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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) DependencyPairsProof [EQUIVALENT, 13 ms]
(2) QDP
(3) QDPOrderProof [EQUIVALENT, 104 ms]
(4) QDP
(5) QDPOrderProof [EQUIVALENT, 38 ms]
(6) QDP
(7) QDPOrderProof [EQUIVALENT, 28 ms]
(8) QDP
(9) DependencyGraphProof [EQUIVALENT, 0 ms]
(10) QDP
(11) QDPOrderProof [EQUIVALENT, 311 ms]
(12) QDP
(13) PisEmptyProof [EQUIVALENT, 0 ms]
(14) YES


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

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

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

Q is empty.

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

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

(2)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   A(a(a(x1))) -> B(x1)
   B(b(x1)) -> A(a(x1))
   B(b(x1)) -> A(x1)
   A(a(x1)) -> A(b(a(x1)))
   A(a(x1)) -> B(a(x1))

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

   A(a(a(x1))) -> B(x1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

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

   <<<
 POL(a(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[1A, 0A, 0A], [0A, -I, -I], [1A, 0A, -I]] 	* 	x_1
>>>

   <<<
 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
>>>

   <<<
 POL(b(x_1)) =  	[[-I], [0A], [0A]] 	 +  	[[0A, 1A, -I], [1A, 0A, 0A], [0A, 1A, 0A]] 	* 	x_1
>>>


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

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


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

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

   B(b(x1)) -> A(a(x1))
   B(b(x1)) -> A(x1)
   A(a(x1)) -> A(b(a(x1)))
   A(a(x1)) -> B(a(x1))

The TRS R consists of the following rules:

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

   B(b(x1)) -> A(x1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
>>>

   <<<
 POL(b(x_1)) =  	[[1A], [1A], [-I]] 	 +  	[[0A, 1A, 1A], [1A, -I, 0A], [-I, -I, -I]] 	* 	x_1
>>>

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

   <<<
 POL(a(x_1)) =  	[[0A], [1A], [-I]] 	 +  	[[-I, 0A, 0A], [0A, 1A, 1A], [-I, -I, -I]] 	* 	x_1
>>>


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

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


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

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

   B(b(x1)) -> A(a(x1))
   A(a(x1)) -> A(b(a(x1)))
   A(a(x1)) -> B(a(x1))

The TRS R consists of the following rules:

   a(a(a(x1))) -> b(x1)
   b(b(x1)) -> a(a(x1))
   a(a(x1)) -> a(b(a(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(x1)) -> B(a(x1))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
>>>

   <<<
 POL(b(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[0A, 1A, 0A], [1A, 0A, 0A], [-I, 0A, -I]] 	* 	x_1
>>>

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

   <<<
 POL(a(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[1A, 0A, 0A], [0A, -I, -I], [0A, -I, 0A]] 	* 	x_1
>>>


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

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


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

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

   B(b(x1)) -> A(a(x1))
   A(a(x1)) -> A(b(a(x1)))

The TRS R consists of the following rules:

   a(a(a(x1))) -> b(x1)
   b(b(x1)) -> a(a(x1))
   a(a(x1)) -> a(b(a(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 1 less node.
----------------------------------------

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

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

The TRS R consists of the following rules:

   a(a(a(x1))) -> b(x1)
   b(b(x1)) -> a(a(x1))
   a(a(x1)) -> a(b(a(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(x1)) -> A(b(a(x1)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic integers [ARCTIC,STERNAGEL_THIEMANN_RTA14]:

   <<<
 POL(A(x_1)) =  	[[0A]] 	 +  	[[-1A, -1A, 2A]] 	* 	x_1
>>>

   <<<
 POL(a(x_1)) =  	[[0A], [-1A], [2A]] 	 +  	[[-I, -1A, -1A], [-I, 0A, -I], [-1A, 2A, 1A]] 	* 	x_1
>>>

   <<<
 POL(b(x_1)) =  	[[2A], [-1A], [-I]] 	 +  	[[-I, 2A, 1A], [-I, 0A, -I], [1A, -1A, -1A]] 	* 	x_1
>>>


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

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


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

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

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

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

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

(14)
YES