/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 0 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, e) -> x
   i(:(x, y)) -> :(y, x)
   :(:(x, y), z) -> :(x, :(z, i(y)))
   :(e, x) -> i(x)
   i(i(x)) -> x
   i(e) -> e
   :(x, :(y, i(x))) -> i(y)
   :(x, :(y, :(i(x), z))) -> :(i(z), y)
   :(i(x), :(y, x)) -> i(y)
   :(i(x), :(y, :(x, z))) -> :(i(z), y)

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:i_1 > :_2 > e

and weight map:

   e=2
   i_1=0
   :_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:

   :(x, x) -> e
   :(x, e) -> x
   i(:(x, y)) -> :(y, x)
   :(:(x, y), z) -> :(x, :(z, i(y)))
   :(e, x) -> i(x)
   i(i(x)) -> x
   i(e) -> e
   :(x, :(y, i(x))) -> i(y)
   :(x, :(y, :(i(x), z))) -> :(i(z), y)
   :(i(x), :(y, x)) -> i(y)
   :(i(x), :(y, :(x, z))) -> :(i(z), 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