(0) Obligation:

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

active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(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(sel(s(N), cons(X, XS))) → SEL(N, XS)
ACTIVE(minus(s(X), s(Y))) → MINUS(X, Y)
ACTIVE(quot(s(X), s(Y))) → S(quot(minus(X, Y), s(Y)))
ACTIVE(quot(s(X), s(Y))) → QUOT(minus(X, Y), s(Y))
ACTIVE(quot(s(X), s(Y))) → MINUS(X, Y)
ACTIVE(zWquot(cons(X, XS), cons(Y, YS))) → CONS(quot(X, Y), zWquot(XS, YS))
ACTIVE(zWquot(cons(X, XS), cons(Y, YS))) → QUOT(X, Y)
ACTIVE(zWquot(cons(X, XS), cons(Y, YS))) → ZWQUOT(XS, YS)
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(sel(X1, X2)) → SEL(active(X1), X2)
ACTIVE(sel(X1, X2)) → ACTIVE(X1)
ACTIVE(sel(X1, X2)) → SEL(X1, active(X2))
ACTIVE(sel(X1, X2)) → ACTIVE(X2)
ACTIVE(minus(X1, X2)) → MINUS(active(X1), X2)
ACTIVE(minus(X1, X2)) → ACTIVE(X1)
ACTIVE(minus(X1, X2)) → MINUS(X1, active(X2))
ACTIVE(minus(X1, X2)) → ACTIVE(X2)
ACTIVE(quot(X1, X2)) → QUOT(active(X1), X2)
ACTIVE(quot(X1, X2)) → ACTIVE(X1)
ACTIVE(quot(X1, X2)) → QUOT(X1, active(X2))
ACTIVE(quot(X1, X2)) → ACTIVE(X2)
ACTIVE(zWquot(X1, X2)) → ZWQUOT(active(X1), X2)
ACTIVE(zWquot(X1, X2)) → ACTIVE(X1)
ACTIVE(zWquot(X1, X2)) → ZWQUOT(X1, active(X2))
ACTIVE(zWquot(X1, X2)) → ACTIVE(X2)
FROM(mark(X)) → FROM(X)
CONS(mark(X1), X2) → CONS(X1, X2)
S(mark(X)) → S(X)
SEL(mark(X1), X2) → SEL(X1, X2)
SEL(X1, mark(X2)) → SEL(X1, X2)
MINUS(mark(X1), X2) → MINUS(X1, X2)
MINUS(X1, mark(X2)) → MINUS(X1, X2)
QUOT(mark(X1), X2) → QUOT(X1, X2)
QUOT(X1, mark(X2)) → QUOT(X1, X2)
ZWQUOT(mark(X1), X2) → ZWQUOT(X1, X2)
ZWQUOT(X1, mark(X2)) → ZWQUOT(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(sel(X1, X2)) → SEL(proper(X1), proper(X2))
PROPER(sel(X1, X2)) → PROPER(X1)
PROPER(sel(X1, X2)) → PROPER(X2)
PROPER(minus(X1, X2)) → MINUS(proper(X1), proper(X2))
PROPER(minus(X1, X2)) → PROPER(X1)
PROPER(minus(X1, X2)) → PROPER(X2)
PROPER(quot(X1, X2)) → QUOT(proper(X1), proper(X2))
PROPER(quot(X1, X2)) → PROPER(X1)
PROPER(quot(X1, X2)) → PROPER(X2)
PROPER(zWquot(X1, X2)) → ZWQUOT(proper(X1), proper(X2))
PROPER(zWquot(X1, X2)) → PROPER(X1)
PROPER(zWquot(X1, X2)) → PROPER(X2)
FROM(ok(X)) → FROM(X)
CONS(ok(X1), ok(X2)) → CONS(X1, X2)
S(ok(X)) → S(X)
SEL(ok(X1), ok(X2)) → SEL(X1, X2)
MINUS(ok(X1), ok(X2)) → MINUS(X1, X2)
QUOT(ok(X1), ok(X2)) → QUOT(X1, X2)
ZWQUOT(ok(X1), ok(X2)) → ZWQUOT(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(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(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 10 SCCs with 31 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

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

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(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.


ZWQUOT(X1, mark(X2)) → ZWQUOT(X1, X2)
ZWQUOT(ok(X1), ok(X2)) → ZWQUOT(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ZWQUOT(x1, x2)  =  x2
mark(x1)  =  mark(x1)
ok(x1)  =  ok(x1)

Recursive Path Order [RPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

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

ZWQUOT(mark(X1), X2) → ZWQUOT(X1, X2)

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(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.


ZWQUOT(mark(X1), X2) → ZWQUOT(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ZWQUOT(x1, x2)  =  x1
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(9) Obligation:

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

active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(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) 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:

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

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(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.

(13) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


QUOT(X1, mark(X2)) → QUOT(X1, X2)
QUOT(ok(X1), ok(X2)) → QUOT(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
QUOT(x1, x2)  =  x2
mark(x1)  =  mark(x1)
ok(x1)  =  ok(x1)

Recursive Path Order [RPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(14) Obligation:

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

QUOT(mark(X1), X2) → QUOT(X1, X2)

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(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.


QUOT(mark(X1), X2) → QUOT(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
QUOT(x1, x2)  =  x1
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(16) 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(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(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) PisEmptyProof (EQUIVALENT transformation)

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

(18) TRUE

(19) Obligation:

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

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

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(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.

(20) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MINUS(X1, mark(X2)) → MINUS(X1, X2)
MINUS(ok(X1), ok(X2)) → MINUS(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MINUS(x1, x2)  =  x2
mark(x1)  =  mark(x1)
ok(x1)  =  ok(x1)

Recursive Path Order [RPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(21) Obligation:

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

MINUS(mark(X1), X2) → MINUS(X1, X2)

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(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.


MINUS(mark(X1), X2) → MINUS(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MINUS(x1, x2)  =  x1
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(23) 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(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(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) PisEmptyProof (EQUIVALENT transformation)

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

(25) TRUE

(26) Obligation:

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

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

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(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.

(27) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


SEL(X1, mark(X2)) → SEL(X1, X2)
SEL(ok(X1), ok(X2)) → SEL(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SEL(x1, x2)  =  x2
mark(x1)  =  mark(x1)
ok(x1)  =  ok(x1)

Recursive Path Order [RPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(28) Obligation:

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

SEL(mark(X1), X2) → SEL(X1, X2)

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(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.


SEL(mark(X1), X2) → SEL(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SEL(x1, x2)  =  x1
mark(x1)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(30) 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(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(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) PisEmptyProof (EQUIVALENT transformation)

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

(32) TRUE

(33) 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(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(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.

(34) 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)  =  x1
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive Path Order [RPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(35) 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(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(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.


S(mark(X)) → S(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive Path Order [RPO].
Precedence:
mark1 > S1

The following usable rules [FROCOS05] were oriented: none

(37) 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(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(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) PisEmptyProof (EQUIVALENT transformation)

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

(39) TRUE

(40) 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(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(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.

(41) 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)  =  mark(x1)

Recursive Path Order [RPO].
Precedence:
ok1 > CONS1
mark1 > CONS1

The following usable rules [FROCOS05] were oriented: none

(42) 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(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(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.


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)

Recursive Path Order [RPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(44) 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(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(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) PisEmptyProof (EQUIVALENT transformation)

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

(46) TRUE

(47) 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(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(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.

(48) 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)  =  x1
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive Path Order [RPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(49) 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(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(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.

(50) 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: Recursive Path Order [RPO].
Precedence:
mark1 > FROM1

The following usable rules [FROCOS05] were oriented: none

(51) 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(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(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.

(52) PisEmptyProof (EQUIVALENT transformation)

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

(53) TRUE

(54) 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(sel(X1, X2)) → PROPER(X1)
PROPER(sel(X1, X2)) → PROPER(X2)
PROPER(minus(X1, X2)) → PROPER(X1)
PROPER(minus(X1, X2)) → PROPER(X2)
PROPER(quot(X1, X2)) → PROPER(X1)
PROPER(quot(X1, X2)) → PROPER(X2)
PROPER(zWquot(X1, X2)) → PROPER(X1)
PROPER(zWquot(X1, X2)) → PROPER(X2)

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(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.

(55) 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(from(X)) → PROPER(X)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(s(X)) → PROPER(X)
PROPER(sel(X1, X2)) → PROPER(X1)
PROPER(sel(X1, X2)) → PROPER(X2)
PROPER(minus(X1, X2)) → PROPER(X1)
PROPER(minus(X1, X2)) → PROPER(X2)
PROPER(quot(X1, X2)) → PROPER(X1)
PROPER(quot(X1, X2)) → PROPER(X2)
PROPER(zWquot(X1, X2)) → PROPER(X1)
PROPER(zWquot(X1, X2)) → PROPER(X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive Path Order [RPO].
Precedence:
cons2 > PROPER1
from1 > PROPER1
s1 > PROPER1
sel2 > PROPER1
minus2 > PROPER1
quot2 > PROPER1
zWquot2 > PROPER1

The following usable rules [FROCOS05] were oriented: none

(56) Obligation:

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

active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(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.

(57) PisEmptyProof (EQUIVALENT transformation)

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

(58) TRUE

(59) Obligation:

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

ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(from(X)) → ACTIVE(X)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(sel(X1, X2)) → ACTIVE(X1)
ACTIVE(sel(X1, X2)) → ACTIVE(X2)
ACTIVE(minus(X1, X2)) → ACTIVE(X1)
ACTIVE(minus(X1, X2)) → ACTIVE(X2)
ACTIVE(quot(X1, X2)) → ACTIVE(X1)
ACTIVE(quot(X1, X2)) → ACTIVE(X2)
ACTIVE(zWquot(X1, X2)) → ACTIVE(X1)
ACTIVE(zWquot(X1, X2)) → ACTIVE(X2)

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(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.

(60) 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)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(sel(X1, X2)) → ACTIVE(X1)
ACTIVE(sel(X1, X2)) → ACTIVE(X2)
ACTIVE(minus(X1, X2)) → ACTIVE(X1)
ACTIVE(minus(X1, X2)) → ACTIVE(X2)
ACTIVE(quot(X1, X2)) → ACTIVE(X1)
ACTIVE(quot(X1, X2)) → ACTIVE(X2)
ACTIVE(zWquot(X1, X2)) → ACTIVE(X1)
ACTIVE(zWquot(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)  =  cons(x1)
from(x1)  =  from(x1)
s(x1)  =  s(x1)
sel(x1, x2)  =  sel(x1, x2)
minus(x1, x2)  =  minus(x1, x2)
quot(x1, x2)  =  quot(x1, x2)
zWquot(x1, x2)  =  zWquot(x1, x2)

Recursive Path Order [RPO].
Precedence:
cons1 > ACTIVE1
from1 > ACTIVE1
s1 > ACTIVE1
sel2 > ACTIVE1
minus2 > ACTIVE1
quot2 > ACTIVE1
zWquot2 > ACTIVE1

The following usable rules [FROCOS05] were oriented: none

(61) 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(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(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.

(62) PisEmptyProof (EQUIVALENT transformation)

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

(63) TRUE

(64) 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(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(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.