/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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


YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 43 ms]
(4) CSR
(5) CSRRRRProof [EQUIVALENT, 0 ms]
(6) CSR
(7) RisEmptyProof [EQUIVALENT, 0 ms]
(8) YES


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

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

   active(zeros) -> mark(cons(0, zeros))
   active(tail(cons(X, XS))) -> mark(XS)
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(tail(X)) -> tail(active(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   tail(mark(X)) -> mark(tail(X))
   proper(zeros) -> ok(zeros)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(tail(X)) -> tail(proper(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   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(zeros)
   active(cons(x0, x1))
   active(tail(x0))
   cons(mark(x0), x1)
   tail(mark(x0))
   proper(zeros)
   proper(cons(x0, x1))
   proper(0)
   proper(tail(x0))
   cons(ok(x0), ok(x1))
   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(zeros) -> mark(cons(0, zeros))
   active(tail(cons(X, XS))) -> mark(XS)
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(tail(X)) -> tail(active(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   tail(mark(X)) -> mark(tail(X))
   proper(zeros) -> ok(zeros)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(tail(X)) -> tail(proper(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   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(zeros)
   active(cons(x0, x1))
   active(tail(x0))
   cons(mark(x0), x1)
   tail(mark(x0))
   proper(zeros)
   proper(cons(x0, x1))
   proper(0)
   proper(tail(x0))
   cons(ok(x0), ok(x1))
   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:

zeros: empty set
cons: {1}
0: empty set
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:

   zeros -> cons(0, zeros)
   tail(cons(X, XS)) -> XS

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
tail: {1}

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

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

   zeros -> cons(0, zeros)
   tail(cons(X, XS)) -> XS

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
tail: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(cons(x_1, x_2)) = x_1 + x_2
   POL(tail(x_1)) = 1 + 2*x_1
   POL(zeros) = 2
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




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

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

   zeros -> cons(0, zeros)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set

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

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

   zeros -> cons(0, zeros)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
Used ordering:
Polynomial interpretation [POLO]:

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

   zeros -> cons(0, zeros)




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

(6)
Obligation:
Context-sensitive rewrite system:
R is empty.

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

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

(8)
YES