/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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NO
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


Termination of the given CSR could be disproven:

(0) CSR
(1) CSRRRRProof [EQUIVALENT, 64 ms]
(2) CSR
(3) ContextSensitiveLoopProof [COMPLETE, 10 ms]
(4) NO


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(0)
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))
   length(nil) -> 0
   length(cons(N, L)) -> U11(tt, L)

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}
nil: empty set

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(1) 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))
   length(nil) -> 0
   length(cons(N, L)) -> U11(tt, L)

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}
nil: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(U11(x_1, x_2)) = x_1 + 2*x_2
   POL(U12(x_1, x_2)) = x_1 + 2*x_2
   POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(length(x_1)) = x_1
   POL(nil) = 1
   POL(s(x_1)) = 2*x_1
   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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(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))
   length(cons(N, L)) -> U11(tt, L)

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}

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(3) ContextSensitiveLoopProof (COMPLETE)

   zeros -> cons(0, zeros)
   U11(tt, L) -> U12(tt, L)
   U12(tt, L) -> s(length(L))
   length(cons(N, L)) -> U11(tt, L)

---------- 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.
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(4)
NO