21.68/6.53	YES
21.68/6.55	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
21.68/6.55	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
21.68/6.55	
21.68/6.55	
21.68/6.55	Termination of the given RelTRS could be proven:
21.68/6.55	
21.68/6.55	(0) RelTRS
21.68/6.55	(1) RelTRS Reverse [EQUIVALENT, 0 ms]
21.68/6.55	(2) RelTRS
21.68/6.55	(3) RootLabelingProof [EQUIVALENT, 0 ms]
21.68/6.55	(4) RelTRS
21.68/6.55	(5) RelTRSRRRProof [EQUIVALENT, 955 ms]
21.68/6.55	(6) RelTRS
21.68/6.55	(7) RelTRSRRRProof [EQUIVALENT, 303 ms]
21.68/6.55	(8) RelTRS
21.68/6.55	(9) RIsEmptyProof [EQUIVALENT, 0 ms]
21.68/6.55	(10) YES
21.68/6.55	
21.68/6.55	
21.68/6.55	----------------------------------------
21.68/6.55	
21.68/6.55	(0)
21.68/6.55	Obligation:
21.68/6.55	Relative term rewrite system:
21.68/6.55	The relative TRS consists of the following R rules:
21.68/6.55	
21.68/6.55	   b(b(a(a(b(a(b(x1))))))) -> b(a(b(b(a(b(b(x1)))))))
21.68/6.55	   b(b(b(x1))) -> b(b(a(b(b(x1)))))
21.68/6.55	
21.68/6.55	The relative TRS consists of the following S rules:
21.68/6.55	
21.68/6.55	   b(a(b(x1))) -> b(a(b(a(a(b(x1))))))
21.68/6.55	
21.68/6.55	
21.68/6.55	----------------------------------------
21.68/6.55	
21.68/6.55	(1) RelTRS Reverse (EQUIVALENT)
21.68/6.55	We have reversed the following relative TRS [REVERSE]:
21.68/6.55	The set of rules R is 
21.68/6.55	   b(b(a(a(b(a(b(x1))))))) -> b(a(b(b(a(b(b(x1)))))))
21.68/6.55	   b(b(b(x1))) -> b(b(a(b(b(x1)))))
21.68/6.55	
21.68/6.55	The set of rules S is 
21.68/6.55	   b(a(b(x1))) -> b(a(b(a(a(b(x1))))))
21.68/6.55	
21.68/6.55	We have obtained the following relative TRS:
21.68/6.55	The set of rules R is 
21.68/6.55	   b(a(b(a(a(b(b(x1))))))) -> b(b(a(b(b(a(b(x1)))))))
21.68/6.55	   b(b(b(x1))) -> b(b(a(b(b(x1)))))
21.68/6.55	
21.68/6.55	The set of rules S is 
21.68/6.55	   b(a(b(x1))) -> b(a(a(b(a(b(x1))))))
21.68/6.55	
21.68/6.55	
21.68/6.55	----------------------------------------
21.68/6.55	
21.68/6.55	(2)
21.68/6.55	Obligation:
21.68/6.55	Relative term rewrite system:
21.68/6.55	The relative TRS consists of the following R rules:
21.68/6.55	
21.68/6.55	   b(a(b(a(a(b(b(x1))))))) -> b(b(a(b(b(a(b(x1)))))))
21.68/6.55	   b(b(b(x1))) -> b(b(a(b(b(x1)))))
21.68/6.55	
21.68/6.55	The relative TRS consists of the following S rules:
21.68/6.55	
21.68/6.55	   b(a(b(x1))) -> b(a(a(b(a(b(x1))))))
21.68/6.55	
21.68/6.55	
21.68/6.55	----------------------------------------
21.68/6.55	
21.68/6.55	(3) RootLabelingProof (EQUIVALENT)
21.68/6.55	We used plain root labeling [ROOTLAB] with the following heuristic:
21.68/6.55	LabelAll: All function symbols get labeled
21.68/6.55	
21.68/6.55	
21.68/6.55	----------------------------------------
21.68/6.55	
21.68/6.55	(4)
21.68/6.55	Obligation:
21.68/6.55	Relative term rewrite system:
21.68/6.55	The relative TRS consists of the following R rules:
21.68/6.55	
21.68/6.55	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))
21.68/6.55	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))
21.68/6.55	   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
21.68/6.55	   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
21.68/6.55	
21.68/6.55	The relative TRS consists of the following S rules:
21.68/6.55	
21.68/6.55	   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))
21.68/6.55	   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))))
21.68/6.55	
21.68/6.55	
21.68/6.55	----------------------------------------
21.68/6.55	
21.68/6.55	(5) RelTRSRRRProof (EQUIVALENT)
21.68/6.55	We used the following monotonic ordering for rule removal:
21.68/6.55	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
21.68/6.55	
21.68/6.55	   <<<
21.68/6.55	 POL(b_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 2], [1, 2]] 	* 	x_1
21.68/6.55	>>>
21.68/6.55	
21.68/6.55	   <<<
21.68/6.55	 POL(a_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
21.68/6.55	>>>
21.68/6.55	
21.68/6.55	   <<<
21.68/6.55	 POL(a_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
21.68/6.55	>>>
21.68/6.55	
21.68/6.55	   <<<
21.68/6.55	 POL(b_{b_1}(x_1)) =  	[[1], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
21.68/6.55	>>>
21.68/6.55	
21.68/6.55	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
21.68/6.55	Rules from R:
21.68/6.55	
21.68/6.55	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))
21.68/6.55	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))
21.68/6.55	Rules from S:
21.68/6.55	none
21.68/6.55	
21.68/6.55	
21.68/6.55	
21.68/6.55	
21.68/6.55	----------------------------------------
21.68/6.55	
21.68/6.55	(6)
21.68/6.55	Obligation:
21.68/6.55	Relative term rewrite system:
21.68/6.55	The relative TRS consists of the following R rules:
21.68/6.55	
21.68/6.55	   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
21.68/6.55	   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
21.68/6.55	
21.68/6.55	The relative TRS consists of the following S rules:
21.68/6.55	
21.68/6.55	   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))
21.68/6.55	   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))))
21.68/6.55	
21.68/6.55	
21.68/6.55	----------------------------------------
21.68/6.55	
21.68/6.55	(7) RelTRSRRRProof (EQUIVALENT)
21.68/6.55	We used the following monotonic ordering for rule removal:
21.68/6.55	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
21.68/6.55	
21.68/6.55	   <<<
21.68/6.55	 POL(b_{b_1}(x_1)) =  	[[2], [2]] 	 +  	[[2, 1], [1, 2]] 	* 	x_1
21.68/6.55	>>>
21.68/6.55	
21.68/6.55	   <<<
21.68/6.55	 POL(b_{a_1}(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
21.68/6.55	>>>
21.68/6.55	
21.68/6.55	   <<<
21.68/6.55	 POL(a_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
21.68/6.55	>>>
21.68/6.55	
21.68/6.55	   <<<
21.68/6.55	 POL(a_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
21.68/6.55	>>>
21.68/6.55	
21.68/6.55	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
21.68/6.55	Rules from R:
21.68/6.55	
21.68/6.55	   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
21.68/6.55	   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
21.68/6.55	Rules from S:
21.68/6.55	none
21.68/6.55	
21.68/6.55	
21.68/6.55	
21.68/6.55	
21.68/6.55	----------------------------------------
21.68/6.55	
21.68/6.55	(8)
21.68/6.55	Obligation:
21.68/6.55	Relative term rewrite system:
21.68/6.55	R is empty.
21.68/6.55	The relative TRS consists of the following S rules:
21.68/6.55	
21.68/6.55	   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))
21.68/6.55	   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))))
21.68/6.55	
21.68/6.55	
21.68/6.55	----------------------------------------
21.68/6.55	
21.68/6.55	(9) RIsEmptyProof (EQUIVALENT)
21.68/6.55	The TRS R is empty. Hence, termination is trivially proven.
21.68/6.55	----------------------------------------
21.68/6.55	
21.68/6.55	(10)
21.68/6.55	YES
22.03/6.62	EOF