(0) Obligation:

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

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

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

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

(2) Obligation:

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

ACTIVE(app(cons(X, XS), YS)) → CONS(X, app(XS, YS))
ACTIVE(app(cons(X, XS), YS)) → APP(XS, YS)
ACTIVE(from(X)) → CONS(X, from(s(X)))
ACTIVE(from(X)) → FROM(s(X))
ACTIVE(from(X)) → S(X)
ACTIVE(zWadr(cons(X, XS), cons(Y, YS))) → CONS(app(Y, cons(X, nil)), zWadr(XS, YS))
ACTIVE(zWadr(cons(X, XS), cons(Y, YS))) → APP(Y, cons(X, nil))
ACTIVE(zWadr(cons(X, XS), cons(Y, YS))) → CONS(X, nil)
ACTIVE(zWadr(cons(X, XS), cons(Y, YS))) → ZWADR(XS, YS)
ACTIVE(prefix(L)) → CONS(nil, zWadr(L, prefix(L)))
ACTIVE(prefix(L)) → ZWADR(L, prefix(L))
ACTIVE(app(X1, X2)) → APP(active(X1), X2)
ACTIVE(app(X1, X2)) → ACTIVE(X1)
ACTIVE(app(X1, X2)) → APP(X1, active(X2))
ACTIVE(app(X1, X2)) → ACTIVE(X2)
ACTIVE(cons(X1, X2)) → CONS(active(X1), X2)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(from(X)) → FROM(active(X))
ACTIVE(from(X)) → ACTIVE(X)
ACTIVE(s(X)) → S(active(X))
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(zWadr(X1, X2)) → ZWADR(active(X1), X2)
ACTIVE(zWadr(X1, X2)) → ACTIVE(X1)
ACTIVE(zWadr(X1, X2)) → ZWADR(X1, active(X2))
ACTIVE(zWadr(X1, X2)) → ACTIVE(X2)
ACTIVE(prefix(X)) → PREFIX(active(X))
ACTIVE(prefix(X)) → ACTIVE(X)
APP(mark(X1), X2) → APP(X1, X2)
APP(X1, mark(X2)) → APP(X1, X2)
CONS(mark(X1), X2) → CONS(X1, X2)
FROM(mark(X)) → FROM(X)
S(mark(X)) → S(X)
ZWADR(mark(X1), X2) → ZWADR(X1, X2)
ZWADR(X1, mark(X2)) → ZWADR(X1, X2)
PREFIX(mark(X)) → PREFIX(X)
PROPER(app(X1, X2)) → APP(proper(X1), proper(X2))
PROPER(app(X1, X2)) → PROPER(X1)
PROPER(app(X1, X2)) → PROPER(X2)
PROPER(cons(X1, X2)) → CONS(proper(X1), proper(X2))
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(from(X)) → FROM(proper(X))
PROPER(from(X)) → PROPER(X)
PROPER(s(X)) → S(proper(X))
PROPER(s(X)) → PROPER(X)
PROPER(zWadr(X1, X2)) → ZWADR(proper(X1), proper(X2))
PROPER(zWadr(X1, X2)) → PROPER(X1)
PROPER(zWadr(X1, X2)) → PROPER(X2)
PROPER(prefix(X)) → PREFIX(proper(X))
PROPER(prefix(X)) → PROPER(X)
APP(ok(X1), ok(X2)) → APP(X1, X2)
CONS(ok(X1), ok(X2)) → CONS(X1, X2)
FROM(ok(X)) → FROM(X)
S(ok(X)) → S(X)
ZWADR(ok(X1), ok(X2)) → ZWADR(X1, X2)
PREFIX(ok(X)) → PREFIX(X)
TOP(mark(X)) → TOP(proper(X))
TOP(mark(X)) → PROPER(X)
TOP(ok(X)) → TOP(active(X))
TOP(ok(X)) → ACTIVE(X)

The TRS R consists of the following rules:

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

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 9 SCCs with 27 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic path order with status [LPO].
Quasi-Precedence:
top > [app2, nil, zWadr2, prefix1, proper1] > [ok1, s1]

Status:
prefix1: [1]
PREFIX1: [1]
ok1: [1]
app2: [1,2]
s1: [1]
proper1: [1]
top: []
zWadr2: [2,1]
nil: []


The following usable rules [FROCOS05] were oriented:

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

(7) Obligation:

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

PREFIX(mark(X)) → PREFIX(X)

The TRS R consists of the following rules:

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

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

(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: Combined order from the following AFS and order.
PREFIX(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
app(x1, x2)  =  app(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
from(x1)  =  x1
s(x1)  =  x1
zWadr(x1, x2)  =  zWadr(x1, x2)
prefix(x1)  =  x1
proper(x1)  =  proper(x1)
ok(x1)  =  ok
top(x1)  =  top

Lexicographic path order with status [LPO].
Quasi-Precedence:
top > proper1 > [active1, app2] > [zWadr2, ok] > [nil, cons2] > mark1

Status:
active1: [1]
cons2: [1,2]
mark1: [1]
app2: [1,2]
ok: []
proper1: [1]
top: []
zWadr2: [2,1]
nil: []


The following usable rules [FROCOS05] were oriented:

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

(9) Obligation:

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

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

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

(10) PisEmptyProof (EQUIVALENT transformation)

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

(11) TRUE

(12) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(13) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ZWADR(X1, mark(X2)) → ZWADR(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ZWADR(x1, x2)  =  ZWADR(x2)
mark(x1)  =  mark(x1)
ok(x1)  =  x1
active(x1)  =  active(x1)
app(x1, x2)  =  app(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
from(x1)  =  from(x1)
s(x1)  =  x1
zWadr(x1, x2)  =  zWadr(x1, x2)
prefix(x1)  =  prefix(x1)
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic path order with status [LPO].
Quasi-Precedence:
ZWADR1 > mark1
[proper1, top] > [active1, prefix1] > from1 > mark1
[proper1, top] > [active1, prefix1] > zWadr2 > cons2 > app2 > mark1
[proper1, top] > [active1, prefix1] > zWadr2 > cons2 > nil > mark1

Status:
prefix1: [1]
active1: [1]
cons2: [2,1]
from1: [1]
mark1: [1]
ZWADR1: [1]
app2: [2,1]
proper1: [1]
top: []
zWadr2: [2,1]
nil: []


The following usable rules [FROCOS05] were oriented:

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

(14) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ZWADR(ok(X1), ok(X2)) → ZWADR(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ZWADR(x1, x2)  =  ZWADR(x1)
mark(x1)  =  x1
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
app(x1, x2)  =  app(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x2)
from(x1)  =  from(x1)
s(x1)  =  x1
zWadr(x1, x2)  =  x2
prefix(x1)  =  prefix(x1)
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic path order with status [LPO].
Quasi-Precedence:
[active1, app2, nil, prefix1, proper1, top] > [ok1, cons1, from1]

Status:
prefix1: [1]
active1: [1]
from1: [1]
cons1: [1]
ok1: [1]
ZWADR1: [1]
app2: [2,1]
proper1: [1]
top: []
nil: []


The following usable rules [FROCOS05] were oriented:

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

(16) Obligation:

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

ZWADR(mark(X1), X2) → ZWADR(X1, X2)

The TRS R consists of the following rules:

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

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

(17) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ZWADR(mark(X1), X2) → ZWADR(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ZWADR(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
app(x1, x2)  =  app(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
from(x1)  =  from(x1)
s(x1)  =  x1
zWadr(x1, x2)  =  zWadr(x1, x2)
prefix(x1)  =  x1
proper(x1)  =  proper(x1)
ok(x1)  =  ok
top(x1)  =  top

Lexicographic path order with status [LPO].
Quasi-Precedence:
[active1, app2, from1, proper1] > ok > [nil, zWadr2] > cons2 > mark1
top > mark1

Status:
active1: [1]
cons2: [2,1]
from1: [1]
mark1: [1]
app2: [2,1]
ok: []
proper1: [1]
top: []
zWadr2: [2,1]
nil: []


The following usable rules [FROCOS05] were oriented:

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

(18) Obligation:

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

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

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

(19) PisEmptyProof (EQUIVALENT transformation)

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

(20) TRUE

(21) Obligation:

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

S(ok(X)) → S(X)
S(mark(X)) → S(X)

The TRS R consists of the following rules:

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

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

(22) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic path order with status [LPO].
Quasi-Precedence:
top > [app2, nil, zWadr2, prefix1, proper1] > [ok1, s1]

Status:
prefix1: [1]
ok1: [1]
app2: [1,2]
s1: [1]
proper1: [1]
top: []
zWadr2: [2,1]
nil: []
S1: [1]


The following usable rules [FROCOS05] were oriented:

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

(23) Obligation:

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

S(mark(X)) → S(X)

The TRS R consists of the following rules:

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

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

(24) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic path order with status [LPO].
Quasi-Precedence:
top > proper1 > [active1, app2] > [zWadr2, ok] > [nil, cons2] > mark1

Status:
active1: [1]
cons2: [1,2]
mark1: [1]
app2: [1,2]
ok: []
proper1: [1]
top: []
zWadr2: [2,1]
nil: []


The following usable rules [FROCOS05] were oriented:

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

(25) Obligation:

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

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

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

(26) PisEmptyProof (EQUIVALENT transformation)

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

(27) TRUE

(28) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(29) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic path order with status [LPO].
Quasi-Precedence:
top > [app2, nil, zWadr2, prefix1, proper1] > [ok1, s1]

Status:
prefix1: [1]
ok1: [1]
app2: [1,2]
s1: [1]
proper1: [1]
top: []
zWadr2: [2,1]
nil: []
FROM1: [1]


The following usable rules [FROCOS05] were oriented:

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

(30) Obligation:

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

FROM(mark(X)) → FROM(X)

The TRS R consists of the following rules:

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

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

(31) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic path order with status [LPO].
Quasi-Precedence:
top > proper1 > [active1, app2] > [zWadr2, ok] > [nil, cons2] > mark1

Status:
active1: [1]
cons2: [1,2]
mark1: [1]
app2: [1,2]
ok: []
proper1: [1]
top: []
zWadr2: [2,1]
nil: []


The following usable rules [FROCOS05] were oriented:

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

(32) Obligation:

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

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

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

(33) PisEmptyProof (EQUIVALENT transformation)

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

(34) TRUE

(35) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(36) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


CONS(ok(X1), ok(X2)) → CONS(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2)  =  x2
ok(x1)  =  ok(x1)
mark(x1)  =  x1
active(x1)  =  active(x1)
app(x1, x2)  =  app(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x2)
from(x1)  =  from(x1)
s(x1)  =  x1
zWadr(x1, x2)  =  x1
prefix(x1)  =  x1
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic path order with status [LPO].
Quasi-Precedence:
top > [active1, nil] > app2 > [ok1, cons1]
top > [active1, nil] > from1 > [ok1, cons1]
top > proper1 > app2 > [ok1, cons1]
top > proper1 > from1 > [ok1, cons1]

Status:
from1: [1]
active1: [1]
cons1: [1]
ok1: [1]
app2: [1,2]
proper1: [1]
top: []
nil: []


The following usable rules [FROCOS05] were oriented:

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

(37) Obligation:

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

CONS(mark(X1), X2) → CONS(X1, X2)

The TRS R consists of the following rules:

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

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

(38) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


CONS(mark(X1), X2) → CONS(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
app(x1, x2)  =  app(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
from(x1)  =  from(x1)
s(x1)  =  x1
zWadr(x1, x2)  =  zWadr(x1, x2)
prefix(x1)  =  x1
proper(x1)  =  x1
ok(x1)  =  ok
top(x1)  =  top

Lexicographic path order with status [LPO].
Quasi-Precedence:
[active1, app2, from1] > nil > ok > zWadr2 > mark1
[active1, app2, from1] > cons2 > ok > zWadr2 > mark1

Status:
active1: [1]
cons2: [2,1]
from1: [1]
mark1: [1]
app2: [1,2]
ok: []
top: []
zWadr2: [2,1]
nil: []


The following usable rules [FROCOS05] were oriented:

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

(39) Obligation:

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

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

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

(40) PisEmptyProof (EQUIVALENT transformation)

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

(41) TRUE

(42) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(43) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


APP(X1, mark(X2)) → APP(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
APP(x1, x2)  =  APP(x2)
mark(x1)  =  mark(x1)
ok(x1)  =  x1
active(x1)  =  active(x1)
app(x1, x2)  =  app(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
from(x1)  =  from(x1)
s(x1)  =  s(x1)
zWadr(x1, x2)  =  zWadr(x1, x2)
prefix(x1)  =  prefix(x1)
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic path order with status [LPO].
Quasi-Precedence:
APP1 > mark1
top > [active1, s1, prefix1, proper1] > nil > mark1
top > [active1, s1, prefix1, proper1] > from1 > mark1
top > [active1, s1, prefix1, proper1] > zWadr2 > cons2 > app2 > mark1

Status:
APP1: [1]
from1: [1]
mark1: [1]
prefix1: [1]
active1: [1]
cons2: [2,1]
app2: [2,1]
s1: [1]
proper1: [1]
top: []
zWadr2: [2,1]
nil: []


The following usable rules [FROCOS05] were oriented:

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

(44) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(45) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


APP(ok(X1), ok(X2)) → APP(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
APP(x1, x2)  =  APP(x1)
mark(x1)  =  x1
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
app(x1, x2)  =  app(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x2)
from(x1)  =  from(x1)
s(x1)  =  x1
zWadr(x1, x2)  =  x2
prefix(x1)  =  prefix(x1)
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic path order with status [LPO].
Quasi-Precedence:
[active1, app2, nil, prefix1, proper1, top] > [ok1, cons1, from1]

Status:
APP1: [1]
prefix1: [1]
active1: [1]
from1: [1]
cons1: [1]
ok1: [1]
app2: [2,1]
proper1: [1]
top: []
nil: []


The following usable rules [FROCOS05] were oriented:

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

(46) Obligation:

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

APP(mark(X1), X2) → APP(X1, X2)

The TRS R consists of the following rules:

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

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

(47) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


APP(mark(X1), X2) → APP(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
APP(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
app(x1, x2)  =  app(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
from(x1)  =  from(x1)
s(x1)  =  x1
zWadr(x1, x2)  =  zWadr(x1, x2)
prefix(x1)  =  x1
proper(x1)  =  x1
ok(x1)  =  ok
top(x1)  =  top

Lexicographic path order with status [LPO].
Quasi-Precedence:
[active1, app2, from1] > nil > ok > zWadr2 > mark1
[active1, app2, from1] > cons2 > ok > zWadr2 > mark1

Status:
active1: [1]
cons2: [2,1]
from1: [1]
mark1: [1]
app2: [1,2]
ok: []
top: []
zWadr2: [2,1]
nil: []


The following usable rules [FROCOS05] were oriented:

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

(48) Obligation:

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

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

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

(49) PisEmptyProof (EQUIVALENT transformation)

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

(50) TRUE

(51) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(52) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(app(X1, X2)) → PROPER(X2)
PROPER(app(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(zWadr(X1, X2)) → PROPER(X1)
PROPER(zWadr(X1, X2)) → PROPER(X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
app(x1, x2)  =  app(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
from(x1)  =  x1
s(x1)  =  x1
zWadr(x1, x2)  =  zWadr(x1, x2)
prefix(x1)  =  x1
active(x1)  =  active(x1)
nil  =  nil
mark(x1)  =  x1
proper(x1)  =  proper(x1)
ok(x1)  =  x1
top(x1)  =  top

Lexicographic path order with status [LPO].
Quasi-Precedence:
[app2, active1, proper1] > [cons2, zWadr2] > nil

Status:
active1: [1]
cons2: [2,1]
PROPER1: [1]
app2: [2,1]
proper1: [1]
top: []
zWadr2: [1,2]
nil: []


The following usable rules [FROCOS05] were oriented:

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

(53) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(54) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic path order with status [LPO].
Quasi-Precedence:
[active1, nil] > ok > [cons1, proper1] > [prefix1, zWadr1] > app2
[active1, nil] > ok > top

Status:
prefix1: [1]
PROPER1: [1]
active1: [1]
cons1: [1]
app2: [1,2]
ok: []
proper1: [1]
zWadr1: [1]
top: []
nil: []


The following usable rules [FROCOS05] were oriented:

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

(55) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(56) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic path order with status [LPO].
Quasi-Precedence:
[PROPER1, s1] > [prefix, ok] > cons2 > [app2, mark]
nil > [prefix, ok] > cons2 > [app2, mark]
top > [app2, mark]

Status:
PROPER1: [1]
cons2: [1,2]
prefix: []
mark: []
s1: [1]
app2: [1,2]
ok: []
top: []
nil: []


The following usable rules [FROCOS05] were oriented:

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

(57) Obligation:

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

PROPER(from(X)) → PROPER(X)

The TRS R consists of the following rules:

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

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

(58) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic path order with status [LPO].
Quasi-Precedence:
[from1, cons, prefix, proper1, top] > [s, ok] > app2
[from1, cons, prefix, proper1, top] > [s, ok] > [nil, zWadr1]

Status:
PROPER1: [1]
from1: [1]
cons: []
prefix: []
app2: [1,2]
ok: []
s: []
proper1: [1]
zWadr1: [1]
top: []
nil: []


The following usable rules [FROCOS05] were oriented:

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

(59) Obligation:

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

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

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

(60) PisEmptyProof (EQUIVALENT transformation)

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

(61) TRUE

(62) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(63) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(app(X1, X2)) → ACTIVE(X2)
ACTIVE(app(X1, X2)) → ACTIVE(X1)
ACTIVE(zWadr(X1, X2)) → ACTIVE(X1)
ACTIVE(zWadr(X1, X2)) → ACTIVE(X2)
ACTIVE(prefix(X)) → ACTIVE(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  x1
app(x1, x2)  =  app(x1, x2)
cons(x1, x2)  =  x1
from(x1)  =  x1
s(x1)  =  x1
zWadr(x1, x2)  =  zWadr(x1, x2)
prefix(x1)  =  prefix(x1)
active(x1)  =  active(x1)
nil  =  nil
mark(x1)  =  x1
proper(x1)  =  proper(x1)
ok(x1)  =  ok
top(x1)  =  top

Lexicographic path order with status [LPO].
Quasi-Precedence:
top > [active1, nil] > ok > app2
top > [active1, nil] > ok > zWadr2
top > [active1, nil] > ok > prefix1
top > proper1 > ok > app2
top > proper1 > ok > zWadr2
top > proper1 > ok > prefix1

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


The following usable rules [FROCOS05] were oriented:

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

(64) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(65) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic path order with status [LPO].
Quasi-Precedence:
[nil, prefix] > proper1 > [s1, ok] > zWadr2 > app2

Status:
prefix: []
s1: [1]
app2: [1,2]
ok: []
proper1: [1]
top: []
zWadr2: [1,2]
ACTIVE1: [1]
nil: []


The following usable rules [FROCOS05] were oriented:

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

(66) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(67) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(from(X)) → ACTIVE(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
cons(x1, x2)  =  cons(x1)
from(x1)  =  from(x1)
active(x1)  =  x1
app(x1, x2)  =  app(x1, x2)
nil  =  nil
mark(x1)  =  mark(x1)
s(x1)  =  s(x1)
zWadr(x1, x2)  =  zWadr(x1, x2)
prefix(x1)  =  prefix(x1)
proper(x1)  =  x1
ok(x1)  =  x1
top(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:
[ACTIVE1, from1] > cons1 > nil > mark1
[ACTIVE1, from1] > s1 > mark1
[app2, zWadr2, prefix1] > cons1 > nil > mark1

Status:
prefix1: [1]
from1: [1]
cons1: [1]
mark1: [1]
app2: [1,2]
s1: [1]
zWadr2: [2,1]
ACTIVE1: [1]
nil: []


The following usable rules [FROCOS05] were oriented:

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

(68) Obligation:

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

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

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

(69) PisEmptyProof (EQUIVALENT transformation)

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

(70) TRUE

(71) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(72) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic path order with status [LPO].
Quasi-Precedence:
from1 > cons1 > app2 > [mark1, nil, s1]
[zWadr2, prefix1] > cons1 > app2 > [mark1, nil, s1]

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


The following usable rules [FROCOS05] were oriented:

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

(73) Obligation:

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

TOP(ok(X)) → TOP(active(X))

The TRS R consists of the following rules:

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

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

(74) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic path order with status [LPO].
Quasi-Precedence:
TOP1 > active1 > mark
[ok1, nil, proper1, top] > active1 > mark

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


The following usable rules [FROCOS05] were oriented:

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

(75) Obligation:

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

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

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

(76) PisEmptyProof (EQUIVALENT transformation)

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

(77) TRUE