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


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YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


Termination w.r.t. Q of the given QTRS could be proven:

(0) QTRS
(1) QTRS Reverse [EQUIVALENT, 0 ms]
(2) QTRS
(3) DependencyPairsProof [EQUIVALENT, 14 ms]
(4) QDP
(5) QDPOrderProof [EQUIVALENT, 38 ms]
(6) QDP
(7) DependencyGraphProof [EQUIVALENT, 0 ms]
(8) AND
    (9) QDP
        (10) QDPOrderProof [EQUIVALENT, 13 ms]
        (11) QDP
        (12) DependencyGraphProof [EQUIVALENT, 0 ms]
        (13) TRUE
    (14) QDP
        (15) QDPOrderProof [EQUIVALENT, 363 ms]
        (16) QDP
        (17) PisEmptyProof [EQUIVALENT, 0 ms]
        (18) YES


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

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

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

Q is empty.

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

(1) QTRS Reverse (EQUIVALENT)
We applied the QTRS Reverse Processor [REVERSE].
----------------------------------------

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

   b(c(c(x1))) -> b(c(a(x1)))
   a(b(c(a(x1)))) -> c(c(b(x1)))
   c(a(b(x1))) -> a(c(b(a(x1))))
   a(c(b(x1))) -> b(a(c(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:

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

The TRS R consists of the following rules:

   b(c(c(x1))) -> b(c(a(x1)))
   a(b(c(a(x1)))) -> c(c(b(x1)))
   c(a(b(x1))) -> a(c(b(a(x1))))
   a(c(b(x1))) -> b(a(c(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(b(c(a(x1)))) -> B(x1)
   C(a(b(x1))) -> B(a(x1))
   C(a(b(x1))) -> A(x1)
   A(c(b(x1))) -> B(a(c(x1)))
   A(c(b(x1))) -> A(c(x1))
   A(c(b(x1))) -> C(x1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(A(x_1)) = x_1
   POL(B(x_1)) = x_1
   POL(C(x_1)) = x_1
   POL(a(x_1)) = x_1
   POL(b(x_1)) = 1 + x_1
   POL(c(x_1)) = x_1

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

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


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

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

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

The TRS R consists of the following rules:

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

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

(7) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 2 less nodes.
----------------------------------------

(8)
Complex Obligation (AND)

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

(9)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

   C(a(b(x1))) -> A(c(b(a(x1))))
   C(a(b(x1))) -> C(b(a(x1)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(A(x_1)) = 0
   POL(C(x_1)) = x_1
   POL(a(x_1)) = 1
   POL(b(x_1)) = 0
   POL(c(x_1)) = x_1

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

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


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

(11)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

(12) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
----------------------------------------

(13)
TRUE

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

(14)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   B(c(c(x1))) -> B(c(a(x1)))

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

   B(c(c(x1))) -> B(c(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]] 	 +  	[[1A, 0A, -I]] 	* 	x_1
>>>

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

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

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


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

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


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

(16)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:

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

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

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

(18)
YES