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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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

   app(app(app(compose, f), g), x) -> app(g, app(f, x))
   app(reverse, l) -> app(app(reverse2, l), nil)
   app(app(reverse2, nil), l) -> l
   app(app(reverse2, app(app(cons, x), xs)), l) -> app(app(reverse2, xs), app(app(cons, x), l))
   app(hd, app(app(cons, x), xs)) -> x
   app(tl, app(app(cons, x), xs)) -> xs
   last -> app(app(compose, hd), reverse)
   init -> app(app(compose, reverse), app(app(compose, tl), reverse))

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:init > last > tl > hd > cons > nil > reverse2 > reverse > compose > app_2

and weight map:

   compose=1
   reverse=4
   reverse2=1
   nil=2
   cons=1
   hd=1
   tl=1
   last=6
   init=11
   app_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:

   app(app(app(compose, f), g), x) -> app(g, app(f, x))
   app(reverse, l) -> app(app(reverse2, l), nil)
   app(app(reverse2, nil), l) -> l
   app(app(reverse2, app(app(cons, x), xs)), l) -> app(app(reverse2, xs), app(app(cons, x), l))
   app(hd, app(app(cons, x), xs)) -> x
   app(tl, app(app(cons, x), xs)) -> xs
   last -> app(app(compose, hd), reverse)
   init -> app(app(compose, reverse), app(app(compose, tl), reverse))




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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