YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) TransformationProof [EQUIVALENT, 0 ms]
(6) QDP
(7) DependencyGraphProof [EQUIVALENT, 0 ms]
(8) QDP
(9) UsableRulesProof [EQUIVALENT, 0 ms]
(10) QDP
(11) TransformationProof [EQUIVALENT, 0 ms]
(12) QDP
(13) DependencyGraphProof [EQUIVALENT, 0 ms]
(14) TRUE


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

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

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

Q is empty.

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

(1) Overlay + Local Confluence (EQUIVALENT)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
----------------------------------------

(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

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

The set Q consists of the following terms:

   f(f(f(a, f(a, a)), a), x0)


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

(3) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
----------------------------------------

(4)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

   f(f(f(a, f(a, a)), a), x0)

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

(5) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule F(f(f(a, f(a, a)), a), x) -> F(x, f(x, a)) at position [] we obtained the following new rules [LPAR04]:

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


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

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

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

The set Q consists of the following terms:

   f(f(f(a, f(a, a)), a), x0)

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

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

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

   f(f(f(a, f(a, a)), a), x0)

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

(9) 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.
----------------------------------------

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

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

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

   f(f(f(a, f(a, a)), a), x0)

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

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

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


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

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

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

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

   f(f(f(a, f(a, 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.
----------------------------------------

(14)
TRUE