/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) Overlay + Local Confluence [EQUIVALENT, 4 ms]
(2) QTRS
(3) DependencyPairsProof [EQUIVALENT, 0 ms]
(4) QDP
(5) DependencyGraphProof [EQUIVALENT, 0 ms]
(6) QDP
(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
(8) YES


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

   app(app(twice, f), x) -> app(f, app(f, x))
   app(app(map, f), nil) -> nil
   app(app(map, f), app(app(cons, h), t)) -> app(app(cons, app(f, h)), app(app(map, f), t))
   app(app(fmap, nil), x) -> nil
   app(app(fmap, app(app(cons, f), t_f)), x) -> app(app(cons, app(f, x)), app(app(fmap, t_f), x))

Q is empty.

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(1) Overlay + Local Confluence (EQUIVALENT)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
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(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   app(app(twice, f), x) -> app(f, app(f, x))
   app(app(map, f), nil) -> nil
   app(app(map, f), app(app(cons, h), t)) -> app(app(cons, app(f, h)), app(app(map, f), t))
   app(app(fmap, nil), x) -> nil
   app(app(fmap, app(app(cons, f), t_f)), x) -> app(app(cons, app(f, x)), app(app(fmap, t_f), x))

The set Q consists of the following terms:

   app(app(twice, x0), x1)
   app(app(map, x0), nil)
   app(app(map, x0), app(app(cons, x1), x2))
   app(app(fmap, nil), x0)
   app(app(fmap, app(app(cons, x0), t_f)), x1)


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(3) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
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(4)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   APP(app(twice, f), x) -> APP(f, app(f, x))
   APP(app(twice, f), x) -> APP(f, x)
   APP(app(map, f), app(app(cons, h), t)) -> APP(app(cons, app(f, h)), app(app(map, f), t))
   APP(app(map, f), app(app(cons, h), t)) -> APP(cons, app(f, h))
   APP(app(map, f), app(app(cons, h), t)) -> APP(f, h)
   APP(app(map, f), app(app(cons, h), t)) -> APP(app(map, f), t)
   APP(app(fmap, app(app(cons, f), t_f)), x) -> APP(app(cons, app(f, x)), app(app(fmap, t_f), x))
   APP(app(fmap, app(app(cons, f), t_f)), x) -> APP(cons, app(f, x))
   APP(app(fmap, app(app(cons, f), t_f)), x) -> APP(f, x)
   APP(app(fmap, app(app(cons, f), t_f)), x) -> APP(app(fmap, t_f), x)
   APP(app(fmap, app(app(cons, f), t_f)), x) -> APP(fmap, t_f)

The TRS R consists of the following rules:

   app(app(twice, f), x) -> app(f, app(f, x))
   app(app(map, f), nil) -> nil
   app(app(map, f), app(app(cons, h), t)) -> app(app(cons, app(f, h)), app(app(map, f), t))
   app(app(fmap, nil), x) -> nil
   app(app(fmap, app(app(cons, f), t_f)), x) -> app(app(cons, app(f, x)), app(app(fmap, t_f), x))

The set Q consists of the following terms:

   app(app(twice, x0), x1)
   app(app(map, x0), nil)
   app(app(map, x0), app(app(cons, x1), x2))
   app(app(fmap, nil), x0)
   app(app(fmap, app(app(cons, x0), t_f)), x1)

We have to consider all minimal (P,Q,R)-chains.
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(5) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes.
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(6)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   APP(app(twice, f), x) -> APP(f, x)
   APP(app(twice, f), x) -> APP(f, app(f, x))
   APP(app(map, f), app(app(cons, h), t)) -> APP(f, h)
   APP(app(map, f), app(app(cons, h), t)) -> APP(app(map, f), t)
   APP(app(fmap, app(app(cons, f), t_f)), x) -> APP(f, x)

The TRS R consists of the following rules:

   app(app(twice, f), x) -> app(f, app(f, x))
   app(app(map, f), nil) -> nil
   app(app(map, f), app(app(cons, h), t)) -> app(app(cons, app(f, h)), app(app(map, f), t))
   app(app(fmap, nil), x) -> nil
   app(app(fmap, app(app(cons, f), t_f)), x) -> app(app(cons, app(f, x)), app(app(fmap, t_f), x))

The set Q consists of the following terms:

   app(app(twice, x0), x1)
   app(app(map, x0), nil)
   app(app(map, x0), app(app(cons, x1), x2))
   app(app(fmap, nil), x0)
   app(app(fmap, app(app(cons, x0), t_f)), x1)

We have to consider all minimal (P,Q,R)-chains.
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(7) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*APP(app(map, f), app(app(cons, h), t)) -> APP(app(map, f), t)
The graph contains the following edges 1 >= 1, 2 > 2


*APP(app(map, f), app(app(cons, h), t)) -> APP(f, h)
The graph contains the following edges 1 > 1, 2 > 2


*APP(app(twice, f), x) -> APP(f, x)
The graph contains the following edges 1 > 1, 2 >= 2


*APP(app(twice, f), x) -> APP(f, app(f, x))
The graph contains the following edges 1 > 1


*APP(app(fmap, app(app(cons, f), t_f)), x) -> APP(f, x)
The graph contains the following edges 1 > 1, 2 >= 2


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(8)
YES