/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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, 94 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 19 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) -> g1(x, x, y)
   f(x, y) -> g1(y, x, x)
   f(x, y) -> g2(x, y, y)
   f(x, y) -> g2(y, y, x)
   g1(x, x, y) -> h(x, y)
   g1(y, x, x) -> h(x, y)
   g2(x, y, y) -> h(x, y)
   g2(y, y, x) -> h(x, y)
   h(x, x) -> x

Q is empty.

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

   POL(f(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
   POL(g1(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
   POL(g2(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
   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:

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

Q is empty.

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(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Quasi precedence:
f_2 > g1_3 > h_2
f_2 > g2_3 > h_2


Status:
f_2: [1,2]
g1_3: [3,2,1]
g2_3: multiset status
h_2: multiset status

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

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




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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