/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) DependencyGraphProof [EQUIVALENT, 0 ms]
(10) QDP
(11) TransformationProof [EQUIVALENT, 0 ms]
(12) QDP
(13) DependencyGraphProof [EQUIVALENT, 0 ms]
(14) TRUE


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

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

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

The set Q consists of the following terms:

   f(f(a, a), x0)


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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

   f(f(a, a), x0)

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

   f(f(a, a), x0)

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

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

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


----------------------------------------

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

   f(f(a, a), x0)

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

(9) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
----------------------------------------

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

   f(f(a, a), x0)

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

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

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


----------------------------------------

(12)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

   f(f(a, a), x0)

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

(13) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
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(14)
TRUE