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


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NO
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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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 disproven:

(0) QTRS
(1) QTRSToCSRProof [EQUIVALENT, 0 ms]
(2) CSR
(3) CSRRRRProof [EQUIVALENT, 30 ms]
(4) CSR
(5) CSRRRRProof [EQUIVALENT, 0 ms]
(6) CSR
(7) CSRRRRProof [EQUIVALENT, 10 ms]
(8) CSR
(9) ContextSensitiveLoopProof [COMPLETE, 0 ms]
(10) NO


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

   active(zeros) -> mark(cons(0, zeros))
   active(and(tt, X)) -> mark(X)
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(s(length(L)))
   active(take(0, IL)) -> mark(nil)
   active(take(s(M), cons(N, IL))) -> mark(cons(N, take(M, IL)))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(and(X1, X2)) -> and(active(X1), X2)
   active(length(X)) -> length(active(X))
   active(s(X)) -> s(active(X))
   active(take(X1, X2)) -> take(active(X1), X2)
   active(take(X1, X2)) -> take(X1, active(X2))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   and(mark(X1), X2) -> mark(and(X1, X2))
   length(mark(X)) -> mark(length(X))
   s(mark(X)) -> mark(s(X))
   take(mark(X1), X2) -> mark(take(X1, X2))
   take(X1, mark(X2)) -> mark(take(X1, X2))
   proper(zeros) -> ok(zeros)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(length(X)) -> length(proper(X))
   proper(nil) -> ok(nil)
   proper(s(X)) -> s(proper(X))
   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   length(ok(X)) -> ok(length(X))
   s(ok(X)) -> ok(s(X))
   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

Q is empty.

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(1) QTRSToCSRProof (EQUIVALENT)
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(and(tt, X)) -> mark(X)
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(s(length(L)))
   active(take(0, IL)) -> mark(nil)
   active(take(s(M), cons(N, IL))) -> mark(cons(N, take(M, IL)))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(and(X1, X2)) -> and(active(X1), X2)
   active(length(X)) -> length(active(X))
   active(s(X)) -> s(active(X))
   active(take(X1, X2)) -> take(active(X1), X2)
   active(take(X1, X2)) -> take(X1, active(X2))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   and(mark(X1), X2) -> mark(and(X1, X2))
   length(mark(X)) -> mark(length(X))
   s(mark(X)) -> mark(s(X))
   take(mark(X1), X2) -> mark(take(X1, X2))
   take(X1, mark(X2)) -> mark(take(X1, X2))
   proper(zeros) -> ok(zeros)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(length(X)) -> length(proper(X))
   proper(nil) -> ok(nil)
   proper(s(X)) -> s(proper(X))
   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   length(ok(X)) -> ok(length(X))
   s(ok(X)) -> ok(s(X))
   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

Q is empty.
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
and: {1}
tt: empty set
length: {1}
nil: empty set
s: {1}
take: {1, 2}
The QTRS contained all rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is complete (and sound).
----------------------------------------

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

   zeros -> cons(0, zeros)
   and(tt, X) -> X
   length(nil) -> 0
   length(cons(N, L)) -> s(length(L))
   take(0, IL) -> nil
   take(s(M), cons(N, IL)) -> cons(N, take(M, IL))

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
and: {1}
tt: empty set
length: {1}
nil: empty set
s: {1}
take: {1, 2}

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

   zeros -> cons(0, zeros)
   and(tt, X) -> X
   length(nil) -> 0
   length(cons(N, L)) -> s(length(L))
   take(0, IL) -> nil
   take(s(M), cons(N, IL)) -> cons(N, take(M, IL))

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
and: {1}
tt: empty set
length: {1}
nil: empty set
s: {1}
take: {1, 2}
Used ordering:
Polynomial interpretation [POLO]:

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

   length(nil) -> 0




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

   zeros -> cons(0, zeros)
   and(tt, X) -> X
   length(cons(N, L)) -> s(length(L))
   take(0, IL) -> nil
   take(s(M), cons(N, IL)) -> cons(N, take(M, IL))

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
and: {1}
tt: empty set
length: {1}
nil: empty set
s: {1}
take: {1, 2}

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

   zeros -> cons(0, zeros)
   and(tt, X) -> X
   length(cons(N, L)) -> s(length(L))
   take(0, IL) -> nil
   take(s(M), cons(N, IL)) -> cons(N, take(M, IL))

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
and: {1}
tt: empty set
length: {1}
nil: empty set
s: {1}
take: {1, 2}
Used ordering:
Polynomial interpretation [POLO]:

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

   and(tt, X) -> X




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

   zeros -> cons(0, zeros)
   length(cons(N, L)) -> s(length(L))
   take(0, IL) -> nil
   take(s(M), cons(N, IL)) -> cons(N, take(M, IL))

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
length: {1}
nil: empty set
s: {1}
take: {1, 2}

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

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

   zeros -> cons(0, zeros)
   length(cons(N, L)) -> s(length(L))
   take(0, IL) -> nil
   take(s(M), cons(N, IL)) -> cons(N, take(M, IL))

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
length: {1}
nil: empty set
s: {1}
take: {1, 2}
Used ordering:
Polynomial interpretation [POLO]:

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

   take(0, IL) -> nil




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

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

   zeros -> cons(0, zeros)
   length(cons(N, L)) -> s(length(L))
   take(s(M), cons(N, IL)) -> cons(N, take(M, IL))

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
length: {1}
s: {1}
take: {1, 2}

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

(9) ContextSensitiveLoopProof (COMPLETE)

   zeros -> cons(0, zeros)
   length(cons(N, L)) -> s(length(L))
   take(s(M), cons(N, IL)) -> cons(N, take(M, IL))

---------- Loop: ----------

length(zeros) -> length(cons(0, zeros)) with rule zeros -> cons(0, zeros) at position [0] and matcher [ ]

length(cons(0, zeros)) -> s(length(zeros)) with rule length(cons(N, L)) -> s(length(L)) at position [] and matcher [N / 0, L / zeros]

Now an instance of the first term with Matcher [ ] occurs in the last term at position [0].

Context: s([])

We used [[THIEMANN_LOOPS_UNDER_STRATEGIES], Theorem 1] to show that this loop is an context-sensitive loop.
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(10)
NO