YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) QTRS
(1) QTRSToCSRProof [SOUND, 0 ms]
(2) CSR
(3) CSDependencyPairsProof [EQUIVALENT, 0 ms]
(4) QCSDP
(5) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
(6) AND
    (7) QCSDP
        (8) QCSDPSubtermProof [EQUIVALENT, 0 ms]
        (9) QCSDP
        (10) PIsEmptyProof [EQUIVALENT, 0 ms]
        (11) YES
    (12) QCSDP
        (13) QCSDPReductionPairProof [EQUIVALENT, 14 ms]
        (14) QCSDP
        (15) PIsEmptyProof [EQUIVALENT, 0 ms]
        (16) YES
    (17) QCSDP
        (18) QCSDPSubtermProof [EQUIVALENT, 0 ms]
        (19) QCSDP
        (20) PIsEmptyProof [EQUIVALENT, 0 ms]
        (21) YES
    (22) QCSDP
        (23) QCSDPSubtermProof [EQUIVALENT, 0 ms]
        (24) QCSDP
        (25) PIsEmptyProof [EQUIVALENT, 0 ms]
        (26) YES


----------------------------------------

(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))

The set Q consists of the following terms:

   active(from(x0))
   active(cons(x0, x1))
   active(s(x0))
   active(sel(x0, x1))
   active(minus(x0, x1))
   active(quot(x0, x1))
   active(zWquot(x0, x1))
   from(mark(x0))
   cons(mark(x0), x1)
   s(mark(x0))
   sel(mark(x0), x1)
   sel(x0, mark(x1))
   minus(mark(x0), x1)
   minus(x0, mark(x1))
   quot(mark(x0), x1)
   quot(x0, mark(x1))
   zWquot(mark(x0), x1)
   zWquot(x0, mark(x1))
   proper(from(x0))
   proper(cons(x0, x1))
   proper(s(x0))
   proper(sel(x0, x1))
   proper(0)
   proper(minus(x0, x1))
   proper(quot(x0, x1))
   proper(zWquot(x0, x1))
   proper(nil)
   from(ok(x0))
   cons(ok(x0), ok(x1))
   s(ok(x0))
   sel(ok(x0), ok(x1))
   minus(ok(x0), ok(x1))
   quot(ok(x0), ok(x1))
   zWquot(ok(x0), ok(x1))
   top(mark(x0))
   top(ok(x0))


----------------------------------------

(1) QTRSToCSRProof (SOUND)
The following Q TRS is given: 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))

The set Q consists of the following terms:

   active(from(x0))
   active(cons(x0, x1))
   active(s(x0))
   active(sel(x0, x1))
   active(minus(x0, x1))
   active(quot(x0, x1))
   active(zWquot(x0, x1))
   from(mark(x0))
   cons(mark(x0), x1)
   s(mark(x0))
   sel(mark(x0), x1)
   sel(x0, mark(x1))
   minus(mark(x0), x1)
   minus(x0, mark(x1))
   quot(mark(x0), x1)
   quot(x0, mark(x1))
   zWquot(mark(x0), x1)
   zWquot(x0, mark(x1))
   proper(from(x0))
   proper(cons(x0, x1))
   proper(s(x0))
   proper(sel(x0, x1))
   proper(0)
   proper(minus(x0, x1))
   proper(quot(x0, x1))
   proper(zWquot(x0, x1))
   proper(nil)
   from(ok(x0))
   cons(ok(x0), ok(x1))
   s(ok(x0))
   sel(ok(x0), ok(x1))
   minus(ok(x0), ok(x1))
   quot(ok(x0), ok(x1))
   zWquot(ok(x0), ok(x1))
   top(mark(x0))
   top(ok(x0))

Special symbols used for the transformation (see [GM04]):
top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
The replacement map contains the following entries:

from: {1}
cons: {1}
s: {1}
sel: {1, 2}
0: empty set
minus: {1, 2}
quot: {1, 2}
zWquot: {1, 2}
nil: empty set
The QTRS contained just a subset of rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is sound, but not necessarily complete.
----------------------------------------

(2)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   from(X) -> cons(X, from(s(X)))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, XS)
   minus(X, 0) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
   zWquot(XS, nil) -> nil
   zWquot(nil, XS) -> nil
   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), zWquot(XS, YS))

The replacement map contains the following entries:

from: {1}
cons: {1}
s: {1}
sel: {1, 2}
0: empty set
minus: {1, 2}
quot: {1, 2}
zWquot: {1, 2}
nil: empty set

----------------------------------------

(3) CSDependencyPairsProof (EQUIVALENT)
Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
----------------------------------------

(4)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {from_1, s_1, sel_2, minus_2, quot_2, zWquot_2, SEL_2, MINUS_2, QUOT_2, ZWQUOT_2, FROM_1} are replacing on all positions.
For all symbols f in {cons_2} we have mu(f) = {1}.
The symbols in {U_1} are not replacing on any position.

The ordinary context-sensitive dependency pairs DP_o are:
   SEL(s(N), cons(X, XS)) -> SEL(N, XS)
   MINUS(s(X), s(Y)) -> MINUS(X, Y)
   QUOT(s(X), s(Y)) -> QUOT(minus(X, Y), s(Y))
   QUOT(s(X), s(Y)) -> MINUS(X, Y)
   ZWQUOT(cons(X, XS), cons(Y, YS)) -> QUOT(X, Y)

The collapsing dependency pairs are DP_c:
   SEL(s(N), cons(X, XS)) -> XS


The hidden terms of R are:

   from(s(x0))
   zWquot(x0, x1)

Every hiding context is built from:
   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@2c24230a
   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@16363af5
   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@24e8bf76

Hence, the new unhiding pairs DP_u are :
   SEL(s(N), cons(X, XS)) -> U(XS)
   U(s(x_0)) -> U(x_0)
   U(from(x_0)) -> U(x_0)
   U(zWquot(x_0, x_1)) -> U(x_0)
   U(zWquot(x_0, x_1)) -> U(x_1)
   U(from(s(x0))) -> FROM(s(x0))
   U(zWquot(x0, x1)) -> ZWQUOT(x0, x1)

The TRS R consists of the following rules:

   from(X) -> cons(X, from(s(X)))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, XS)
   minus(X, 0) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
   zWquot(XS, nil) -> nil
   zWquot(nil, XS) -> nil
   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), zWquot(XS, YS))

Q is empty.

----------------------------------------

(5) QCSDependencyGraphProof (EQUIVALENT)
The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 4 SCCs with 5 less nodes.

----------------------------------------

(6)
Complex Obligation (AND)

----------------------------------------

(7)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {from_1, s_1, sel_2, minus_2, quot_2, zWquot_2, MINUS_2} are replacing on all positions.
For all symbols f in {cons_2} we have mu(f) = {1}.

The TRS P consists of the following rules:

   MINUS(s(X), s(Y)) -> MINUS(X, Y)

The TRS R consists of the following rules:

   from(X) -> cons(X, from(s(X)))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, XS)
   minus(X, 0) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
   zWquot(XS, nil) -> nil
   zWquot(nil, XS) -> nil
   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), zWquot(XS, YS))

Q is empty.

----------------------------------------

(8) QCSDPSubtermProof (EQUIVALENT)
We use the subterm processor [DA_EMMES].


The following pairs can be oriented strictly and are deleted.

   MINUS(s(X), s(Y)) -> MINUS(X, Y)
The remaining pairs can at least be oriented weakly.
none
Used ordering:  Combined order from the following AFS and order.
MINUS(x1, x2)  =  x1


Subterm Order

----------------------------------------

(9)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {from_1, s_1, sel_2, minus_2, quot_2, zWquot_2} are replacing on all positions.
For all symbols f in {cons_2} we have mu(f) = {1}.

The TRS P consists of the following rules:
none

The TRS R consists of the following rules:

   from(X) -> cons(X, from(s(X)))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, XS)
   minus(X, 0) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
   zWquot(XS, nil) -> nil
   zWquot(nil, XS) -> nil
   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), zWquot(XS, YS))

Q is empty.

----------------------------------------

(10) PIsEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
----------------------------------------

(11)
YES

----------------------------------------

(12)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {from_1, s_1, sel_2, minus_2, quot_2, zWquot_2, QUOT_2} are replacing on all positions.
For all symbols f in {cons_2} we have mu(f) = {1}.

The TRS P consists of the following rules:

   QUOT(s(X), s(Y)) -> QUOT(minus(X, Y), s(Y))

The TRS R consists of the following rules:

   from(X) -> cons(X, from(s(X)))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, XS)
   minus(X, 0) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
   zWquot(XS, nil) -> nil
   zWquot(nil, XS) -> nil
   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), zWquot(XS, YS))

Q is empty.

----------------------------------------

(13) QCSDPReductionPairProof (EQUIVALENT)
Using the order

Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(QUOT(x_1, x_2)) = 2*x_1
   POL(cons(x_1, x_2)) = 0
   POL(from(x_1)) = 0
   POL(minus(x_1, x_2)) = 0
   POL(nil) = 0
   POL(quot(x_1, x_2)) = x_1 + x_2
   POL(s(x_1)) = 2
   POL(zWquot(x_1, x_2)) = 2


the following usable rules


   minus(X, 0) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
   from(X) -> cons(X, from(s(X)))
   zWquot(XS, nil) -> nil
   zWquot(nil, XS) -> nil
   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), zWquot(XS, YS))


could all be oriented weakly.

Furthermore, the pairs


   QUOT(s(X), s(Y)) -> QUOT(minus(X, Y), s(Y))


could be oriented strictly and thus removed by the CS-Reduction Pair Processor [LPAR08,DA_EMMES].


----------------------------------------

(14)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {from_1, s_1, sel_2, minus_2, quot_2, zWquot_2} are replacing on all positions.
For all symbols f in {cons_2} we have mu(f) = {1}.

The TRS P consists of the following rules:
none

The TRS R consists of the following rules:

   from(X) -> cons(X, from(s(X)))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, XS)
   minus(X, 0) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
   zWquot(XS, nil) -> nil
   zWquot(nil, XS) -> nil
   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), zWquot(XS, YS))

Q is empty.

----------------------------------------

(15) PIsEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
----------------------------------------

(16)
YES

----------------------------------------

(17)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {from_1, s_1, sel_2, minus_2, quot_2, zWquot_2} are replacing on all positions.
For all symbols f in {cons_2} we have mu(f) = {1}.
The symbols in {U_1} are not replacing on any position.

The TRS P consists of the following rules:

   U(s(x_0)) -> U(x_0)
   U(from(x_0)) -> U(x_0)
   U(zWquot(x_0, x_1)) -> U(x_0)
   U(zWquot(x_0, x_1)) -> U(x_1)

The TRS R consists of the following rules:

   from(X) -> cons(X, from(s(X)))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, XS)
   minus(X, 0) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
   zWquot(XS, nil) -> nil
   zWquot(nil, XS) -> nil
   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), zWquot(XS, YS))

Q is empty.

----------------------------------------

(18) QCSDPSubtermProof (EQUIVALENT)
We use the subterm processor [DA_EMMES].


The following pairs can be oriented strictly and are deleted.

   U(s(x_0)) -> U(x_0)
   U(from(x_0)) -> U(x_0)
   U(zWquot(x_0, x_1)) -> U(x_0)
   U(zWquot(x_0, x_1)) -> U(x_1)
The remaining pairs can at least be oriented weakly.
none
Used ordering:  Combined order from the following AFS and order.
U(x1)  =  x1


Subterm Order

----------------------------------------

(19)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {from_1, s_1, sel_2, minus_2, quot_2, zWquot_2} are replacing on all positions.
For all symbols f in {cons_2} we have mu(f) = {1}.

The TRS P consists of the following rules:
none

The TRS R consists of the following rules:

   from(X) -> cons(X, from(s(X)))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, XS)
   minus(X, 0) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
   zWquot(XS, nil) -> nil
   zWquot(nil, XS) -> nil
   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), zWquot(XS, YS))

Q is empty.

----------------------------------------

(20) PIsEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
----------------------------------------

(21)
YES

----------------------------------------

(22)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {from_1, s_1, sel_2, minus_2, quot_2, zWquot_2, SEL_2} are replacing on all positions.
For all symbols f in {cons_2} we have mu(f) = {1}.

The TRS P consists of the following rules:

   SEL(s(N), cons(X, XS)) -> SEL(N, XS)

The TRS R consists of the following rules:

   from(X) -> cons(X, from(s(X)))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, XS)
   minus(X, 0) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
   zWquot(XS, nil) -> nil
   zWquot(nil, XS) -> nil
   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), zWquot(XS, YS))

Q is empty.

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(23) QCSDPSubtermProof (EQUIVALENT)
We use the subterm processor [DA_EMMES].


The following pairs can be oriented strictly and are deleted.

   SEL(s(N), cons(X, XS)) -> SEL(N, XS)
The remaining pairs can at least be oriented weakly.
none
Used ordering:  Combined order from the following AFS and order.
SEL(x1, x2)  =  x1


Subterm Order

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(24)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {from_1, s_1, sel_2, minus_2, quot_2, zWquot_2} are replacing on all positions.
For all symbols f in {cons_2} we have mu(f) = {1}.

The TRS P consists of the following rules:
none

The TRS R consists of the following rules:

   from(X) -> cons(X, from(s(X)))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, XS)
   minus(X, 0) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
   zWquot(XS, nil) -> nil
   zWquot(nil, XS) -> nil
   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), zWquot(XS, YS))

Q is empty.

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(25) PIsEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
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(26)
YES