/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 0 ms]
(2) QTRS
(3) DependencyPairsProof [EQUIVALENT, 0 ms]
(4) QDP
(5) DependencyGraphProof [EQUIVALENT, 0 ms]
(6) QDP
(7) TransformationProof [EQUIVALENT, 0 ms]
(8) QDP
(9) TransformationProof [EQUIVALENT, 0 ms]
(10) QDP
(11) DependencyGraphProof [EQUIVALENT, 0 ms]
(12) QDP
(13) TransformationProof [EQUIVALENT, 0 ms]
(14) QDP
(15) DependencyGraphProof [EQUIVALENT, 0 ms]
(16) QDP
(17) UsableRulesProof [EQUIVALENT, 0 ms]
(18) QDP
(19) TransformationProof [EQUIVALENT, 0 ms]
(20) QDP
(21) QDPSizeChangeProof [EQUIVALENT, 0 ms]
(22) YES


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

   f(x, f(y, a)) -> f(f(f(f(a, a), y), h(a)), 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:

   f(x, f(y, a)) -> f(f(f(f(a, a), y), h(a)), x)

The set Q consists of the following terms:

   f(x0, f(x1, a))


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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:

   F(x, f(y, a)) -> F(f(f(f(a, a), y), h(a)), x)
   F(x, f(y, a)) -> F(f(f(a, a), y), h(a))
   F(x, f(y, a)) -> F(f(a, a), y)
   F(x, f(y, a)) -> F(a, a)

The TRS R consists of the following rules:

   f(x, f(y, a)) -> f(f(f(f(a, a), y), h(a)), x)

The set Q consists of the following terms:

   f(x0, f(x1, a))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(5) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
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(6)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F(x, f(y, a)) -> F(f(a, a), y)
   F(x, f(y, a)) -> F(f(f(f(a, a), y), h(a)), x)

The TRS R consists of the following rules:

   f(x, f(y, a)) -> f(f(f(f(a, a), y), h(a)), x)

The set Q consists of the following terms:

   f(x0, f(x1, a))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(7) TransformationProof (EQUIVALENT)
By instantiating [LPAR04] the rule F(x, f(y, a)) -> F(f(a, a), y) we obtained the following new rules [LPAR04]:

   (F(f(a, a), f(x1, a)) -> F(f(a, a), x1),F(f(a, a), f(x1, a)) -> F(f(a, a), x1))
   (F(f(y_0, h(a)), f(x1, a)) -> F(f(a, a), x1),F(f(y_0, h(a)), f(x1, a)) -> F(f(a, a), x1))


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(8)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F(x, f(y, a)) -> F(f(f(f(a, a), y), h(a)), x)
   F(f(a, a), f(x1, a)) -> F(f(a, a), x1)
   F(f(y_0, h(a)), f(x1, a)) -> F(f(a, a), x1)

The TRS R consists of the following rules:

   f(x, f(y, a)) -> f(f(f(f(a, a), y), h(a)), x)

The set Q consists of the following terms:

   f(x0, f(x1, a))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(9) TransformationProof (EQUIVALENT)
By instantiating [LPAR04] the rule F(x, f(y, a)) -> F(f(f(f(a, a), y), h(a)), x) we obtained the following new rules [LPAR04]:

   (F(f(y_0, h(a)), f(x1, a)) -> F(f(f(f(a, a), x1), h(a)), f(y_0, h(a))),F(f(y_0, h(a)), f(x1, a)) -> F(f(f(f(a, a), x1), h(a)), f(y_0, h(a))))
   (F(f(a, a), f(x1, a)) -> F(f(f(f(a, a), x1), h(a)), f(a, a)),F(f(a, a), f(x1, a)) -> F(f(f(f(a, a), x1), h(a)), f(a, a)))


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(10)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F(f(a, a), f(x1, a)) -> F(f(a, a), x1)
   F(f(y_0, h(a)), f(x1, a)) -> F(f(a, a), x1)
   F(f(y_0, h(a)), f(x1, a)) -> F(f(f(f(a, a), x1), h(a)), f(y_0, h(a)))
   F(f(a, a), f(x1, a)) -> F(f(f(f(a, a), x1), h(a)), f(a, a))

The TRS R consists of the following rules:

   f(x, f(y, a)) -> f(f(f(f(a, a), y), h(a)), x)

The set Q consists of the following terms:

   f(x0, f(x1, a))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(11) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
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(12)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F(f(a, a), f(x1, a)) -> F(f(a, a), x1)
   F(f(a, a), f(x1, a)) -> F(f(f(f(a, a), x1), h(a)), f(a, a))
   F(f(y_0, h(a)), f(x1, a)) -> F(f(a, a), x1)

The TRS R consists of the following rules:

   f(x, f(y, a)) -> f(f(f(f(a, a), y), h(a)), x)

The set Q consists of the following terms:

   f(x0, f(x1, a))

We have to consider all minimal (P,Q,R)-chains.
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(13) TransformationProof (EQUIVALENT)
By instantiating [LPAR04] the rule F(f(y_0, h(a)), f(x1, a)) -> F(f(a, a), x1) we obtained the following new rules [LPAR04]:

   (F(f(y_0, h(a)), f(a, a)) -> F(f(a, a), a),F(f(y_0, h(a)), f(a, a)) -> F(f(a, a), a))


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(14)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F(f(a, a), f(x1, a)) -> F(f(a, a), x1)
   F(f(a, a), f(x1, a)) -> F(f(f(f(a, a), x1), h(a)), f(a, a))
   F(f(y_0, h(a)), f(a, a)) -> F(f(a, a), a)

The TRS R consists of the following rules:

   f(x, f(y, a)) -> f(f(f(f(a, a), y), h(a)), x)

The set Q consists of the following terms:

   f(x0, f(x1, a))

We have to consider all minimal (P,Q,R)-chains.
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(15) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
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(16)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F(f(a, a), f(x1, a)) -> F(f(a, a), x1)

The TRS R consists of the following rules:

   f(x, f(y, a)) -> f(f(f(f(a, a), y), h(a)), x)

The set Q consists of the following terms:

   f(x0, f(x1, a))

We have to consider all minimal (P,Q,R)-chains.
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(17) 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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(18)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F(f(a, a), f(x1, a)) -> F(f(a, a), x1)

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

   f(x0, f(x1, a))

We have to consider all minimal (P,Q,R)-chains.
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(19) TransformationProof (EQUIVALENT)
By forward instantiating [JAR06] the rule F(f(a, a), f(x1, a)) -> F(f(a, a), x1) we obtained the following new rules [LPAR04]:

   (F(f(a, a), f(f(y_0, a), a)) -> F(f(a, a), f(y_0, a)),F(f(a, a), f(f(y_0, a), a)) -> F(f(a, a), f(y_0, a)))


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(20)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F(f(a, a), f(f(y_0, a), a)) -> F(f(a, a), f(y_0, a))

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

   f(x0, f(x1, a))

We have to consider all minimal (P,Q,R)-chains.
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(21) 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:
*F(f(a, a), f(f(y_0, a), a)) -> F(f(a, a), f(y_0, a))
The graph contains the following edges 1 >= 1, 2 > 2


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(22)
YES