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


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YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


Termination of the given CSR could be proven:

(0) CSR
(1) CSRRRRProof [EQUIVALENT, 43 ms]
(2) CSR
(3) CSRRRRProof [EQUIVALENT, 0 ms]
(4) CSR
(5) CSRRRRProof [EQUIVALENT, 0 ms]
(6) CSR
(7) RisEmptyProof [EQUIVALENT, 0 ms]
(8) YES


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(0)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   from_2(x) -> cons_0(x, from_1(s_0(x)))
   from_3(x) -> cons_0(x, from_2(s_0(x)))
   from_4(x) -> cons_0(x, from_2(s_0(x)))
   from_1(x) -> cons_1(x, from_1(s_0(x)))
   cons_1(s_0(s_0(x)), xs) -> nil_0

The replacement map contains the following entries:

from_2: empty set
cons_0: {1, 2}
from_1: empty set
s_0: {1}
from_3: empty set
from_4: empty set
cons_1: empty set
nil_0: 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:

   from_2(x) -> cons_0(x, from_1(s_0(x)))
   from_3(x) -> cons_0(x, from_2(s_0(x)))
   from_4(x) -> cons_0(x, from_2(s_0(x)))
   from_1(x) -> cons_1(x, from_1(s_0(x)))
   cons_1(s_0(s_0(x)), xs) -> nil_0

The replacement map contains the following entries:

from_2: empty set
cons_0: {1, 2}
from_1: empty set
s_0: {1}
from_3: empty set
from_4: empty set
cons_1: empty set
nil_0: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(cons_0(x_1, x_2)) = x_1 + x_2
   POL(cons_1(x_1, x_2)) = 0
   POL(from_1(x_1)) = 0
   POL(from_2(x_1)) = x_1
   POL(from_3(x_1)) = 2 + 2*x_1
   POL(from_4(x_1)) = 2 + 2*x_1
   POL(nil_0) = 0
   POL(s_0(x_1)) = x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   from_3(x) -> cons_0(x, from_2(s_0(x)))
   from_4(x) -> cons_0(x, from_2(s_0(x)))




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

   from_2(x) -> cons_0(x, from_1(s_0(x)))
   from_1(x) -> cons_1(x, from_1(s_0(x)))
   cons_1(s_0(s_0(x)), xs) -> nil_0

The replacement map contains the following entries:

from_2: empty set
cons_0: {1, 2}
from_1: empty set
s_0: {1}
cons_1: empty set
nil_0: empty set

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

   from_2(x) -> cons_0(x, from_1(s_0(x)))
   from_1(x) -> cons_1(x, from_1(s_0(x)))
   cons_1(s_0(s_0(x)), xs) -> nil_0

The replacement map contains the following entries:

from_2: empty set
cons_0: {1, 2}
from_1: empty set
s_0: {1}
cons_1: empty set
nil_0: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(cons_0(x_1, x_2)) = 2*x_1 + x_2
   POL(cons_1(x_1, x_2)) = 0
   POL(from_1(x_1)) = 1
   POL(from_2(x_1)) = 2 + 2*x_1
   POL(nil_0) = 0
   POL(s_0(x_1)) = x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   from_2(x) -> cons_0(x, from_1(s_0(x)))
   from_1(x) -> cons_1(x, from_1(s_0(x)))




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

   cons_1(s_0(s_0(x)), xs) -> nil_0

The replacement map contains the following entries:

s_0: {1}
cons_1: empty set
nil_0: empty set

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

   cons_1(s_0(s_0(x)), xs) -> nil_0

The replacement map contains the following entries:

s_0: {1}
cons_1: empty set
nil_0: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(cons_1(x_1, x_2)) = 1 + 2*x_1
   POL(nil_0) = 0
   POL(s_0(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:

   cons_1(s_0(s_0(x)), xs) -> nil_0




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

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(7) RisEmptyProof (EQUIVALENT)
The CSR R is empty. Hence, termination is trivially proven.
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(8)
YES