(0) Obligation:

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

active(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
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(from(X)) → CONS(X, from(s(X)))
ACTIVE(from(X)) → FROM(s(X))
ACTIVE(from(X)) → S(X)
ACTIVE(after(s(N), cons(X, XS))) → AFTER(N, XS)
ACTIVE(from(X)) → FROM(active(X))
ACTIVE(from(X)) → ACTIVE(X)
ACTIVE(cons(X1, X2)) → CONS(active(X1), X2)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(s(X)) → S(active(X))
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(after(X1, X2)) → AFTER(active(X1), X2)
ACTIVE(after(X1, X2)) → ACTIVE(X1)
ACTIVE(after(X1, X2)) → AFTER(X1, active(X2))
ACTIVE(after(X1, X2)) → ACTIVE(X2)
FROM(mark(X)) → FROM(X)
CONS(mark(X1), X2) → CONS(X1, X2)
S(mark(X)) → S(X)
AFTER(mark(X1), X2) → AFTER(X1, X2)
AFTER(X1, mark(X2)) → AFTER(X1, X2)
PROPER(from(X)) → FROM(proper(X))
PROPER(from(X)) → PROPER(X)
PROPER(cons(X1, X2)) → CONS(proper(X1), proper(X2))
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(s(X)) → S(proper(X))
PROPER(s(X)) → PROPER(X)
PROPER(after(X1, X2)) → AFTER(proper(X1), proper(X2))
PROPER(after(X1, X2)) → PROPER(X1)
PROPER(after(X1, X2)) → PROPER(X2)
FROM(ok(X)) → FROM(X)
CONS(ok(X1), ok(X2)) → CONS(X1, X2)
S(ok(X)) → S(X)
AFTER(ok(X1), ok(X2)) → AFTER(X1, X2)
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(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
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 7 SCCs with 15 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

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

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
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.


AFTER(ok(X1), ok(X2)) → AFTER(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
AFTER(x1, x2)  =  AFTER(x2)
mark(x1)  =  x1
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
from(x1)  =  x1
cons(x1, x2)  =  x2
s(x1)  =  x1
after(x1, x2)  =  x2
0  =  0
proper(x1)  =  proper
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:
proper > ok1 > AFTER1
proper > 0

Status:
AFTER1: [1]
ok1: multiset
active1: multiset
0: multiset
proper: multiset
top: multiset

The following usable rules [FROCOS05] were oriented:

active(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
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:

AFTER(X1, mark(X2)) → AFTER(X1, X2)
AFTER(mark(X1), X2) → AFTER(X1, X2)

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
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.


AFTER(X1, mark(X2)) → AFTER(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
AFTER(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
from(x1)  =  from(x1)
cons(x1, x2)  =  cons(x1, x2)
s(x1)  =  x1
after(x1, x2)  =  after(x1, x2)
0  =  0
proper(x1)  =  proper(x1)
ok(x1)  =  ok
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:
0 > ok > active1 > from1 > mark1
0 > ok > active1 > cons2 > mark1
0 > ok > active1 > after2 > mark1
0 > ok > top
proper1 > from1 > mark1
proper1 > cons2 > mark1
proper1 > after2 > mark1

Status:
mark1: [1]
active1: [1]
from1: multiset
cons2: [2,1]
after2: multiset
0: multiset
proper1: [1]
ok: []
top: []

The following usable rules [FROCOS05] were oriented:

active(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(9) Obligation:

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

AFTER(mark(X1), X2) → AFTER(X1, X2)

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
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) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


AFTER(mark(X1), X2) → AFTER(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
AFTER(x1, x2)  =  AFTER(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
from(x1)  =  x1
cons(x1, x2)  =  cons(x1, x2)
s(x1)  =  x1
after(x1, x2)  =  after(x1, x2)
0  =  0
proper(x1)  =  proper(x1)
ok(x1)  =  x1
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:
proper1 > cons2 > after2 > mark1
proper1 > 0 > mark1
top > active1 > cons2 > after2 > mark1

Status:
AFTER2: [1,2]
mark1: multiset
active1: multiset
cons2: multiset
after2: [2,1]
0: multiset
proper1: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(11) Obligation:

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

active(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
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.

(12) PisEmptyProof (EQUIVALENT transformation)

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

(13) TRUE

(14) 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(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
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.


S(mark(X)) → S(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S(x1)  =  S(x1)
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
from(x1)  =  x1
cons(x1, x2)  =  cons(x1, x2)
s(x1)  =  x1
after(x1, x2)  =  after(x1, x2)
0  =  0
proper(x1)  =  x1
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:
active1 > cons2 > mark1
active1 > after2 > mark1
0 > mark1

Status:
S1: [1]
mark1: multiset
active1: multiset
cons2: [1,2]
after2: [2,1]
0: multiset
top: []

The following usable rules [FROCOS05] were oriented:

active(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
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:

S(ok(X)) → S(X)

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
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.


S(ok(X)) → S(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S(x1)  =  S(x1)
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
from(x1)  =  x1
mark(x1)  =  mark(x1)
cons(x1, x2)  =  cons(x1, x2)
s(x1)  =  s(x1)
after(x1, x2)  =  after(x1, x2)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:
S1 > ok1
active1 > cons2 > mark1 > ok1
active1 > s1 > after2 > mark1 > ok1
top > proper1 > cons2 > mark1 > ok1
top > proper1 > s1 > after2 > mark1 > ok1
top > proper1 > 0 > ok1

Status:
S1: multiset
ok1: multiset
active1: [1]
mark1: [1]
cons2: multiset
s1: [1]
after2: multiset
0: multiset
proper1: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
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(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
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:

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

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
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.


CONS(ok(X1), ok(X2)) → CONS(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2)  =  CONS(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  x1
active(x1)  =  active(x1)
from(x1)  =  from(x1)
cons(x1, x2)  =  cons(x1, x2)
s(x1)  =  s(x1)
after(x1, x2)  =  x2
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:
active1 > from1 > ok1 > CONS1 > top
active1 > cons2 > ok1 > CONS1 > top
active1 > s1 > ok1 > CONS1 > top
proper1 > from1 > ok1 > CONS1 > top
proper1 > cons2 > ok1 > CONS1 > top
proper1 > s1 > ok1 > CONS1 > top
proper1 > 0 > top

Status:
CONS1: multiset
ok1: multiset
active1: multiset
from1: [1]
cons2: [1,2]
s1: [1]
0: multiset
proper1: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
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:

CONS(mark(X1), X2) → CONS(X1, X2)

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
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.


CONS(mark(X1), X2) → CONS(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2)  =  CONS(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
from(x1)  =  x1
cons(x1, x2)  =  cons(x1, x2)
s(x1)  =  x1
after(x1, x2)  =  after(x1, x2)
0  =  0
proper(x1)  =  proper(x1)
ok(x1)  =  x1
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:
proper1 > cons2 > after2 > mark1
proper1 > 0 > mark1
top > active1 > cons2 > after2 > mark1

Status:
CONS2: [1,2]
mark1: multiset
active1: multiset
cons2: multiset
after2: [2,1]
0: multiset
proper1: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
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(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
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(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(29) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


FROM(mark(X)) → FROM(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
FROM(x1)  =  FROM(x1)
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
from(x1)  =  x1
cons(x1, x2)  =  cons(x1, x2)
s(x1)  =  x1
after(x1, x2)  =  after(x1, x2)
0  =  0
proper(x1)  =  x1
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:
active1 > cons2 > mark1
active1 > after2 > mark1
0 > mark1

Status:
FROM1: [1]
mark1: multiset
active1: multiset
cons2: [1,2]
after2: [2,1]
0: multiset
top: []

The following usable rules [FROCOS05] were oriented:

active(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(30) Obligation:

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

FROM(ok(X)) → FROM(X)

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(31) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


FROM(ok(X)) → FROM(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
FROM(x1)  =  FROM(x1)
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
from(x1)  =  x1
mark(x1)  =  mark(x1)
cons(x1, x2)  =  cons(x1, x2)
s(x1)  =  s(x1)
after(x1, x2)  =  after(x1, x2)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:
FROM1 > ok1
active1 > cons2 > mark1 > ok1
active1 > s1 > after2 > mark1 > ok1
top > proper1 > cons2 > mark1 > ok1
top > proper1 > s1 > after2 > mark1 > ok1
top > proper1 > 0 > ok1

Status:
FROM1: multiset
ok1: multiset
active1: [1]
mark1: [1]
cons2: multiset
s1: [1]
after2: multiset
0: multiset
proper1: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
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(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
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:

PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(from(X)) → PROPER(X)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(s(X)) → PROPER(X)
PROPER(after(X1, X2)) → PROPER(X1)
PROPER(after(X1, X2)) → PROPER(X2)

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
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.


PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(s(X)) → PROPER(X)
PROPER(after(X1, X2)) → PROPER(X1)
PROPER(after(X1, X2)) → PROPER(X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
cons(x1, x2)  =  cons(x1, x2)
from(x1)  =  x1
s(x1)  =  s(x1)
after(x1, x2)  =  after(x1, x2)
active(x1)  =  active(x1)
mark(x1)  =  mark(x1)
0  =  0
proper(x1)  =  x1
ok(x1)  =  x1
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:
active1 > cons2 > PROPER1 > top
active1 > cons2 > mark1 > top
active1 > s1 > after2 > PROPER1 > top
active1 > s1 > after2 > mark1 > top
0 > top

Status:
PROPER1: multiset
cons2: multiset
s1: [1]
after2: multiset
active1: multiset
mark1: [1]
0: multiset
top: []

The following usable rules [FROCOS05] were oriented:

active(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
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:

PROPER(from(X)) → PROPER(X)

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
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.


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)  =  x1
from(x1)  =  from(x1)
active(x1)  =  x1
mark(x1)  =  mark
cons(x1, x2)  =  cons(x1, x2)
s(x1)  =  s(x1)
after(x1, x2)  =  after(x2)
0  =  0
proper(x1)  =  proper(x1)
ok(x1)  =  x1
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:
0 > mark > s1
proper1 > from1 > mark > s1
proper1 > from1 > cons2 > s1
proper1 > after1 > mark > s1
top > s1

Status:
from1: multiset
mark: multiset
cons2: multiset
s1: multiset
after1: multiset
0: multiset
proper1: [1]
top: multiset

The following usable rules [FROCOS05] were oriented:

active(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
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(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
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:

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

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
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.


ACTIVE(after(X1, X2)) → ACTIVE(X1)
ACTIVE(after(X1, X2)) → ACTIVE(X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
cons(x1, x2)  =  x1
from(x1)  =  x1
s(x1)  =  x1
after(x1, x2)  =  after(x1, x2)
active(x1)  =  active(x1)
mark(x1)  =  mark
0  =  0
proper(x1)  =  proper(x1)
ok(x1)  =  ok
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:
proper1 > after2 > mark > ACTIVE1
proper1 > after2 > ok > ACTIVE1
proper1 > 0 > mark > ACTIVE1
proper1 > 0 > ok > ACTIVE1
top > active1 > after2 > mark > ACTIVE1
top > active1 > after2 > ok > ACTIVE1

Status:
ACTIVE1: multiset
after2: multiset
active1: multiset
mark: multiset
0: multiset
proper1: multiset
ok: multiset
top: multiset

The following usable rules [FROCOS05] were oriented:

active(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
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:

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(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
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.


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)  =  active(x1)
mark(x1)  =  mark
after(x1, x2)  =  after(x1, x2)
0  =  0
proper(x1)  =  proper(x1)
ok(x1)  =  x1
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:
0 > mark > s1
0 > mark > after2
proper1 > s1
proper1 > after2
top > active1 > mark > s1
top > active1 > mark > after2

Status:
ACTIVE1: [1]
s1: multiset
active1: [1]
mark: []
after2: multiset
0: multiset
proper1: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
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:

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

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
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.


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)  =  x1
cons(x1, x2)  =  x1
from(x1)  =  from(x1)
active(x1)  =  active(x1)
mark(x1)  =  mark
s(x1)  =  x1
after(x1, x2)  =  after(x1, x2)
0  =  0
proper(x1)  =  proper(x1)
ok(x1)  =  x1
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:
top > active1 > mark > proper1 > from1
top > active1 > mark > proper1 > after2
top > active1 > mark > proper1 > 0

Status:
from1: multiset
active1: multiset
mark: []
after2: multiset
0: multiset
proper1: multiset
top: []

The following usable rules [FROCOS05] were oriented:

active(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(48) Obligation:

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

ACTIVE(cons(X1, X2)) → ACTIVE(X1)

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(49) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(cons(X1, X2)) → ACTIVE(X1)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  x1
cons(x1, x2)  =  cons(x1, x2)
active(x1)  =  active(x1)
from(x1)  =  from
mark(x1)  =  mark
s(x1)  =  s
after(x1, x2)  =  after
0  =  0
proper(x1)  =  x1
ok(x1)  =  ok
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:
active1 > cons2 > mark > ok
active1 > from > mark > ok
active1 > s > mark > ok
after > mark > ok
0 > ok
top > ok

Status:
cons2: [1,2]
active1: [1]
from: multiset
mark: multiset
s: multiset
after: []
0: multiset
ok: multiset
top: multiset

The following usable rules [FROCOS05] were oriented:

active(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(50) Obligation:

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

active(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(51) PisEmptyProof (EQUIVALENT transformation)

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

(52) TRUE

(53) Obligation:

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

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

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(X1, X2))
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.