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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) DependencyGraphProof [EQUIVALENT, 0 ms]
(6) QDP
(7) QDPOrderProof [EQUIVALENT, 27 ms]
(8) QDP
(9) PisEmptyProof [EQUIVALENT, 0 ms]
(10) YES


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

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

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

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

The set Q consists of the following terms:

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(7) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   F(a, f(f(a, a), x)) -> F(a, x)
   F(a, f(f(a, a), x)) -> F(a, f(a, x))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

   <<<
 POL(F(x_1, x_2)) =  	[[0]] 	 +  	[[1, 1]] 	* 	x_1 	 +  	[[0, 1]] 	* 	x_2
>>>

   <<<
 POL(a) =  	[[0], [0]]
>>>

   <<<
 POL(f(x_1, x_2)) =  	[[1], [0]] 	 +  	[[0, 0], [1, 0]] 	* 	x_1 	 +  	[[0, 0], [0, 1]] 	* 	x_2
>>>


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

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


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

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

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

The set Q consists of the following terms:

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

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

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

(10)
YES