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


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


YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination of the given RelTRS could be proven:

(0) RelTRS
(1) FlatCCProof [EQUIVALENT, 0 ms]
(2) RelTRS
(3) RootLabelingProof [EQUIVALENT, 0 ms]
(4) RelTRS
(5) RelTRSRRRProof [EQUIVALENT, 1710 ms]
(6) RelTRS
(7) RelTRSRRRProof [EQUIVALENT, 14 ms]
(8) RelTRS
(9) RIsEmptyProof [EQUIVALENT, 0 ms]
(10) YES


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

(0)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

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

The relative TRS consists of the following S rules:

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


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

(1) FlatCCProof (EQUIVALENT)
We used flat context closure [ROOTLAB]

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

(2)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

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

The relative TRS consists of the following S rules:

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


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

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


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

(4)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
   b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{b_1}(b_{b_1}(b_{c_1}(x1)))
   b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(x1)))
   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{c_1}(x1)))
   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(x1)))
   a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1))))
   a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1))))
   a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1))))
   b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1))))
   b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1))))
   b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1))))
   c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1))))
   c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1))))
   c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1))))
   a_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
   a_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
   a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
   b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
   b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
   b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
   c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
   c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
   c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))

The relative TRS consists of the following S rules:

   a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
   a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(x1)))
   a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))
   a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))
   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))
   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))
   b_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))
   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))
   c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))
   c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))
   c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))


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

(5) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

   <<<
 POL(b_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(c_{c_1}(x_1)) =  	[[0], [1]] 	 +  	[[2, 0], [1, 0]] 	* 	x_1
>>>

   <<<
 POL(c_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(c_{a_1}(x_1)) =  	[[0], [1]] 	 +  	[[2, 0], [2, 2]] 	* 	x_1
>>>

   <<<
 POL(b_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(a_{b_1}(x_1)) =  	[[1], [1]] 	 +  	[[2, 0], [2, 0]] 	* 	x_1
>>>

   <<<
 POL(a_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 2], [0, 2]] 	* 	x_1
>>>

   <<<
 POL(a_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 2]] 	* 	x_1
>>>

With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1))))
   a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1))))
   a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1))))
   b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1))))
   b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1))))
   b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1))))
   c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1))))
   c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1))))
   c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1))))
Rules from S:

   a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
   a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(x1)))
   a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))
   b_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))
   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))
   c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))
   c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))
   c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))




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

(6)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
   b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{b_1}(b_{b_1}(b_{c_1}(x1)))
   b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(x1)))
   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{c_1}(x1)))
   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(x1)))
   a_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
   a_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
   a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
   b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
   b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
   b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
   c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
   c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
   c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))

The relative TRS consists of the following S rules:

   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))
   a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))
   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))


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

(7) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Knuth-Bendix order [KBO] with precedence:c_{a_1}_1 > a_{b_1}_1 > b_{a_1}_1 > b_{b_1}_1 > a_{a_1}_1 > c_{c_1}_1 > c_{b_1}_1 > a_{c_1}_1 > b_{c_1}_1

and weight map:

   b_{c_1}_1=6
   c_{c_1}_1=8
   c_{b_1}_1=6
   b_{b_1}_1=5
   c_{a_1}_1=1
   b_{a_1}_1=4
   a_{b_1}_1=16
   a_{c_1}_1=4
   a_{a_1}_1=1

The variable weight is 1With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
   b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{b_1}(b_{b_1}(b_{c_1}(x1)))
   b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(x1)))
   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{c_1}(x1)))
   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(x1)))
   a_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
   a_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
   a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
   b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
   b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
   b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
   c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
   c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
   c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
Rules from S:

   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))
   a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))
   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))




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

(8)
Obligation:
Relative term rewrite system:
R is empty.
S is empty.

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

(9) RIsEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
----------------------------------------

(10)
YES