Termination w.r.t. Q proof of /tmp/tmpuly4DA/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 QDPOrderProof (⇔)

        ↳20 QDP

        ↳21 PisEmptyProof (⇔)

        ↳22 TRUE

    ↳23 QDP

        ↳24 QDPOrderProof (⇔)

        ↳25 QDP

        ↳26 QDPOrderProof (⇔)

        ↳27 QDP

        ↳28 PisEmptyProof (⇔)

        ↳29 TRUE

    ↳30 QDP

        ↳31 QDPOrderProof (⇔)

        ↳32 QDP

        ↳33 QDPOrderProof (⇔)

        ↳34 QDP

        ↳35 PisEmptyProof (⇔)

        ↳36 TRUE

    ↳37 QDP

        ↳38 QDPOrderProof (⇔)

        ↳39 QDP

        ↳40 QDPOrderProof (⇔)

        ↳41 QDP

        ↳42 QDPOrderProof (⇔)

        ↳43 QDP

        ↳44 QDPOrderProof (⇔)

        ↳45 QDP

        ↳46 PisEmptyProof (⇔)

        ↳47 TRUE

    ↳48 QDP

        ↳49 QDPOrderProof (⇔)

        ↳50 QDP

        ↳51 QDPOrderProof (⇔)

        ↳52 QDP

        ↳53 QDPOrderProof (⇔)

        ↳54 QDP

        ↳55 QDPOrderProof (⇔)

        ↳56 QDP

        ↳57 PisEmptyProof (⇔)

        ↳58 TRUE

    ↳59 QDP

        ↳60 QDPOrderProof (⇔)

        ↳61 QDP

        ↳62 QDPOrderProof (⇔)

        ↳63 QDP

        ↳64 QDPOrderProof (⇔)

        ↳65 QDP

        ↳66 QDPOrderProof (⇔)

        ↳67 QDP

        ↳68 QDPOrderProof (⇔)

        ↳69 QDP

        ↳70 QDPOrderProof (⇔)

        ↳71 QDP

        ↳72 QDPOrderProof (⇔)

        ↳73 QDP

        ↳74 QDPOrderProof (⇔)

        ↳75 QDP

        ↳76 QDPOrderProof (⇔)

        ↳77 QDP

        ↳78 QDPOrderProof (⇔)

        ↳79 QDP

        ↳80 PisEmptyProof (⇔)

        ↳81 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))))
mark(app(X1, X2)) → active(app(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(zWadr(X1, X2)) → active(zWadr(mark(X1), mark(X2)))
mark(prefix(X)) → active(prefix(mark(X)))
app(mark(X1), X2) → app(X1, X2)
app(X1, mark(X2)) → app(X1, X2)
app(active(X1), X2) → app(X1, X2)
app(X1, active(X2)) → app(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
zWadr(mark(X1), X2) → zWadr(X1, X2)
zWadr(X1, mark(X2)) → zWadr(X1, X2)
zWadr(active(X1), X2) → zWadr(X1, X2)
zWadr(X1, active(X2)) → zWadr(X1, X2)
prefix(mark(X)) → prefix(X)
prefix(active(X)) → prefix(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(nil, YS)) → MARK(YS)
ACTIVE(app(cons(X, XS), YS)) → MARK(cons(X, app(XS, YS)))
ACTIVE(app(cons(X, XS), YS)) → CONS(X, app(XS, YS))
ACTIVE(app(cons(X, XS), YS)) → APP(XS, YS)
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
ACTIVE(from(X)) → CONS(X, from(s(X)))
ACTIVE(from(X)) → FROM(s(X))
ACTIVE(from(X)) → 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(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)) → MARK(cons(nil, zWadr(L, prefix(L))))
ACTIVE(prefix(L)) → CONS(nil, zWadr(L, prefix(L)))
ACTIVE(prefix(L)) → ZWADR(L, prefix(L))
MARK(app(X1, X2)) → ACTIVE(app(mark(X1), mark(X2)))
MARK(app(X1, X2)) → APP(mark(X1), mark(X2))
MARK(app(X1, X2)) → MARK(X1)
MARK(app(X1, X2)) → MARK(X2)
MARK(nil) → ACTIVE(nil)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(cons(X1, X2)) → CONS(mark(X1), X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(from(X)) → ACTIVE(from(mark(X)))
MARK(from(X)) → FROM(mark(X))
MARK(from(X)) → MARK(X)
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(s(X)) → S(mark(X))
MARK(s(X)) → MARK(X)
MARK(zWadr(X1, X2)) → ACTIVE(zWadr(mark(X1), mark(X2)))
MARK(zWadr(X1, X2)) → ZWADR(mark(X1), mark(X2))
MARK(zWadr(X1, X2)) → MARK(X1)
MARK(zWadr(X1, X2)) → MARK(X2)
MARK(prefix(X)) → ACTIVE(prefix(mark(X)))
MARK(prefix(X)) → PREFIX(mark(X))
MARK(prefix(X)) → MARK(X)
APP(mark(X1), X2) → APP(X1, X2)
APP(X1, mark(X2)) → APP(X1, X2)
APP(active(X1), X2) → APP(X1, X2)
APP(X1, active(X2)) → APP(X1, X2)
CONS(mark(X1), X2) → CONS(X1, X2)
CONS(X1, mark(X2)) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)
FROM(mark(X)) → FROM(X)
FROM(active(X)) → FROM(X)
S(mark(X)) → S(X)
S(active(X)) → S(X)
ZWADR(mark(X1), X2) → ZWADR(X1, X2)
ZWADR(X1, mark(X2)) → ZWADR(X1, X2)
ZWADR(active(X1), X2) → ZWADR(X1, X2)
ZWADR(X1, active(X2)) → ZWADR(X1, X2)
PREFIX(mark(X)) → PREFIX(X)
PREFIX(active(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))))
mark(app(X1, X2)) → active(app(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(zWadr(X1, X2)) → active(zWadr(mark(X1), mark(X2)))
mark(prefix(X)) → active(prefix(mark(X)))
app(mark(X1), X2) → app(X1, X2)
app(X1, mark(X2)) → app(X1, X2)
app(active(X1), X2) → app(X1, X2)
app(X1, active(X2)) → app(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
zWadr(mark(X1), X2) → zWadr(X1, X2)
zWadr(X1, mark(X2)) → zWadr(X1, X2)
zWadr(active(X1), X2) → zWadr(X1, X2)
zWadr(X1, active(X2)) → zWadr(X1, X2)
prefix(mark(X)) → prefix(X)
prefix(active(X)) → prefix(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 7 SCCs with 20 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

PREFIX(active(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))))
mark(app(X1, X2)) → active(app(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(zWadr(X1, X2)) → active(zWadr(mark(X1), mark(X2)))
mark(prefix(X)) → active(prefix(mark(X)))
app(mark(X1), X2) → app(X1, X2)
app(X1, mark(X2)) → app(X1, X2)
app(active(X1), X2) → app(X1, X2)
app(X1, active(X2)) → app(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
zWadr(mark(X1), X2) → zWadr(X1, X2)
zWadr(X1, mark(X2)) → zWadr(X1, X2)
zWadr(active(X1), X2) → zWadr(X1, X2)
zWadr(X1, active(X2)) → zWadr(X1, X2)
prefix(mark(X)) → prefix(X)
prefix(active(X)) → prefix(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(active(X)) → PREFIX(X)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
PREFIX(x0, x1) = PREFIX(x1)

Tags:
PREFIX has argument tags [1,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
PREFIX(x1) = PREFIX
active(x1) = active(x1)
mark(x1) = x1

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

trivial

Status:

PREFIX: []
active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

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

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))))
mark(app(X1, X2)) → active(app(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(zWadr(X1, X2)) → active(zWadr(mark(X1), mark(X2)))
mark(prefix(X)) → active(prefix(mark(X)))
app(mark(X1), X2) → app(X1, X2)
app(X1, mark(X2)) → app(X1, X2)
app(active(X1), X2) → app(X1, X2)
app(X1, active(X2)) → app(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
zWadr(mark(X1), X2) → zWadr(X1, X2)
zWadr(X1, mark(X2)) → zWadr(X1, X2)
zWadr(active(X1), X2) → zWadr(X1, X2)
zWadr(X1, active(X2)) → zWadr(X1, X2)
prefix(mark(X)) → prefix(X)
prefix(active(X)) → prefix(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(mark(X)) → PREFIX(X)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
PREFIX(x0, x1) = PREFIX(x1)

Tags:
PREFIX has argument tags [1,1] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
PREFIX(x1) = PREFIX
mark(x1) = mark(x1)

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

trivial

Status:

PREFIX: []
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(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))))
mark(app(X1, X2)) → active(app(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(zWadr(X1, X2)) → active(zWadr(mark(X1), mark(X2)))
mark(prefix(X)) → active(prefix(mark(X)))
app(mark(X1), X2) → app(X1, X2)
app(X1, mark(X2)) → app(X1, X2)
app(active(X1), X2) → app(X1, X2)
app(X1, active(X2)) → app(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
zWadr(mark(X1), X2) → zWadr(X1, X2)
zWadr(X1, mark(X2)) → zWadr(X1, X2)
zWadr(active(X1), X2) → zWadr(X1, X2)
zWadr(X1, active(X2)) → zWadr(X1, X2)
prefix(mark(X)) → prefix(X)
prefix(active(X)) → prefix(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(active(X1), X2) → ZWADR(X1, X2)
ZWADR(X1, active(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))))
mark(app(X1, X2)) → active(app(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(zWadr(X1, X2)) → active(zWadr(mark(X1), mark(X2)))
mark(prefix(X)) → active(prefix(mark(X)))
app(mark(X1), X2) → app(X1, X2)
app(X1, mark(X2)) → app(X1, X2)
app(active(X1), X2) → app(X1, X2)
app(X1, active(X2)) → app(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
zWadr(mark(X1), X2) → zWadr(X1, X2)
zWadr(X1, mark(X2)) → zWadr(X1, X2)
zWadr(active(X1), X2) → zWadr(X1, X2)
zWadr(X1, active(X2)) → zWadr(X1, X2)
prefix(mark(X)) → prefix(X)
prefix(active(X)) → prefix(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(X1, mark(X2)) → ZWADR(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
ZWADR(x0, x1, x2) = ZWADR(x2)

Tags:
ZWADR has argument tags [2,0,3] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
ZWADR(x1, x2) = ZWADR(x1, x2)
mark(x1) = mark(x1)
active(x1) = x1

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

trivial

Status:

ZWADR₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(14) Obligation:

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

ZWADR(mark(X1), X2) → ZWADR(X1, X2)
ZWADR(active(X1), X2) → ZWADR(X1, X2)
ZWADR(X1, active(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))))
mark(app(X1, X2)) → active(app(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(zWadr(X1, X2)) → active(zWadr(mark(X1), mark(X2)))
mark(prefix(X)) → active(prefix(mark(X)))
app(mark(X1), X2) → app(X1, X2)
app(X1, mark(X2)) → app(X1, X2)
app(active(X1), X2) → app(X1, X2)
app(X1, active(X2)) → app(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
zWadr(mark(X1), X2) → zWadr(X1, X2)
zWadr(X1, mark(X2)) → zWadr(X1, X2)
zWadr(active(X1), X2) → zWadr(X1, X2)
zWadr(X1, active(X2)) → zWadr(X1, X2)
prefix(mark(X)) → prefix(X)
prefix(active(X)) → prefix(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: SCNP Order with the following components:
Level mapping:
Top level AFS:
ZWADR(x0, x1, x2) = ZWADR(x1)

Tags:
ZWADR has argument tags [0,3,2] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
ZWADR(x1, x2) = ZWADR(x1)
mark(x1) = mark(x1)
active(x1) = x1

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

[ZWADR₁, mark₁]

Status:

ZWADR₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(16) Obligation:

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

ZWADR(active(X1), X2) → ZWADR(X1, X2)
ZWADR(X1, active(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))))
mark(app(X1, X2)) → active(app(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(zWadr(X1, X2)) → active(zWadr(mark(X1), mark(X2)))
mark(prefix(X)) → active(prefix(mark(X)))
app(mark(X1), X2) → app(X1, X2)
app(X1, mark(X2)) → app(X1, X2)
app(active(X1), X2) → app(X1, X2)
app(X1, active(X2)) → app(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
zWadr(mark(X1), X2) → zWadr(X1, X2)
zWadr(X1, mark(X2)) → zWadr(X1, X2)
zWadr(active(X1), X2) → zWadr(X1, X2)
zWadr(X1, active(X2)) → zWadr(X1, X2)
prefix(mark(X)) → prefix(X)
prefix(active(X)) → prefix(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, active(X2)) → ZWADR(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
ZWADR(x0, x1, x2) = ZWADR(x2)

Tags:
ZWADR has argument tags [0,3,1] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
ZWADR(x1, x2) = ZWADR(x1)
active(x1) = active(x1)

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

trivial

Status:

ZWADR₁: [1]
active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(18) Obligation:

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

ZWADR(active(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))))
mark(app(X1, X2)) → active(app(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(zWadr(X1, X2)) → active(zWadr(mark(X1), mark(X2)))
mark(prefix(X)) → active(prefix(mark(X)))
app(mark(X1), X2) → app(X1, X2)
app(X1, mark(X2)) → app(X1, X2)
app(active(X1), X2) → app(X1, X2)
app(X1, active(X2)) → app(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
zWadr(mark(X1), X2) → zWadr(X1, X2)
zWadr(X1, mark(X2)) → zWadr(X1, X2)
zWadr(active(X1), X2) → zWadr(X1, X2)
zWadr(X1, active(X2)) → zWadr(X1, X2)
prefix(mark(X)) → prefix(X)
prefix(active(X)) → prefix(X)

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

(19) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ZWADR(active(X1), X2) → ZWADR(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
ZWADR(x0, x1, x2) = ZWADR(x1)

Tags:
ZWADR has argument tags [3,3,3] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
ZWADR(x1, x2) = x2
active(x1) = active(x1)

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

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(20) 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))))
mark(app(X1, X2)) → active(app(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(zWadr(X1, X2)) → active(zWadr(mark(X1), mark(X2)))
mark(prefix(X)) → active(prefix(mark(X)))
app(mark(X1), X2) → app(X1, X2)
app(X1, mark(X2)) → app(X1, X2)
app(active(X1), X2) → app(X1, X2)
app(X1, active(X2)) → app(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
zWadr(mark(X1), X2) → zWadr(X1, X2)
zWadr(X1, mark(X2)) → zWadr(X1, X2)
zWadr(active(X1), X2) → zWadr(X1, X2)
zWadr(X1, active(X2)) → zWadr(X1, X2)
prefix(mark(X)) → prefix(X)
prefix(active(X)) → prefix(X)

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

(21) PisEmptyProof (EQUIVALENT transformation)

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

(22) TRUE

(23) Obligation:

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

S(active(X)) → S(X)
S(mark(X)) → S(X)

The TRS R consists of the following rules:

active(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))))
mark(app(X1, X2)) → active(app(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(zWadr(X1, X2)) → active(zWadr(mark(X1), mark(X2)))
mark(prefix(X)) → active(prefix(mark(X)))
app(mark(X1), X2) → app(X1, X2)
app(X1, mark(X2)) → app(X1, X2)
app(active(X1), X2) → app(X1, X2)
app(X1, active(X2)) → app(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
zWadr(mark(X1), X2) → zWadr(X1, X2)
zWadr(X1, mark(X2)) → zWadr(X1, X2)
zWadr(active(X1), X2) → zWadr(X1, X2)
zWadr(X1, active(X2)) → zWadr(X1, X2)
prefix(mark(X)) → prefix(X)
prefix(active(X)) → prefix(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(active(X)) → S(X)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
S(x0, x1) = S(x1)

Tags:
S has argument tags [1,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
S(x1) = S
active(x1) = active(x1)
mark(x1) = x1

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

trivial

Status:

S: []
active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(25) Obligation:

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

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))))
mark(app(X1, X2)) → active(app(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(zWadr(X1, X2)) → active(zWadr(mark(X1), mark(X2)))
mark(prefix(X)) → active(prefix(mark(X)))
app(mark(X1), X2) → app(X1, X2)
app(X1, mark(X2)) → app(X1, X2)
app(active(X1), X2) → app(X1, X2)
app(X1, active(X2)) → app(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
zWadr(mark(X1), X2) → zWadr(X1, X2)
zWadr(X1, mark(X2)) → zWadr(X1, X2)
zWadr(active(X1), X2) → zWadr(X1, X2)
zWadr(X1, active(X2)) → zWadr(X1, X2)
prefix(mark(X)) → prefix(X)
prefix(active(X)) → prefix(X)

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

(26) 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: SCNP Order with the following components:
Level mapping:
Top level AFS:
S(x0, x1) = S(x1)

Tags:
S has argument tags [1,1] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
S(x1) = S
mark(x1) = mark(x1)

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

trivial

Status:

S: []
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(27) 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))))
mark(app(X1, X2)) → active(app(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(zWadr(X1, X2)) → active(zWadr(mark(X1), mark(X2)))
mark(prefix(X)) → active(prefix(mark(X)))
app(mark(X1), X2) → app(X1, X2)
app(X1, mark(X2)) → app(X1, X2)
app(active(X1), X2) → app(X1, X2)
app(X1, active(X2)) → app(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
zWadr(mark(X1), X2) → zWadr(X1, X2)
zWadr(X1, mark(X2)) → zWadr(X1, X2)
zWadr(active(X1), X2) → zWadr(X1, X2)
zWadr(X1, active(X2)) → zWadr(X1, X2)
prefix(mark(X)) → prefix(X)
prefix(active(X)) → prefix(X)

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

(28) PisEmptyProof (EQUIVALENT transformation)

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

(29) TRUE

(30) Obligation:

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

FROM(active(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))))
mark(app(X1, X2)) → active(app(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(zWadr(X1, X2)) → active(zWadr(mark(X1), mark(X2)))
mark(prefix(X)) → active(prefix(mark(X)))
app(mark(X1), X2) → app(X1, X2)
app(X1, mark(X2)) → app(X1, X2)
app(active(X1), X2) → app(X1, X2)
app(X1, active(X2)) → app(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
zWadr(mark(X1), X2) → zWadr(X1, X2)
zWadr(X1, mark(X2)) → zWadr(X1, X2)
zWadr(active(X1), X2) → zWadr(X1, X2)
zWadr(X1, active(X2)) → zWadr(X1, X2)
prefix(mark(X)) → prefix(X)
prefix(active(X)) → prefix(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(active(X)) → FROM(X)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
FROM(x0, x1) = FROM(x1)

Tags:
FROM has argument tags [1,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
FROM(x1) = FROM
active(x1) = active(x1)
mark(x1) = x1

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

trivial

Status:

FROM: []
active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(32) Obligation:

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

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))))
mark(app(X1, X2)) → active(app(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(zWadr(X1, X2)) → active(zWadr(mark(X1), mark(X2)))
mark(prefix(X)) → active(prefix(mark(X)))
app(mark(X1), X2) → app(X1, X2)
app(X1, mark(X2)) → app(X1, X2)
app(active(X1), X2) → app(X1, X2)
app(X1, active(X2)) → app(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
zWadr(mark(X1), X2) → zWadr(X1, X2)
zWadr(X1, mark(X2)) → zWadr(X1, X2)
zWadr(active(X1), X2) → zWadr(X1, X2)
zWadr(X1, active(X2)) → zWadr(X1, X2)
prefix(mark(X)) → prefix(X)
prefix(active(X)) → prefix(X)

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

(33) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FROM(mark(X)) → FROM(X)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
FROM(x0, x1) = FROM(x1)

Tags:
FROM has argument tags [1,1] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
FROM(x1) = FROM
mark(x1) = mark(x1)

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

trivial

Status:

FROM: []
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(34) 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))))
mark(app(X1, X2)) → active(app(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(zWadr(X1, X2)) → active(zWadr(mark(X1), mark(X2)))
mark(prefix(X)) → active(prefix(mark(X)))
app(mark(X1), X2) → app(X1, X2)
app(X1, mark(X2)) → app(X1, X2)
app(active(X1), X2) → app(X1, X2)
app(X1, active(X2)) → app(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
zWadr(mark(X1), X2) → zWadr(X1, X2)
zWadr(X1, mark(X2)) → zWadr(X1, X2)
zWadr(active(X1), X2) → zWadr(X1, X2)
zWadr(X1, active(X2)) → zWadr(X1, X2)
prefix(mark(X)) → prefix(X)
prefix(active(X)) → prefix(X)

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

(35) PisEmptyProof (EQUIVALENT transformation)

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

(36) TRUE

(37) Obligation:

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

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

The TRS R consists of the following rules:

active(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))))
mark(app(X1, X2)) → active(app(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(zWadr(X1, X2)) → active(zWadr(mark(X1), mark(X2)))
mark(prefix(X)) → active(prefix(mark(X)))
app(mark(X1), X2) → app(X1, X2)
app(X1, mark(X2)) → app(X1, X2)
app(active(X1), X2) → app(X1, X2)
app(X1, active(X2)) → app(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
zWadr(mark(X1), X2) → zWadr(X1, X2)
zWadr(X1, mark(X2)) → zWadr(X1, X2)
zWadr(active(X1), X2) → zWadr(X1, X2)
zWadr(X1, active(X2)) → zWadr(X1, X2)
prefix(mark(X)) → prefix(X)
prefix(active(X)) → prefix(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(X1, mark(X2)) → CONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
CONS(x0, x1, x2) = CONS(x2)

Tags:
CONS has argument tags [2,0,3] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
CONS(x1, x2) = CONS(x1, x2)
mark(x1) = mark(x1)
active(x1) = x1

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

trivial

Status:

CONS₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(39) Obligation:

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

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

The TRS R consists of the following rules:

active(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))))
mark(app(X1, X2)) → active(app(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(zWadr(X1, X2)) → active(zWadr(mark(X1), mark(X2)))
mark(prefix(X)) → active(prefix(mark(X)))
app(mark(X1), X2) → app(X1, X2)
app(X1, mark(X2)) → app(X1, X2)
app(active(X1), X2) → app(X1, X2)
app(X1, active(X2)) → app(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
zWadr(mark(X1), X2) → zWadr(X1, X2)
zWadr(X1, mark(X2)) → zWadr(X1, X2)
zWadr(active(X1), X2) → zWadr(X1, X2)
zWadr(X1, active(X2)) → zWadr(X1, X2)
prefix(mark(X)) → prefix(X)
prefix(active(X)) → prefix(X)

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

(40) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(mark(X1), X2) → CONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
CONS(x0, x1, x2) = CONS(x1)

Tags:
CONS has argument tags [0,3,2] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
CONS(x1, x2) = CONS(x1)
mark(x1) = mark(x1)
active(x1) = x1

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

[CONS₁, mark₁]

Status:

CONS₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(41) Obligation:

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

CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, active(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))))
mark(app(X1, X2)) → active(app(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(zWadr(X1, X2)) → active(zWadr(mark(X1), mark(X2)))
mark(prefix(X)) → active(prefix(mark(X)))
app(mark(X1), X2) → app(X1, X2)
app(X1, mark(X2)) → app(X1, X2)
app(active(X1), X2) → app(X1, X2)
app(X1, active(X2)) → app(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
zWadr(mark(X1), X2) → zWadr(X1, X2)
zWadr(X1, mark(X2)) → zWadr(X1, X2)
zWadr(active(X1), X2) → zWadr(X1, X2)
zWadr(X1, active(X2)) → zWadr(X1, X2)
prefix(mark(X)) → prefix(X)
prefix(active(X)) → prefix(X)

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

(42) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(X1, active(X2)) → CONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
CONS(x0, x1, x2) = CONS(x2)

Tags:
CONS has argument tags [0,3,1] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
CONS(x1, x2) = CONS(x1)
active(x1) = active(x1)

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

trivial

Status:

CONS₁: [1]
active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(43) Obligation:

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

CONS(active(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))))
mark(app(X1, X2)) → active(app(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(zWadr(X1, X2)) → active(zWadr(mark(X1), mark(X2)))
mark(prefix(X)) → active(prefix(mark(X)))
app(mark(X1), X2) → app(X1, X2)
app(X1, mark(X2)) → app(X1, X2)
app(active(X1), X2) → app(X1, X2)
app(X1, active(X2)) → app(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
zWadr(mark(X1), X2) → zWadr(X1, X2)
zWadr(X1, mark(X2)) → zWadr(X1, X2)
zWadr(active(X1), X2) → zWadr(X1, X2)
zWadr(X1, active(X2)) → zWadr(X1, X2)
prefix(mark(X)) → prefix(X)
prefix(active(X)) → prefix(X)

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

(44) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(active(X1), X2) → CONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
CONS(x0, x1, x2) = CONS(x1)

Tags:
CONS has argument tags [3,3,3] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
CONS(x1, x2) = x2
active(x1) = active(x1)

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

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(45) 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))))
mark(app(X1, X2)) → active(app(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(zWadr(X1, X2)) → active(zWadr(mark(X1), mark(X2)))
mark(prefix(X)) → active(prefix(mark(X)))
app(mark(X1), X2) → app(X1, X2)
app(X1, mark(X2)) → app(X1, X2)
app(active(X1), X2) → app(X1, X2)
app(X1, active(X2)) → app(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
zWadr(mark(X1), X2) → zWadr(X1, X2)
zWadr(X1, mark(X2)) → zWadr(X1, X2)
zWadr(active(X1), X2) → zWadr(X1, X2)
zWadr(X1, active(X2)) → zWadr(X1, X2)
prefix(mark(X)) → prefix(X)
prefix(active(X)) → prefix(X)

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

(46) PisEmptyProof (EQUIVALENT transformation)

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

(47) TRUE

(48) Obligation:

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

APP(X1, mark(X2)) → APP(X1, X2)
APP(mark(X1), X2) → APP(X1, X2)
APP(active(X1), X2) → APP(X1, X2)
APP(X1, active(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))))
mark(app(X1, X2)) → active(app(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(zWadr(X1, X2)) → active(zWadr(mark(X1), mark(X2)))
mark(prefix(X)) → active(prefix(mark(X)))
app(mark(X1), X2) → app(X1, X2)
app(X1, mark(X2)) → app(X1, X2)
app(active(X1), X2) → app(X1, X2)
app(X1, active(X2)) → app(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
zWadr(mark(X1), X2) → zWadr(X1, X2)
zWadr(X1, mark(X2)) → zWadr(X1, X2)
zWadr(active(X1), X2) → zWadr(X1, X2)
zWadr(X1, active(X2)) → zWadr(X1, X2)
prefix(mark(X)) → prefix(X)
prefix(active(X)) → prefix(X)

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

(49) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

APP(X1, mark(X2)) → APP(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
APP(x0, x1, x2) = APP(x2)

Tags:
APP has argument tags [2,0,3] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
APP(x1, x2) = APP(x1, x2)
mark(x1) = mark(x1)
active(x1) = x1

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

trivial

Status:

APP₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(50) Obligation:

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

APP(mark(X1), X2) → APP(X1, X2)
APP(active(X1), X2) → APP(X1, X2)
APP(X1, active(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))))
mark(app(X1, X2)) → active(app(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(zWadr(X1, X2)) → active(zWadr(mark(X1), mark(X2)))
mark(prefix(X)) → active(prefix(mark(X)))
app(mark(X1), X2) → app(X1, X2)
app(X1, mark(X2)) → app(X1, X2)
app(active(X1), X2) → app(X1, X2)
app(X1, active(X2)) → app(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
zWadr(mark(X1), X2) → zWadr(X1, X2)
zWadr(X1, mark(X2)) → zWadr(X1, X2)
zWadr(active(X1), X2) → zWadr(X1, X2)
zWadr(X1, active(X2)) → zWadr(X1, X2)
prefix(mark(X)) → prefix(X)
prefix(active(X)) → prefix(X)

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

(51) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

APP(mark(X1), X2) → APP(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
APP(x0, x1, x2) = APP(x1)

Tags:
APP has argument tags [0,3,2] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
APP(x1, x2) = APP(x1)
mark(x1) = mark(x1)
active(x1) = x1

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

[APP₁, mark₁]

Status:

APP₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(52) Obligation:

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

APP(active(X1), X2) → APP(X1, X2)
APP(X1, active(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))))
mark(app(X1, X2)) → active(app(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(zWadr(X1, X2)) → active(zWadr(mark(X1), mark(X2)))
mark(prefix(X)) → active(prefix(mark(X)))
app(mark(X1), X2) → app(X1, X2)
app(X1, mark(X2)) → app(X1, X2)
app(active(X1), X2) → app(X1, X2)
app(X1, active(X2)) → app(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
zWadr(mark(X1), X2) → zWadr(X1, X2)
zWadr(X1, mark(X2)) → zWadr(X1, X2)
zWadr(active(X1), X2) → zWadr(X1, X2)
zWadr(X1, active(X2)) → zWadr(X1, X2)
prefix(mark(X)) → prefix(X)
prefix(active(X)) → prefix(X)

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

(53) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

APP(X1, active(X2)) → APP(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
APP(x0, x1, x2) = APP(x2)

Tags:
APP has argument tags [0,3,1] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
APP(x1, x2) = APP(x1)
active(x1) = active(x1)

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

trivial

Status:

APP₁: [1]
active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(54) Obligation:

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

APP(active(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))))
mark(app(X1, X2)) → active(app(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(zWadr(X1, X2)) → active(zWadr(mark(X1), mark(X2)))
mark(prefix(X)) → active(prefix(mark(X)))
app(mark(X1), X2) → app(X1, X2)
app(X1, mark(X2)) → app(X1, X2)
app(active(X1), X2) → app(X1, X2)
app(X1, active(X2)) → app(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
zWadr(mark(X1), X2) → zWadr(X1, X2)
zWadr(X1, mark(X2)) → zWadr(X1, X2)
zWadr(active(X1), X2) → zWadr(X1, X2)
zWadr(X1, active(X2)) → zWadr(X1, X2)
prefix(mark(X)) → prefix(X)
prefix(active(X)) → prefix(X)

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

(55) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

APP(active(X1), X2) → APP(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
APP(x0, x1, x2) = APP(x1)

Tags:
APP has argument tags [3,3,3] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
APP(x1, x2) = x2
active(x1) = active(x1)

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

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(56) 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))))
mark(app(X1, X2)) → active(app(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(zWadr(X1, X2)) → active(zWadr(mark(X1), mark(X2)))
mark(prefix(X)) → active(prefix(mark(X)))
app(mark(X1), X2) → app(X1, X2)
app(X1, mark(X2)) → app(X1, X2)
app(active(X1), X2) → app(X1, X2)
app(X1, active(X2)) → app(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
zWadr(mark(X1), X2) → zWadr(X1, X2)
zWadr(X1, mark(X2)) → zWadr(X1, X2)
zWadr(active(X1), X2) → zWadr(X1, X2)
zWadr(X1, active(X2)) → zWadr(X1, X2)
prefix(mark(X)) → prefix(X)
prefix(active(X)) → prefix(X)

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

(57) PisEmptyProof (EQUIVALENT transformation)

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

(58) TRUE

(59) Obligation:

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

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

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

(60) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(app(nil, YS)) → MARK(YS)
MARK(app(X1, X2)) → MARK(X1)
MARK(app(X1, X2)) → MARK(X2)
ACTIVE(app(cons(X, XS), YS)) → MARK(cons(X, app(XS, YS)))
MARK(zWadr(X1, X2)) → MARK(X1)
MARK(zWadr(X1, X2)) → MARK(X2)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
MARK(x0, x1) = MARK(x0, x1)
ACTIVE(x0, x1) = ACTIVE(x1)

Tags:
MARK has argument tags [0,2] and root tag 0
ACTIVE has argument tags [0,2] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
MARK(x1) = x1
app(x1, x2) = app(x1, x2)
ACTIVE(x1) = x1
mark(x1) = x1
nil = nil
cons(x1, x2) = x1
from(x1) = x1
s(x1) = x1
zWadr(x1, x2) = zWadr(x1, x2)
prefix(x1) = x1
active(x1) = x1

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

[app₂, zWadr₂] > nil

Status:

app₂: [2,1]
nil: []
zWadr₂: [1,2]

The following usable rules [FROCOS05] were oriented:

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

(61) Obligation:

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

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

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

(62) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
MARK(x0, x1) = MARK(x0)
ACTIVE(x0, x1) = ACTIVE(x1)

Tags:
MARK has argument tags [3,0] and root tag 0
ACTIVE has argument tags [2,3] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
MARK(x1) = MARK
app(x1, x2) = app
ACTIVE(x1) = ACTIVE(x1)
mark(x1) = mark
cons(x1, x2) = cons
from(x1) = from
s(x1) = s
zWadr(x1, x2) = zWadr
nil = nil
prefix(x1) = prefix
active(x1) = active

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

[mark, nil] > [MARK, app, ACTIVE₁, from, s, zWadr, prefix] > active > cons

Status:

MARK: []
app: []
ACTIVE₁: [1]
mark: []
cons: []
from: []
s: []
zWadr: []
nil: []
prefix: []
active: []

The following usable rules [FROCOS05] were oriented:

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

(63) Obligation:

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

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

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

(64) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(zWadr(cons(X, XS), cons(Y, YS))) → MARK(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
MARK(x0, x1) = MARK(x0)
ACTIVE(x0, x1) = ACTIVE(x0)

Tags:
MARK has argument tags [2,0] and root tag 0
ACTIVE has argument tags [2,2] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
MARK(x1) = x1
app(x1, x2) = app(x1, x2)
ACTIVE(x1) = x1
mark(x1) = x1
cons(x1, x2) = x1
from(x1) = x1
s(x1) = x1
zWadr(x1, x2) = zWadr(x1, x2)
nil = nil
prefix(x1) = x1
active(x1) = x1

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

zWadr₂ > app₂ > nil

Status:

app₂: [2,1]
zWadr₂: [2,1]
nil: []

The following usable rules [FROCOS05] were oriented:

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

(65) Obligation:

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

MARK(app(X1, X2)) → ACTIVE(app(mark(X1), mark(X2)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(from(X)) → ACTIVE(from(mark(X)))
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
MARK(from(X)) → MARK(X)
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(s(X)) → MARK(X)
MARK(zWadr(X1, X2)) → ACTIVE(zWadr(mark(X1), mark(X2)))
ACTIVE(prefix(L)) → MARK(cons(nil, zWadr(L, prefix(L))))
MARK(prefix(X)) → ACTIVE(prefix(mark(X)))
MARK(prefix(X)) → MARK(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))))
mark(app(X1, X2)) → active(app(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(zWadr(X1, X2)) → active(zWadr(mark(X1), mark(X2)))
mark(prefix(X)) → active(prefix(mark(X)))
app(mark(X1), X2) → app(X1, X2)
app(X1, mark(X2)) → app(X1, X2)
app(active(X1), X2) → app(X1, X2)
app(X1, active(X2)) → app(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
zWadr(mark(X1), X2) → zWadr(X1, X2)
zWadr(X1, mark(X2)) → zWadr(X1, X2)
zWadr(active(X1), X2) → zWadr(X1, X2)
zWadr(X1, active(X2)) → zWadr(X1, X2)
prefix(mark(X)) → prefix(X)
prefix(active(X)) → prefix(X)

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

(66) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(s(X)) → ACTIVE(s(mark(X)))

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
MARK(x0, x1) = MARK(x0)
ACTIVE(x0, x1) = ACTIVE(x0, x1)

Tags:
MARK has argument tags [2,0] and root tag 0
ACTIVE has argument tags [3,2] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
MARK(x1) = MARK
app(x1, x2) = app
ACTIVE(x1) = ACTIVE
mark(x1) = mark(x1)
cons(x1, x2) = cons
from(x1) = from
s(x1) = s
zWadr(x1, x2) = zWadr
prefix(x1) = prefix
nil = nil
active(x1) = active(x1)

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

nil > mark₁ > [cons, active₁] > [MARK, app, from, zWadr, prefix] > ACTIVE > s

Status:

MARK: []
app: []
ACTIVE: []
mark₁: [1]
cons: []
from: []
s: []
zWadr: []
prefix: []
nil: []
active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(67) Obligation:

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

MARK(app(X1, X2)) → ACTIVE(app(mark(X1), mark(X2)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(from(X)) → ACTIVE(from(mark(X)))
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
MARK(from(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(zWadr(X1, X2)) → ACTIVE(zWadr(mark(X1), mark(X2)))
ACTIVE(prefix(L)) → MARK(cons(nil, zWadr(L, prefix(L))))
MARK(prefix(X)) → ACTIVE(prefix(mark(X)))
MARK(prefix(X)) → MARK(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))))
mark(app(X1, X2)) → active(app(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(zWadr(X1, X2)) → active(zWadr(mark(X1), mark(X2)))
mark(prefix(X)) → active(prefix(mark(X)))
app(mark(X1), X2) → app(X1, X2)
app(X1, mark(X2)) → app(X1, X2)
app(active(X1), X2) → app(X1, X2)
app(X1, active(X2)) → app(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
zWadr(mark(X1), X2) → zWadr(X1, X2)
zWadr(X1, mark(X2)) → zWadr(X1, X2)
zWadr(active(X1), X2) → zWadr(X1, X2)
zWadr(X1, active(X2)) → zWadr(X1, X2)
prefix(mark(X)) → prefix(X)
prefix(active(X)) → prefix(X)

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

(68) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
MARK(x0, x1) = MARK(x0)
ACTIVE(x0, x1) = ACTIVE(x1)

Tags:
MARK has argument tags [1,0] and root tag 1
ACTIVE has argument tags [2,1] and root tag 1

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
MARK(x1) = MARK
app(x1, x2) = app
ACTIVE(x1) = ACTIVE(x1)
mark(x1) = mark(x1)
cons(x1, x2) = cons
from(x1) = from
s(x1) = x1
zWadr(x1, x2) = zWadr
prefix(x1) = prefix
nil = nil
active(x1) = active(x1)

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

[MARK, ACTIVE₁, mark₁, cons, from, prefix, nil] > app > active₁ > zWadr

Status:

MARK: []
app: []
ACTIVE₁: [1]
mark₁: [1]
cons: []
from: []
zWadr: []
prefix: []
nil: []
active₁: [1]

The following usable rules [FROCOS05] were oriented:

app(X1, mark(X2)) → app(X1, X2)
app(mark(X1), X2) → app(X1, X2)
app(active(X1), X2) → app(X1, X2)
app(X1, active(X2)) → app(X1, X2)
from(active(X)) → from(X)
from(mark(X)) → from(X)
zWadr(X1, mark(X2)) → zWadr(X1, X2)
zWadr(mark(X1), X2) → zWadr(X1, X2)
zWadr(active(X1), X2) → zWadr(X1, X2)
zWadr(X1, active(X2)) → zWadr(X1, X2)
prefix(active(X)) → prefix(X)
prefix(mark(X)) → prefix(X)

(69) Obligation:

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

MARK(cons(X1, X2)) → MARK(X1)
MARK(from(X)) → ACTIVE(from(mark(X)))
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
MARK(from(X)) → MARK(X)
MARK(s(X)) → MARK(X)
ACTIVE(prefix(L)) → MARK(cons(nil, zWadr(L, prefix(L))))
MARK(prefix(X)) → ACTIVE(prefix(mark(X)))
MARK(prefix(X)) → MARK(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))))
mark(app(X1, X2)) → active(app(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(zWadr(X1, X2)) → active(zWadr(mark(X1), mark(X2)))
mark(prefix(X)) → active(prefix(mark(X)))
app(mark(X1), X2) → app(X1, X2)
app(X1, mark(X2)) → app(X1, X2)
app(active(X1), X2) → app(X1, X2)
app(X1, active(X2)) → app(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
zWadr(mark(X1), X2) → zWadr(X1, X2)
zWadr(X1, mark(X2)) → zWadr(X1, X2)
zWadr(active(X1), X2) → zWadr(X1, X2)
zWadr(X1, active(X2)) → zWadr(X1, X2)
prefix(mark(X)) → prefix(X)
prefix(active(X)) → prefix(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.

MARK(s(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
MARK(x0, x1) = MARK(x1)
ACTIVE(x0, x1) = ACTIVE(x0, x1)

Tags:
MARK has argument tags [2,1] and root tag 1
ACTIVE has argument tags [1,1] and root tag 1

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
MARK(x1) = x1
cons(x1, x2) = x1
from(x1) = x1
ACTIVE(x1) = x1
mark(x1) = x1
s(x1) = s(x1)
prefix(x1) = x1
nil = nil
zWadr(x1, x2) = zWadr(x1, x2)
app(x1, x2) = app(x1, x2)
active(x1) = x1

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

s₁ > nil
zWadr₂ > app₂ > nil

Status:

s₁: [1]
nil: []
zWadr₂: [2,1]
app₂: [2,1]

The following usable rules [FROCOS05] were oriented:

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

(71) Obligation:

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

MARK(cons(X1, X2)) → MARK(X1)
MARK(from(X)) → ACTIVE(from(mark(X)))
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
MARK(from(X)) → MARK(X)
ACTIVE(prefix(L)) → MARK(cons(nil, zWadr(L, prefix(L))))
MARK(prefix(X)) → ACTIVE(prefix(mark(X)))
MARK(prefix(X)) → MARK(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))))
mark(app(X1, X2)) → active(app(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(zWadr(X1, X2)) → active(zWadr(mark(X1), mark(X2)))
mark(prefix(X)) → active(prefix(mark(X)))
app(mark(X1), X2) → app(X1, X2)
app(X1, mark(X2)) → app(X1, X2)
app(active(X1), X2) → app(X1, X2)
app(X1, active(X2)) → app(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
zWadr(mark(X1), X2) → zWadr(X1, X2)
zWadr(X1, mark(X2)) → zWadr(X1, X2)
zWadr(active(X1), X2) → zWadr(X1, X2)
zWadr(X1, active(X2)) → zWadr(X1, X2)
prefix(mark(X)) → prefix(X)
prefix(active(X)) → prefix(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.

MARK(cons(X1, X2)) → MARK(X1)
MARK(from(X)) → MARK(X)
MARK(prefix(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
MARK(x0, x1) = MARK(x0, x1)
ACTIVE(x0, x1) = ACTIVE(x1)

Tags:
MARK has argument tags [2,0] and root tag 0
ACTIVE has argument tags [2,2] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
MARK(x1) = x1
cons(x1, x2) = cons(x1)
from(x1) = from(x1)
ACTIVE(x1) = ACTIVE
mark(x1) = x1
s(x1) = x1
prefix(x1) = prefix(x1)
nil = nil
zWadr(x1, x2) = zWadr(x1, x2)
app(x1, x2) = app(x1, x2)
active(x1) = x1

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

ACTIVE > [cons₁, from₁, prefix₁, nil]
[zWadr₂, app₂] > [cons₁, from₁, prefix₁, nil]

Status:

cons₁: [1]
from₁: [1]
ACTIVE: []
prefix₁: [1]
nil: []
zWadr₂: [1,2]
app₂: [2,1]

The following usable rules [FROCOS05] were oriented:

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

(73) Obligation:

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

MARK(from(X)) → ACTIVE(from(mark(X)))
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
ACTIVE(prefix(L)) → MARK(cons(nil, zWadr(L, prefix(L))))
MARK(prefix(X)) → ACTIVE(prefix(mark(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))))
mark(app(X1, X2)) → active(app(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(zWadr(X1, X2)) → active(zWadr(mark(X1), mark(X2)))
mark(prefix(X)) → active(prefix(mark(X)))
app(mark(X1), X2) → app(X1, X2)
app(X1, mark(X2)) → app(X1, X2)
app(active(X1), X2) → app(X1, X2)
app(X1, active(X2)) → app(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
zWadr(mark(X1), X2) → zWadr(X1, X2)
zWadr(X1, mark(X2)) → zWadr(X1, X2)
zWadr(active(X1), X2) → zWadr(X1, X2)
zWadr(X1, active(X2)) → zWadr(X1, X2)
prefix(mark(X)) → prefix(X)
prefix(active(X)) → prefix(X)

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

(74) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(from(X)) → MARK(cons(X, from(s(X))))

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
MARK(x0, x1) = MARK(x1)
ACTIVE(x0, x1) = ACTIVE(x0)

Tags:
MARK has argument tags [0,3] and root tag 0
ACTIVE has argument tags [3,3] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
MARK(x1) = MARK
from(x1) = from(x1)
ACTIVE(x1) = x1
mark(x1) = x1
cons(x1, x2) = cons(x1)
s(x1) = s
prefix(x1) = prefix(x1)
nil = nil
zWadr(x1, x2) = zWadr(x1, x2)
app(x1, x2) = app(x1, x2)
active(x1) = x1

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

MARK > from₁ > [cons₁, prefix₁, nil]
s > [cons₁, prefix₁, nil]
[zWadr₂, app₂] > [cons₁, prefix₁, nil]

Status:

MARK: []
from₁: [1]
cons₁: [1]
s: []
prefix₁: [1]
nil: []
zWadr₂: [1,2]
app₂: [2,1]

The following usable rules [FROCOS05] were oriented:

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

(75) Obligation:

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

MARK(from(X)) → ACTIVE(from(mark(X)))
ACTIVE(prefix(L)) → MARK(cons(nil, zWadr(L, prefix(L))))
MARK(prefix(X)) → ACTIVE(prefix(mark(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))))
mark(app(X1, X2)) → active(app(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(zWadr(X1, X2)) → active(zWadr(mark(X1), mark(X2)))
mark(prefix(X)) → active(prefix(mark(X)))
app(mark(X1), X2) → app(X1, X2)
app(X1, mark(X2)) → app(X1, X2)
app(active(X1), X2) → app(X1, X2)
app(X1, active(X2)) → app(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
zWadr(mark(X1), X2) → zWadr(X1, X2)
zWadr(X1, mark(X2)) → zWadr(X1, X2)
zWadr(active(X1), X2) → zWadr(X1, X2)
zWadr(X1, active(X2)) → zWadr(X1, X2)
prefix(mark(X)) → prefix(X)
prefix(active(X)) → prefix(X)

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

(76) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(from(X)) → ACTIVE(from(mark(X)))

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
MARK(x0, x1) = MARK(x0, x1)
ACTIVE(x0, x1) = ACTIVE(x1)

Tags:
MARK has argument tags [0,0] and root tag 0
ACTIVE has argument tags [3,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
MARK(x1) = MARK
from(x1) = from
ACTIVE(x1) = x1
mark(x1) = mark
prefix(x1) = prefix
cons(x1, x2) = cons
nil = nil
zWadr(x1, x2) = zWadr
app(x1, x2) = app(x1, x2)
active(x1) = active
s(x1) = s(x1)

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

app₂ > [MARK, prefix, cons] > from > mark > nil > [zWadr, active]
s₁ > mark > nil > [zWadr, active]

Status:

MARK: []
from: []
mark: []
prefix: []
cons: []
nil: []
zWadr: []
app₂: [2,1]
active: []
s₁: [1]

The following usable rules [FROCOS05] were oriented:

from(active(X)) → from(X)
from(mark(X)) → from(X)
prefix(active(X)) → prefix(X)
prefix(mark(X)) → prefix(X)

(77) Obligation:

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

ACTIVE(prefix(L)) → MARK(cons(nil, zWadr(L, prefix(L))))
MARK(prefix(X)) → ACTIVE(prefix(mark(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))))
mark(app(X1, X2)) → active(app(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(zWadr(X1, X2)) → active(zWadr(mark(X1), mark(X2)))
mark(prefix(X)) → active(prefix(mark(X)))
app(mark(X1), X2) → app(X1, X2)
app(X1, mark(X2)) → app(X1, X2)
app(active(X1), X2) → app(X1, X2)
app(X1, active(X2)) → app(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
zWadr(mark(X1), X2) → zWadr(X1, X2)
zWadr(X1, mark(X2)) → zWadr(X1, X2)
zWadr(active(X1), X2) → zWadr(X1, X2)
zWadr(X1, active(X2)) → zWadr(X1, X2)
prefix(mark(X)) → prefix(X)
prefix(active(X)) → prefix(X)

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

(78) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(prefix(L)) → MARK(cons(nil, zWadr(L, prefix(L))))
MARK(prefix(X)) → ACTIVE(prefix(mark(X)))

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
ACTIVE(x0, x1) = ACTIVE(x1)
MARK(x0, x1) = MARK(x0, x1)

Tags:
ACTIVE has argument tags [1,2] and root tag 0
MARK has argument tags [2,1] and root tag 1

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
ACTIVE(x1) = x1
prefix(x1) = prefix(x1)
MARK(x1) = x1
cons(x1, x2) = x1
nil = nil
zWadr(x1, x2) = zWadr(x1, x2)
mark(x1) = x1
active(x1) = x1
app(x1, x2) = app(x1, x2)
from(x1) = from(x1)
s(x1) = s

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

from₁ > [prefix₁, nil, zWadr₂, app₂]
s > [prefix₁, nil, zWadr₂, app₂]

Status:

prefix₁: [1]
nil: []
zWadr₂: [2,1]
app₂: [1,2]
from₁: [1]
s: []

The following usable rules [FROCOS05] were oriented:

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

(79) 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))))
mark(app(X1, X2)) → active(app(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(from(X)) → active(from(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(zWadr(X1, X2)) → active(zWadr(mark(X1), mark(X2)))
mark(prefix(X)) → active(prefix(mark(X)))
app(mark(X1), X2) → app(X1, X2)
app(X1, mark(X2)) → app(X1, X2)
app(active(X1), X2) → app(X1, X2)
app(X1, active(X2)) → app(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
from(mark(X)) → from(X)
from(active(X)) → from(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
zWadr(mark(X1), X2) → zWadr(X1, X2)
zWadr(X1, mark(X2)) → zWadr(X1, X2)
zWadr(active(X1), X2) → zWadr(X1, X2)
zWadr(X1, active(X2)) → zWadr(X1, X2)
prefix(mark(X)) → prefix(X)
prefix(active(X)) → prefix(X)

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

(80) PisEmptyProof (EQUIVALENT transformation)

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