NO
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 disproven:

(0) QTRS
(1) QTRSToCSRProof [EQUIVALENT, 0 ms]
(2) CSR
(3) CSRRRRProof [EQUIVALENT, 52 ms]
(4) CSR
(5) CSRRRRProof [EQUIVALENT, 11 ms]
(6) CSR
(7) ContextSensitiveLoopProof [COMPLETE, 44 ms]
(8) NO


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

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

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(U12(tt, L))
   active(U12(tt, L)) -> mark(s(length(L)))
   active(U21(tt, IL, M, N)) -> mark(U22(tt, IL, M, N))
   active(U22(tt, IL, M, N)) -> mark(U23(tt, IL, M, N))
   active(U23(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U11(tt, L))
   active(take(0, IL)) -> mark(nil)
   active(take(s(M), cons(N, IL))) -> mark(U21(tt, IL, M, N))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(U11(X1, X2)) -> U11(active(X1), X2)
   active(U12(X1, X2)) -> U12(active(X1), X2)
   active(s(X)) -> s(active(X))
   active(length(X)) -> length(active(X))
   active(U21(X1, X2, X3, X4)) -> U21(active(X1), X2, X3, X4)
   active(U22(X1, X2, X3, X4)) -> U22(active(X1), X2, X3, X4)
   active(U23(X1, X2, X3, X4)) -> U23(active(X1), X2, X3, X4)
   active(take(X1, X2)) -> take(active(X1), X2)
   active(take(X1, X2)) -> take(X1, active(X2))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   U11(mark(X1), X2) -> mark(U11(X1, X2))
   U12(mark(X1), X2) -> mark(U12(X1, X2))
   s(mark(X)) -> mark(s(X))
   length(mark(X)) -> mark(length(X))
   U21(mark(X1), X2, X3, X4) -> mark(U21(X1, X2, X3, X4))
   U22(mark(X1), X2, X3, X4) -> mark(U22(X1, X2, X3, X4))
   U23(mark(X1), X2, X3, X4) -> mark(U23(X1, X2, X3, X4))
   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(U11(X1, X2)) -> U11(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
   proper(s(X)) -> s(proper(X))
   proper(length(X)) -> length(proper(X))
   proper(U21(X1, X2, X3, X4)) -> U21(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(U22(X1, X2, X3, X4)) -> U22(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(U23(X1, X2, X3, X4)) -> U23(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
   U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
   s(ok(X)) -> ok(s(X))
   length(ok(X)) -> ok(length(X))
   U21(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U21(X1, X2, X3, X4))
   U22(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U22(X1, X2, X3, X4))
   U23(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U23(X1, X2, X3, X4))
   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

Q is empty.

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

(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(U11(tt, L)) -> mark(U12(tt, L))
   active(U12(tt, L)) -> mark(s(length(L)))
   active(U21(tt, IL, M, N)) -> mark(U22(tt, IL, M, N))
   active(U22(tt, IL, M, N)) -> mark(U23(tt, IL, M, N))
   active(U23(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U11(tt, L))
   active(take(0, IL)) -> mark(nil)
   active(take(s(M), cons(N, IL))) -> mark(U21(tt, IL, M, N))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(U11(X1, X2)) -> U11(active(X1), X2)
   active(U12(X1, X2)) -> U12(active(X1), X2)
   active(s(X)) -> s(active(X))
   active(length(X)) -> length(active(X))
   active(U21(X1, X2, X3, X4)) -> U21(active(X1), X2, X3, X4)
   active(U22(X1, X2, X3, X4)) -> U22(active(X1), X2, X3, X4)
   active(U23(X1, X2, X3, X4)) -> U23(active(X1), X2, X3, X4)
   active(take(X1, X2)) -> take(active(X1), X2)
   active(take(X1, X2)) -> take(X1, active(X2))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   U11(mark(X1), X2) -> mark(U11(X1, X2))
   U12(mark(X1), X2) -> mark(U12(X1, X2))
   s(mark(X)) -> mark(s(X))
   length(mark(X)) -> mark(length(X))
   U21(mark(X1), X2, X3, X4) -> mark(U21(X1, X2, X3, X4))
   U22(mark(X1), X2, X3, X4) -> mark(U22(X1, X2, X3, X4))
   U23(mark(X1), X2, X3, X4) -> mark(U23(X1, X2, X3, X4))
   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(U11(X1, X2)) -> U11(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
   proper(s(X)) -> s(proper(X))
   proper(length(X)) -> length(proper(X))
   proper(U21(X1, X2, X3, X4)) -> U21(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(U22(X1, X2, X3, X4)) -> U22(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(U23(X1, X2, X3, X4)) -> U23(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
   U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
   s(ok(X)) -> ok(s(X))
   length(ok(X)) -> ok(length(X))
   U21(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U21(X1, X2, X3, X4))
   U22(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U22(X1, X2, X3, X4))
   U23(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U23(X1, X2, X3, X4))
   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
U11: {1}
tt: empty set
U12: {1}
s: {1}
length: {1}
U21: {1}
U22: {1}
U23: {1}
take: {1, 2}
nil: empty set
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)
   U11(tt, L) -> U12(tt, L)
   U12(tt, L) -> s(length(L))
   U21(tt, IL, M, N) -> U22(tt, IL, M, N)
   U22(tt, IL, M, N) -> U23(tt, IL, M, N)
   U23(tt, IL, M, N) -> cons(N, take(M, IL))
   length(nil) -> 0
   length(cons(N, L)) -> U11(tt, L)
   take(0, IL) -> nil
   take(s(M), cons(N, IL)) -> U21(tt, IL, M, N)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
U12: {1}
s: {1}
length: {1}
U21: {1}
U22: {1}
U23: {1}
take: {1, 2}
nil: empty set

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

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

   zeros -> cons(0, zeros)
   U11(tt, L) -> U12(tt, L)
   U12(tt, L) -> s(length(L))
   U21(tt, IL, M, N) -> U22(tt, IL, M, N)
   U22(tt, IL, M, N) -> U23(tt, IL, M, N)
   U23(tt, IL, M, N) -> cons(N, take(M, IL))
   length(nil) -> 0
   length(cons(N, L)) -> U11(tt, L)
   take(0, IL) -> nil
   take(s(M), cons(N, IL)) -> U21(tt, IL, M, N)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
U12: {1}
s: {1}
length: {1}
U21: {1}
U22: {1}
U23: {1}
take: {1, 2}
nil: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(U11(x_1, x_2)) = x_1 + x_2
   POL(U12(x_1, x_2)) = x_1 + x_2
   POL(U21(x_1, x_2, x_3, x_4)) = 1 + x_1 + 2*x_2 + x_3 + 2*x_4
   POL(U22(x_1, x_2, x_3, x_4)) = 1 + x_1 + 2*x_2 + x_3 + 2*x_4
   POL(U23(x_1, x_2, x_3, x_4)) = 1 + 2*x_1 + 2*x_2 + x_3 + 2*x_4
   POL(cons(x_1, x_2)) = 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 + 2*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




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

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

   zeros -> cons(0, zeros)
   U11(tt, L) -> U12(tt, L)
   U12(tt, L) -> s(length(L))
   U21(tt, IL, M, N) -> U22(tt, IL, M, N)
   U22(tt, IL, M, N) -> U23(tt, IL, M, N)
   U23(tt, IL, M, N) -> cons(N, take(M, IL))
   length(cons(N, L)) -> U11(tt, L)
   take(0, IL) -> nil
   take(s(M), cons(N, IL)) -> U21(tt, IL, M, N)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
U12: {1}
s: {1}
length: {1}
U21: {1}
U22: {1}
U23: {1}
take: {1, 2}
nil: 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)
   U11(tt, L) -> U12(tt, L)
   U12(tt, L) -> s(length(L))
   U21(tt, IL, M, N) -> U22(tt, IL, M, N)
   U22(tt, IL, M, N) -> U23(tt, IL, M, N)
   U23(tt, IL, M, N) -> cons(N, take(M, IL))
   length(cons(N, L)) -> U11(tt, L)
   take(0, IL) -> nil
   take(s(M), cons(N, IL)) -> U21(tt, IL, M, N)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
U12: {1}
s: {1}
length: {1}
U21: {1}
U22: {1}
U23: {1}
take: {1, 2}
nil: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(U11(x_1, x_2)) = x_1 + x_2
   POL(U12(x_1, x_2)) = x_1 + x_2
   POL(U21(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4
   POL(U22(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4
   POL(U23(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4
   POL(cons(x_1, x_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)) = 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:

   take(0, IL) -> nil




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

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

   zeros -> cons(0, zeros)
   U11(tt, L) -> U12(tt, L)
   U12(tt, L) -> s(length(L))
   U21(tt, IL, M, N) -> U22(tt, IL, M, N)
   U22(tt, IL, M, N) -> U23(tt, IL, M, N)
   U23(tt, IL, M, N) -> cons(N, take(M, IL))
   length(cons(N, L)) -> U11(tt, L)
   take(s(M), cons(N, IL)) -> U21(tt, IL, M, N)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
U12: {1}
s: {1}
length: {1}
U21: {1}
U22: {1}
U23: {1}
take: {1, 2}

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

(7) ContextSensitiveLoopProof (COMPLETE)

   zeros -> cons(0, zeros)
   U11(tt, L) -> U12(tt, L)
   U12(tt, L) -> s(length(L))
   U21(tt, IL, M, N) -> U22(tt, IL, M, N)
   U22(tt, IL, M, N) -> U23(tt, IL, M, N)
   U23(tt, IL, M, N) -> cons(N, take(M, IL))
   length(cons(N, L)) -> U11(tt, L)
   take(s(M), cons(N, IL)) -> U21(tt, IL, M, N)

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

U12(tt, zeros) -> s(length(zeros)) with rule U12(tt, L) -> s(length(L)) at position [] and matcher [L / zeros]

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

s(length(cons(0, zeros))) -> s(U11(tt, zeros)) with rule length(cons(N, L)) -> U11(tt, L) at position [0] and matcher [N / 0, L / zeros]

s(U11(tt, zeros)) -> s(U12(tt, zeros)) with rule U11(tt, L) -> U12(tt, L) at position [0] and matcher [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.
----------------------------------------

(8)
NO