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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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


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

   app(app(map, f), nil) -> nil
   app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
   app(app(treemap, f), app(app(node, x), xs)) -> app(app(node, app(f, x)), app(app(map, app(treemap, f)), xs))

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(map, f), nil) -> nil
   app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
   app(app(treemap, f), app(app(node, x), xs)) -> app(app(node, app(f, x)), app(app(map, app(treemap, f)), xs))

The set Q consists of the following terms:

   app(app(map, x0), nil)
   app(app(map, x0), app(app(cons, x1), x2))
   app(app(treemap, x0), app(app(node, x1), x2))


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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(map, f), app(app(cons, x), xs)) -> APP(app(cons, app(f, x)), app(app(map, f), xs))
   APP(app(map, f), app(app(cons, x), xs)) -> APP(cons, app(f, x))
   APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)
   APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)
   APP(app(treemap, f), app(app(node, x), xs)) -> APP(app(node, app(f, x)), app(app(map, app(treemap, f)), xs))
   APP(app(treemap, f), app(app(node, x), xs)) -> APP(node, app(f, x))
   APP(app(treemap, f), app(app(node, x), xs)) -> APP(f, x)
   APP(app(treemap, f), app(app(node, x), xs)) -> APP(app(map, app(treemap, f)), xs)
   APP(app(treemap, f), app(app(node, x), xs)) -> APP(map, app(treemap, f))

The TRS R consists of the following rules:

   app(app(map, f), nil) -> nil
   app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
   app(app(treemap, f), app(app(node, x), xs)) -> app(app(node, app(f, x)), app(app(map, app(treemap, f)), xs))

The set Q consists of the following terms:

   app(app(map, x0), nil)
   app(app(map, x0), app(app(cons, x1), x2))
   app(app(treemap, x0), app(app(node, x1), x2))

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 5 less nodes.
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(6)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)
   APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)
   APP(app(treemap, f), app(app(node, x), xs)) -> APP(f, x)
   APP(app(treemap, f), app(app(node, x), xs)) -> APP(app(map, app(treemap, f)), xs)

The TRS R consists of the following rules:

   app(app(map, f), nil) -> nil
   app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
   app(app(treemap, f), app(app(node, x), xs)) -> app(app(node, app(f, x)), app(app(map, app(treemap, f)), xs))

The set Q consists of the following terms:

   app(app(map, x0), nil)
   app(app(map, x0), app(app(cons, x1), x2))
   app(app(treemap, x0), app(app(node, x1), x2))

We have to consider all minimal (P,Q,R)-chains.
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(7) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
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(8)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)
   APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)
   APP(app(treemap, f), app(app(node, x), xs)) -> APP(f, x)
   APP(app(treemap, f), app(app(node, x), xs)) -> APP(app(map, app(treemap, f)), xs)

R is empty.
The set Q consists of the following terms:

   app(app(map, x0), nil)
   app(app(map, x0), app(app(cons, x1), x2))
   app(app(treemap, x0), app(app(node, x1), x2))

We have to consider all minimal (P,Q,R)-chains.
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(9) 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, x), xs)) -> APP(f, x)
The graph contains the following edges 1 > 1, 2 > 2


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


*APP(app(treemap, f), app(app(node, x), xs)) -> APP(f, x)
The graph contains the following edges 1 > 1, 2 > 2


*APP(app(treemap, f), app(app(node, x), xs)) -> APP(app(map, app(treemap, f)), xs)
The graph contains the following edges 2 > 2


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(10)
YES