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


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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 93 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 13 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:

   f(x, y, w, w, a) -> g1(x, x, y, w)
   f(x, y, w, a, a) -> g1(y, x, x, w)
   f(x, y, a, a, w) -> g2(x, y, y, w)
   f(x, y, a, w, w) -> g2(y, y, x, w)
   g1(x, x, y, a) -> h(x, y)
   g1(y, x, x, a) -> h(x, y)
   g2(x, y, y, a) -> h(x, y)
   g2(y, y, x, a) -> h(x, y)
   h(x, x) -> x

Q is empty.

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

   POL(a) = 0
   POL(f(x_1, x_2, x_3, x_4, x_5)) = 2 + 2*x_1 + 2*x_2 + 2*x_3 + x_4 + 2*x_5
   POL(g1(x_1, x_2, x_3, x_4)) = 2 + x_1 + x_2 + x_3 + 2*x_4
   POL(g2(x_1, x_2, x_3, x_4)) = 2 + x_1 + x_2 + x_3 + 2*x_4
   POL(h(x_1, x_2)) = 1 + x_1 + x_2
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   g1(x, x, y, a) -> h(x, y)
   g1(y, x, x, a) -> h(x, y)
   g2(x, y, y, a) -> h(x, y)
   g2(y, y, x, a) -> h(x, y)
   h(x, x) -> x




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

   f(x, y, w, w, a) -> g1(x, x, y, w)
   f(x, y, w, a, a) -> g1(y, x, x, w)
   f(x, y, a, a, w) -> g2(x, y, y, w)
   f(x, y, a, w, w) -> g2(y, y, x, w)

Q is empty.

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

   POL(a) = 2
   POL(f(x_1, x_2, x_3, x_4, x_5)) = 2 + 2*x_1 + 2*x_2 + 2*x_3 + 2*x_4 + 2*x_5
   POL(g1(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + 2*x_4
   POL(g2(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + 2*x_4
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   f(x, y, w, w, a) -> g1(x, x, y, w)
   f(x, y, w, a, a) -> g1(y, x, x, w)
   f(x, y, a, a, w) -> g2(x, y, y, w)
   f(x, y, a, w, w) -> g2(y, y, x, w)




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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