/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) UsableRulesProof [EQUIVALENT, 0 ms]
(8) QDP
(9) MRRProof [EQUIVALENT, 0 ms]
(10) QDP
(11) PisEmptyProof [EQUIVALENT, 0 ms]
(12) YES


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

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

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

The set Q consists of the following terms:

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(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.
----------------------------------------

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

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

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

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

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

(9) MRRProof (EQUIVALENT)
By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.

Strictly oriented dependency pairs:

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


Used ordering: Polynomial interpretation [POLO]:

   POL(F(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(a) = 1
   POL(f(x_1, x_2)) = x_1 + 2*x_2


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

(10)
Obligation:
Q DP problem:
P is empty.
R is empty.
The set Q consists of the following terms:

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

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

(11) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
----------------------------------------

(12)
YES