/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 37 ms]
(2) QTRS
(3) RisEmptyProof [EQUIVALENT, 0 ms]
(4) YES


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

   \(x, x) -> e
   /(x, x) -> e
   .(e, x) -> x
   .(x, e) -> x
   \(e, x) -> x
   /(x, e) -> x
   .(x, \(x, y)) -> y
   .(/(y, x), x) -> y
   \(x, .(x, y)) -> y
   /(.(y, x), x) -> y
   /(x, \(y, x)) -> y
   \(/(x, y), x) -> y

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Quasi precedence:
._2 > [\_2, e, /_2]


Status:
\_2: [2,1]
e: multiset status
/_2: [2,1]
._2: multiset status

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

   \(x, x) -> e
   /(x, x) -> e
   .(e, x) -> x
   .(x, e) -> x
   \(e, x) -> x
   /(x, e) -> x
   .(x, \(x, y)) -> y
   .(/(y, x), x) -> y
   \(x, .(x, y)) -> y
   /(.(y, x), x) -> y
   /(x, \(y, x)) -> y
   \(/(x, y), x) -> y




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(2)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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(3) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(4)
YES