YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination w.r.t. Q of the given QTRS could be proven:

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 56 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 0 ms]
(4) QTRS
(5) RisEmptyProof [EQUIVALENT, 0 ms]
(6) YES


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

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

The set Q consists of the following terms:

   first(0, x0)
   first(s(x0), cons(x1))
   from(x0)


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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(cons(x_1)) = 2 + 2*x_1
   POL(first(x_1, x_2)) = x_1 + 2*x_2
   POL(from(x_1)) = 2 + 2*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(s(X), cons(Y)) -> cons(Y)




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(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

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

The set Q consists of the following terms:

   first(0, x0)
   first(s(x0), cons(x1))
   from(x0)


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(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:first_2 > from_1 > cons_1 > nil > 0

and weight map:

   0=1
   nil=2
   from_1=1
   cons_1=1
   first_2=0

The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

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




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(4)
Obligation:
Q restricted rewrite system:
R is empty.
The set Q consists of the following terms:

   first(0, x0)
   first(s(x0), cons(x1))
   from(x0)


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(5) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(6)
YES