YES
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) CSRRRRProof [EQUIVALENT, 49 ms]
(4) CSR
(5) CSRRRRProof [EQUIVALENT, 0 ms]
(6) CSR
(7) CSRRRRProof [EQUIVALENT, 0 ms]
(8) CSR
(9) CSRRRRProof [EQUIVALENT, 0 ms]
(10) CSR
(11) RisEmptyProof [EQUIVALENT, 0 ms]
(12) YES


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(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   active(nats) -> mark(cons(0, incr(nats)))
   active(pairs) -> mark(cons(0, incr(odds)))
   active(odds) -> mark(incr(pairs))
   active(incr(cons(X, XS))) -> mark(cons(s(X), incr(XS)))
   active(head(cons(X, XS))) -> mark(X)
   active(tail(cons(X, XS))) -> mark(XS)
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(incr(X)) -> incr(active(X))
   active(s(X)) -> s(active(X))
   active(head(X)) -> head(active(X))
   active(tail(X)) -> tail(active(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   incr(mark(X)) -> mark(incr(X))
   s(mark(X)) -> mark(s(X))
   head(mark(X)) -> mark(head(X))
   tail(mark(X)) -> mark(tail(X))
   proper(nats) -> ok(nats)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(incr(X)) -> incr(proper(X))
   proper(pairs) -> ok(pairs)
   proper(odds) -> ok(odds)
   proper(s(X)) -> s(proper(X))
   proper(head(X)) -> head(proper(X))
   proper(tail(X)) -> tail(proper(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   incr(ok(X)) -> ok(incr(X))
   s(ok(X)) -> ok(s(X))
   head(ok(X)) -> ok(head(X))
   tail(ok(X)) -> ok(tail(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

The set Q consists of the following terms:

   active(nats)
   active(pairs)
   active(odds)
   active(cons(x0, x1))
   active(incr(x0))
   active(s(x0))
   active(head(x0))
   active(tail(x0))
   cons(mark(x0), x1)
   incr(mark(x0))
   s(mark(x0))
   head(mark(x0))
   tail(mark(x0))
   proper(nats)
   proper(cons(x0, x1))
   proper(0)
   proper(incr(x0))
   proper(pairs)
   proper(odds)
   proper(s(x0))
   proper(head(x0))
   proper(tail(x0))
   cons(ok(x0), ok(x1))
   incr(ok(x0))
   s(ok(x0))
   head(ok(x0))
   tail(ok(x0))
   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(nats) -> mark(cons(0, incr(nats)))
   active(pairs) -> mark(cons(0, incr(odds)))
   active(odds) -> mark(incr(pairs))
   active(incr(cons(X, XS))) -> mark(cons(s(X), incr(XS)))
   active(head(cons(X, XS))) -> mark(X)
   active(tail(cons(X, XS))) -> mark(XS)
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(incr(X)) -> incr(active(X))
   active(s(X)) -> s(active(X))
   active(head(X)) -> head(active(X))
   active(tail(X)) -> tail(active(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   incr(mark(X)) -> mark(incr(X))
   s(mark(X)) -> mark(s(X))
   head(mark(X)) -> mark(head(X))
   tail(mark(X)) -> mark(tail(X))
   proper(nats) -> ok(nats)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(incr(X)) -> incr(proper(X))
   proper(pairs) -> ok(pairs)
   proper(odds) -> ok(odds)
   proper(s(X)) -> s(proper(X))
   proper(head(X)) -> head(proper(X))
   proper(tail(X)) -> tail(proper(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   incr(ok(X)) -> ok(incr(X))
   s(ok(X)) -> ok(s(X))
   head(ok(X)) -> ok(head(X))
   tail(ok(X)) -> ok(tail(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

The set Q consists of the following terms:

   active(nats)
   active(pairs)
   active(odds)
   active(cons(x0, x1))
   active(incr(x0))
   active(s(x0))
   active(head(x0))
   active(tail(x0))
   cons(mark(x0), x1)
   incr(mark(x0))
   s(mark(x0))
   head(mark(x0))
   tail(mark(x0))
   proper(nats)
   proper(cons(x0, x1))
   proper(0)
   proper(incr(x0))
   proper(pairs)
   proper(odds)
   proper(s(x0))
   proper(head(x0))
   proper(tail(x0))
   cons(ok(x0), ok(x1))
   incr(ok(x0))
   s(ok(x0))
   head(ok(x0))
   tail(ok(x0))
   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:

nats: empty set
cons: {1}
0: empty set
incr: {1}
pairs: empty set
odds: empty set
s: {1}
head: {1}
tail: {1}
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:

   nats -> cons(0, incr(nats))
   pairs -> cons(0, incr(odds))
   odds -> incr(pairs)
   incr(cons(X, XS)) -> cons(s(X), incr(XS))
   head(cons(X, XS)) -> X
   tail(cons(X, XS)) -> XS

The replacement map contains the following entries:

nats: empty set
cons: {1}
0: empty set
incr: {1}
pairs: empty set
odds: empty set
s: {1}
head: {1}
tail: {1}

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

(3) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   nats -> cons(0, incr(nats))
   pairs -> cons(0, incr(odds))
   odds -> incr(pairs)
   incr(cons(X, XS)) -> cons(s(X), incr(XS))
   head(cons(X, XS)) -> X
   tail(cons(X, XS)) -> XS

The replacement map contains the following entries:

nats: empty set
cons: {1}
0: empty set
incr: {1}
pairs: empty set
odds: empty set
s: {1}
head: {1}
tail: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(cons(x_1, x_2)) = 2*x_1 + x_2
   POL(head(x_1)) = 1 + 2*x_1
   POL(incr(x_1)) = x_1
   POL(nats) = 0
   POL(odds) = 0
   POL(pairs) = 0
   POL(s(x_1)) = x_1
   POL(tail(x_1)) = 2*x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   head(cons(X, XS)) -> X




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

(4)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   nats -> cons(0, incr(nats))
   pairs -> cons(0, incr(odds))
   odds -> incr(pairs)
   incr(cons(X, XS)) -> cons(s(X), incr(XS))
   tail(cons(X, XS)) -> XS

The replacement map contains the following entries:

nats: empty set
cons: {1}
0: empty set
incr: {1}
pairs: empty set
odds: empty set
s: {1}
tail: {1}

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

(5) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   nats -> cons(0, incr(nats))
   pairs -> cons(0, incr(odds))
   odds -> incr(pairs)
   incr(cons(X, XS)) -> cons(s(X), incr(XS))
   tail(cons(X, XS)) -> XS

The replacement map contains the following entries:

nats: empty set
cons: {1}
0: empty set
incr: {1}
pairs: empty set
odds: empty set
s: {1}
tail: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(cons(x_1, x_2)) = x_1 + x_2
   POL(incr(x_1)) = x_1
   POL(nats) = 0
   POL(odds) = 0
   POL(pairs) = 0
   POL(s(x_1)) = x_1
   POL(tail(x_1)) = 1 + x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   tail(cons(X, XS)) -> XS




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

(6)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   nats -> cons(0, incr(nats))
   pairs -> cons(0, incr(odds))
   odds -> incr(pairs)
   incr(cons(X, XS)) -> cons(s(X), incr(XS))

The replacement map contains the following entries:

nats: empty set
cons: {1}
0: empty set
incr: {1}
pairs: empty set
odds: empty set
s: {1}

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

(7) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   nats -> cons(0, incr(nats))
   pairs -> cons(0, incr(odds))
   odds -> incr(pairs)
   incr(cons(X, XS)) -> cons(s(X), incr(XS))

The replacement map contains the following entries:

nats: empty set
cons: {1}
0: empty set
incr: {1}
pairs: empty set
odds: empty set
s: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(cons(x_1, x_2)) = 2*x_1
   POL(incr(x_1)) = 1 + x_1
   POL(nats) = 1
   POL(odds) = 2
   POL(pairs) = 0
   POL(s(x_1)) = x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   nats -> cons(0, incr(nats))
   odds -> incr(pairs)
   incr(cons(X, XS)) -> cons(s(X), incr(XS))




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

(8)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   pairs -> cons(0, incr(odds))

The replacement map contains the following entries:

cons: {1}
0: empty set
incr: {1}
pairs: empty set
odds: empty set

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

(9) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   pairs -> cons(0, incr(odds))

The replacement map contains the following entries:

cons: {1}
0: empty set
incr: {1}
pairs: empty set
odds: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(cons(x_1, x_2)) = x_1 + x_2
   POL(incr(x_1)) = x_1
   POL(odds) = 0
   POL(pairs) = 1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   pairs -> cons(0, incr(odds))




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

(10)
Obligation:
Context-sensitive rewrite system:
R is empty.

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(11) RisEmptyProof (EQUIVALENT)
The CSR R is empty. Hence, termination is trivially proven.
----------------------------------------

(12)
YES