) = proper(x1)
ok(x1) = ok
top(x1) = top

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

[active₁, app₂, zWadr₂, prefix₁, proper₁] > [APP₁, mark₁] > [nil, top]
[active₁, app₂, zWadr₂, prefix₁, proper₁] > ok > [nil, top]

Status:

APP₁: [1]
mark₁: [1]
active₁: [1]
app₂: [1,2]
nil: []
zWadr₂: [1,2]
prefix₁: [1]
proper₁: [1]
ok: []
top: []

The following usable rules [FROCOS05] were oriented:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(48) Obligation:

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

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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) PisEmptyProof (EQUIVALENT transformation)

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

(50) TRUE

(51) Obligation:

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

PROPER(app(X1, X2)) → PROPER(X2)
PROPER(app(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(from(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(zWadr(X1, X2)) → PROPER(X1)
PROPER(zWadr(X1, X2)) → PROPER(X2)
PROPER(prefix(X)) → PROPER(X)

The TRS R consists of the following rules:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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.

(52) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(app(X1, X2)) → PROPER(X2)
PROPER(app(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(zWadr(X1, X2)) → PROPER(X1)
PROPER(zWadr(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)
app(x1, x2) = app(x1, x2)
cons(x1, x2) = cons(x1, x2)
from(x1) = x1
s(x1) = x1
zWadr(x1, x2) = zWadr(x1, x2)
prefix(x1) = x1
active(x1) = x1
nil = nil
mark(x1) = mark
proper(x1) = proper(x1)
ok(x1) = ok
top(x1) = top

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

PROPER₁ > [app₂, cons₂, zWadr₂, mark]
[nil, ok, top] > proper₁ > [app₂, cons₂, zWadr₂, mark]

Status:

PROPER₁: [1]
app₂: [1,2]
cons₂: [2,1]
zWadr₂: [2,1]
nil: []
mark: []
proper₁: [1]
ok: []
top: []

The following usable rules [FROCOS05] were oriented:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(53) Obligation:

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

PROPER(from(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(prefix(X)) → PROPER(X)

The TRS R consists of the following rules:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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.

PROPER(from(X)) → PROPER(X)
PROPER(prefix(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
from(x1) = from(x1)
s(x1) = x1
prefix(x1) = prefix(x1)
active(x1) = x1
app(x1, x2) = app(x1)
nil = nil
mark(x1) = mark
cons(x1, x2) = x1
zWadr(x1, x2) = zWadr(x1, x2)
proper(x1) = proper(x1)
ok(x1) = ok(x1)
top(x1) = top

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

[nil, proper₁] > from₁ > [prefix₁, app₁, mark, zWadr₂] > ok₁ > top

Status:

from₁: [1]
prefix₁: [1]
app₁: [1]
nil: []
mark: []
zWadr₂: [2,1]
proper₁: [1]
ok₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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:

PROPER(s(X)) → PROPER(X)

The TRS R consists of the following rules:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(56) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(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) = x1
s(x1) = s(x1)
active(x1) = x1
app(x1, x2) = app
nil = nil
mark(x1) = mark
cons(x1, x2) = cons(x1, x2)
from(x1) = from
zWadr(x1, x2) = zWadr
prefix(x1) = prefix(x1)
proper(x1) = proper(x1)
ok(x1) = ok
top(x1) = top

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

proper₁ > cons₂ > nil > [s₁, app, ok, top] > [mark, prefix₁]
proper₁ > cons₂ > zWadr > [s₁, app, ok, top] > [mark, prefix₁]
proper₁ > from > [s₁, app, ok, top] > [mark, prefix₁]

Status:

s₁: [1]
app: []
nil: []
mark: []
cons₂: [1,2]
from: []
zWadr: []
prefix₁: [1]
proper₁: [1]
ok: []
top: []

The following usable rules [FROCOS05] were oriented:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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:

ACTIVE(app(X1, X2)) → ACTIVE(X2)
ACTIVE(app(X1, X2)) → ACTIVE(X1)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(from(X)) → ACTIVE(X)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(zWadr(X1, X2)) → ACTIVE(X1)
ACTIVE(zWadr(X1, X2)) → ACTIVE(X2)
ACTIVE(prefix(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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.

ACTIVE(app(X1, X2)) → ACTIVE(X2)
ACTIVE(app(X1, X2)) → ACTIVE(X1)
ACTIVE(zWadr(X1, X2)) → ACTIVE(X1)
ACTIVE(zWadr(X1, X2)) → ACTIVE(X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1) = ACTIVE(x1)
app(x1, x2) = app(x1, x2)
cons(x1, x2) = x1
from(x1) = x1
s(x1) = x1
zWadr(x1, x2) = zWadr(x1, x2)
prefix(x1) = x1
active(x1) = active(x1)
nil = nil
mark(x1) = x1
proper(x1) = x1
ok(x1) = ok
top(x1) = top

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

ACTIVE₁ > app₂
active₁ > nil > ok > zWadr₂ > app₂
top > app₂

Status:

ACTIVE₁: [1]
app₂: [1,2]
zWadr₂: [1,2]
active₁: [1]
nil: []
ok: []
top: []

The following usable rules [FROCOS05] were oriented:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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:

ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(from(X)) → ACTIVE(X)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(prefix(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(63) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(from(X)) → ACTIVE(X)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(prefix(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)
cons(x1, x2) = x1
from(x1) = from(x1)
s(x1) = s(x1)
prefix(x1) = prefix(x1)
active(x1) = x1
app(x1, x2) = app(x1, x2)
nil = nil
mark(x1) = mark
zWadr(x1, x2) = zWadr
proper(x1) = proper(x1)
ok(x1) = x1
top(x1) = top

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

ACTIVE₁ > [s₁, top]
[app₂, proper₁] > nil > [from₁, prefix₁, mark] > [s₁, top]
[app₂, proper₁] > zWadr > [from₁, prefix₁, mark] > [s₁, top]

Status:

ACTIVE₁: [1]
from₁: [1]
s₁: [1]
prefix₁: [1]
app₂: [2,1]
nil: []
mark: []
zWadr: []
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(64) Obligation:

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

ACTIVE(cons(X1, X2)) → ACTIVE(X1)

The TRS R consists of the following rules:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(65) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(cons(X1, X2)) → ACTIVE(X1)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1) = x1
cons(x1, x2) = cons(x1)
active(x1) = active(x1)
app(x1, x2) = app(x1)
nil = nil
mark(x1) = mark
from(x1) = x1
s(x1) = x1
zWadr(x1, x2) = zWadr(x2)
prefix(x1) = prefix
proper(x1) = proper(x1)
ok(x1) = ok
top(x1) = top

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

prefix > [nil, ok, top] > active₁ > cons₁ > [app₁, mark, zWadr₁]
prefix > [nil, ok, top] > proper₁ > cons₁ > [app₁, mark, zWadr₁]

Status:

cons₁: [1]
active₁: [1]
app₁: [1]
nil: []
mark: []
zWadr₁: [1]
prefix: []
proper₁: [1]
ok: []
top: []

The following usable rules [FROCOS05] were oriented:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(66) Obligation:

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

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(67) PisEmptyProof (EQUIVALENT transformation)

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

(68) TRUE

(69) 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(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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.

TOP(mark(X)) → TOP(proper(X))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TOP(x1) = TOP(x1)
ok(x1) = x1
active(x1) = x1
mark(x1) = mark(x1)
proper(x1) = x1
app(x1, x2) = app(x1, x2)
nil = nil
cons(x1, x2) = cons(x1)
from(x1) = from(x1)
s(x1) = s(x1)
zWadr(x1, x2) = zWadr(x1, x2)
prefix(x1) = prefix(x1)
top(x1) = top

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

TOP₁ > top
from₁ > s₁ > [mark₁, cons₁] > top
zWadr₂ > app₂ > [mark₁, cons₁] > top
zWadr₂ > [nil, prefix₁] > [mark₁, cons₁] > top

Status:

TOP₁: [1]
mark₁: [1]
app₂: [1,2]
nil: []
cons₁: [1]
from₁: [1]
s₁: [1]
zWadr₂: [2,1]
prefix₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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:

TOP(ok(X)) → TOP(active(X))

The TRS R consists of the following rules:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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.

TOP(ok(X)) → TOP(active(X))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TOP(x1) = TOP(x1)
ok(x1) = ok(x1)
active(x1) = active(x1)
app(x1, x2) = x1
nil = nil
mark(x1) = mark
cons(x1, x2) = x1
from(x1) = x1
s(x1) = x1
zWadr(x1, x2) = x2
prefix(x1) = x1
proper(x1) = proper(x1)
top(x1) = top

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

TOP₁ > [active₁, nil, mark]
top > [ok₁, proper₁] > [active₁, nil, mark]

Status:

TOP₁: [1]
ok₁: [1]
active₁: [1]
nil: []
mark: []
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(73) Obligation:

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

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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) PisEmptyProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) DependencyPairsProof (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyGraphProof (EQUIVALENT transformation)

(4) Complex Obligation (AND)

(5) Obligation:

(6) QDPOrderProof (EQUIVALENT transformation)

(7) Obligation:

(8) QDPOrderProof (EQUIVALENT transformation)

(9) Obligation:

(10) PisEmptyProof (EQUIVALENT transformation)

(11) TRUE

(12) Obligation:

(13) QDPOrderProof (EQUIVALENT transformation)

(14) Obligation:

(15) QDPOrderProof (EQUIVALENT transformation)

(16) Obligation:

(17) QDPOrderProof (EQUIVALENT transformation)

(18) Obligation:

(19) PisEmptyProof (EQUIVALENT transformation)

(20) TRUE

(21) Obligation:

(22) QDPOrderProof (EQUIVALENT transformation)

(23) Obligation:

(24) QDPOrderProof (EQUIVALENT transformation)

(25) Obligation:

(26) PisEmptyProof (EQUIVALENT transformation)

(27) TRUE

(28) Obligation:

(29) QDPOrderProof (EQUIVALENT transformation)

(30) Obligation:

(31) QDPOrderProof (EQUIVALENT transformation)

(32) Obligation:

(33) PisEmptyProof (EQUIVALENT transformation)

(34) TRUE

(35) Obligation:

(36) QDPOrderProof (EQUIVALENT transformation)

(37) Obligation:

(38) QDPOrderProof (EQUIVALENT transformation)

(39) Obligation:

(40) PisEmptyProof (EQUIVALENT transformation)

(41) TRUE

(42) Obligation:

(43) QDPOrderProof (EQUIVALENT transformation)

(44) Obligation:

(45) QDPOrderProof (EQUIVALENT transformation)

(46) Obligation:

(47) QDPOrderProof (EQUIVALENT transformation)

(48) Obligation:

(49) PisEmptyProof (EQUIVALENT transformation)

(50) TRUE

(51) Obligation:

(52) QDPOrderProof (EQUIVALENT transformation)

(53) Obligation:

(54) QDPOrderProof (EQUIVALENT transformation)

(55) Obligation:

(56) QDPOrderProof (EQUIVALENT transformation)

(57) Obligation:

(58) PisEmptyProof (EQUIVALENT transformation)

(59) TRUE

(60) Obligation:

(61) QDPOrderProof (EQUIVALENT transformation)

(62) Obligation:

(63) QDPOrderProof (EQUIVALENT transformation)

(64) Obligation:

(65) QDPOrderProof (EQUIVALENT transformation)

(66) Obligation:

(67) PisEmptyProof (EQUIVALENT transformation)

(68) TRUE

(69) Obligation:

(70) QDPOrderProof (EQUIVALENT transformation)

(71) Obligation:

(72) QDPOrderProof (EQUIVALENT transformation)

(73) Obligation:

(74) PisEmptyProof (EQUIVALENT transformation)

(75) TRUE