YES
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 proven:

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 55 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:

   g(0, f(x, x)) -> x
   g(x, s(y)) -> g(f(x, y), 0)
   g(s(x), y) -> g(f(x, y), 0)
   g(f(x, y), 0) -> f(g(x, 0), g(y, 0))

Q is empty.

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

   POL(0) = 0
   POL(f(x_1, x_2)) = 1 + x_1 + x_2
   POL(g(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(s(x_1)) = 1 + x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   g(0, f(x, x)) -> x
   g(f(x, y), 0) -> f(g(x, 0), g(y, 0))




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

   g(x, s(y)) -> g(f(x, y), 0)
   g(s(x), y) -> g(f(x, y), 0)

Q is empty.

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(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:0 > f_2 > s_1 > g_2

and weight map:

   0=2
   s_1=3
   f_2=0
   g_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:

   g(x, s(y)) -> g(f(x, y), 0)
   g(s(x), y) -> g(f(x, y), 0)




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(4)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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