YES
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) RootLabelingProof [EQUIVALENT, 0 ms]
(2) QTRS
(3) DependencyPairsProof [EQUIVALENT, 0 ms]
(4) QDP
(5) QDPOrderProof [EQUIVALENT, 338 ms]
(6) QDP
(7) QDPOrderProof [EQUIVALENT, 114 ms]
(8) QDP
(9) DependencyGraphProof [EQUIVALENT, 0 ms]
(10) QDP
(11) QDPOrderProof [EQUIVALENT, 140 ms]
(12) QDP
(13) DependencyGraphProof [EQUIVALENT, 0 ms]
(14) TRUE


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

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

   a(a(b(a(x1)))) -> a(b(b(x1)))
   b(b(x1)) -> b(a(a(a(x1))))

Q is empty.

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

(1) RootLabelingProof (EQUIVALENT)
We used plain root labeling [ROOTLAB] with the following heuristic:
LabelAll: All function symbols get labeled

As Q is empty the root labeling was sound AND complete.

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

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

   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(x1)))
   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(x1)))
   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
   b_{b_1}(b_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))

Q is empty.

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

(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:

   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(x1))
   A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
   A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
   B_{B_1}(b_{a_1}(x1)) -> A_{A_1}(a_{a_1}(a_{a_1}(x1)))
   B_{B_1}(b_{a_1}(x1)) -> A_{A_1}(a_{a_1}(x1))
   B_{B_1}(b_{a_1}(x1)) -> A_{A_1}(x1)
   B_{B_1}(b_{b_1}(x1)) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
   B_{B_1}(b_{b_1}(x1)) -> A_{A_1}(a_{b_1}(x1))

The TRS R consists of the following rules:

   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(x1)))
   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(x1)))
   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
   b_{b_1}(b_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))

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

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


The following pairs can be oriented strictly and are deleted.

   A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(A_{A_1}(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
>>>

   <<<
 POL(a_{b_1}(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, -I, 0A], [0A, -I, -I], [0A, 0A, -I]] 	* 	x_1
>>>

   <<<
 POL(b_{a_1}(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[-I, -I, 0A], [0A, 0A, 1A], [-I, 0A, 0A]] 	* 	x_1
>>>

   <<<
 POL(a_{a_1}(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [-I, -I, 0A]] 	* 	x_1
>>>

   <<<
 POL(B_{B_1}(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
>>>

   <<<
 POL(b_{b_1}(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, -I], [1A, 0A, 0A], [-I, 0A, 0A]] 	* 	x_1
>>>


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

   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
   b_{b_1}(b_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(x1)))
   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(x1)))


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

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

   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(x1))
   A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
   B_{B_1}(b_{a_1}(x1)) -> A_{A_1}(a_{a_1}(a_{a_1}(x1)))
   B_{B_1}(b_{a_1}(x1)) -> A_{A_1}(a_{a_1}(x1))
   B_{B_1}(b_{a_1}(x1)) -> A_{A_1}(x1)
   B_{B_1}(b_{b_1}(x1)) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
   B_{B_1}(b_{b_1}(x1)) -> A_{A_1}(a_{b_1}(x1))

The TRS R consists of the following rules:

   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(x1)))
   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(x1)))
   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
   b_{b_1}(b_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))

Q is empty.
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.

   A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(A_{A_1}(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
>>>

   <<<
 POL(a_{b_1}(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 1A], [0A, 0A, 0A], [0A, 1A, 0A]] 	* 	x_1
>>>

   <<<
 POL(b_{a_1}(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
>>>

   <<<
 POL(a_{a_1}(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, -I], [0A, -I, -I], [0A, 0A, 0A]] 	* 	x_1
>>>

   <<<
 POL(B_{B_1}(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
>>>

   <<<
 POL(b_{b_1}(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 1A, 1A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
>>>


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

   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
   b_{b_1}(b_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(x1)))
   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(x1)))


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

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

   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(x1))
   B_{B_1}(b_{a_1}(x1)) -> A_{A_1}(a_{a_1}(a_{a_1}(x1)))
   B_{B_1}(b_{a_1}(x1)) -> A_{A_1}(a_{a_1}(x1))
   B_{B_1}(b_{a_1}(x1)) -> A_{A_1}(x1)
   B_{B_1}(b_{b_1}(x1)) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
   B_{B_1}(b_{b_1}(x1)) -> A_{A_1}(a_{b_1}(x1))

The TRS R consists of the following rules:

   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(x1)))
   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(x1)))
   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
   b_{b_1}(b_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))

Q is empty.
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:

   B_{B_1}(b_{a_1}(x1)) -> A_{A_1}(a_{a_1}(a_{a_1}(x1)))
   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(x1))
   B_{B_1}(b_{a_1}(x1)) -> A_{A_1}(a_{a_1}(x1))
   B_{B_1}(b_{a_1}(x1)) -> A_{A_1}(x1)

The TRS R consists of the following rules:

   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(x1)))
   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(x1)))
   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
   b_{b_1}(b_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))

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

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


The following pairs can be oriented strictly and are deleted.

   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(x1))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(B_{B_1}(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
>>>

   <<<
 POL(b_{a_1}(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [-I, 0A, 0A]] 	* 	x_1
>>>

   <<<
 POL(A_{A_1}(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
>>>

   <<<
 POL(a_{a_1}(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [-I, 0A, -I], [0A, 0A, 0A]] 	* 	x_1
>>>

   <<<
 POL(a_{b_1}(x_1)) =  	[[-I], [-I], [1A]] 	 +  	[[0A, 0A, 0A], [-I, 0A, -I], [0A, 1A, -I]] 	* 	x_1
>>>

   <<<
 POL(b_{b_1}(x_1)) =  	[[0A], [-I], [1A]] 	 +  	[[0A, 1A, 0A], [0A, 0A, 0A], [0A, 1A, 0A]] 	* 	x_1
>>>


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

   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(x1)))
   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(x1)))
   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
   b_{b_1}(b_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))


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

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

   B_{B_1}(b_{a_1}(x1)) -> A_{A_1}(a_{a_1}(a_{a_1}(x1)))
   B_{B_1}(b_{a_1}(x1)) -> A_{A_1}(a_{a_1}(x1))
   B_{B_1}(b_{a_1}(x1)) -> A_{A_1}(x1)

The TRS R consists of the following rules:

   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(x1)))
   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(x1)))
   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
   b_{b_1}(b_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))

Q is empty.
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 3 less nodes.
----------------------------------------

(14)
TRUE