/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination of the given CSR could be proven:

(0) CSR
(1) CSRRRRProof [EQUIVALENT, 41 ms]
(2) CSR
(3) CSRRRRProof [EQUIVALENT, 0 ms]
(4) CSR
(5) CSRRRRProof [EQUIVALENT, 3 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:

   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
   from(X) -> cons(X, from(s(X)))

The replacement map contains the following entries:

first: {1, 2}
0: empty set
nil: empty set
s: {1}
cons: {1}
from: {1}

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

   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
   from(X) -> cons(X, from(s(X)))

The replacement map contains the following entries:

first: {1, 2}
0: empty set
nil: empty set
s: {1}
cons: {1}
from: {1}
Used ordering:
Polynomial interpretation [POLO]:

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

   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))




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

   first(0, X) -> nil
   from(X) -> cons(X, from(s(X)))

The replacement map contains the following entries:

first: {1, 2}
0: empty set
nil: empty set
s: {1}
cons: {1}
from: {1}

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

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

   first(0, X) -> nil
   from(X) -> cons(X, from(s(X)))

The replacement map contains the following entries:

first: {1, 2}
0: empty set
nil: empty set
s: {1}
cons: {1}
from: {1}
Used ordering:
Polynomial interpretation [POLO]:

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

   first(0, X) -> nil




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

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

   from(X) -> cons(X, from(s(X)))

The replacement map contains the following entries:

s: {1}
cons: {1}
from: {1}

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

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

   from(X) -> cons(X, from(s(X)))

The replacement map contains the following entries:

s: {1}
cons: {1}
from: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(cons(x_1, x_2)) = x_1
   POL(from(x_1)) = 1 + x_1
   POL(s(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(X) -> cons(X, from(s(X)))




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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