4.68/1.86	YES
4.68/1.88	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
4.68/1.88	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.68/1.88	
4.68/1.88	
4.68/1.88	Termination of the given RelTRS could be proven:
4.68/1.88	
4.68/1.88	(0) RelTRS
4.68/1.88	(1) RootLabelingProof [EQUIVALENT, 0 ms]
4.68/1.88	(2) RelTRS
4.68/1.88	(3) RelTRSRRRProof [EQUIVALENT, 106 ms]
4.68/1.88	(4) RelTRS
4.68/1.88	(5) RIsEmptyProof [EQUIVALENT, 0 ms]
4.68/1.88	(6) YES
4.68/1.88	
4.68/1.88	
4.68/1.88	----------------------------------------
4.68/1.88	
4.68/1.88	(0)
4.68/1.88	Obligation:
4.68/1.88	Relative term rewrite system:
4.68/1.88	The relative TRS consists of the following R rules:
4.68/1.88	
4.68/1.88	   a(b(a(x1))) -> a(b(b(a(x1))))
4.68/1.88	
4.68/1.88	The relative TRS consists of the following S rules:
4.68/1.88	
4.68/1.88	   b(x1) -> b(b(x1))
4.68/1.88	
4.68/1.88	
4.68/1.88	----------------------------------------
4.68/1.88	
4.68/1.88	(1) RootLabelingProof (EQUIVALENT)
4.68/1.88	We used plain root labeling [ROOTLAB] with the following heuristic:
4.68/1.88	LabelAll: All function symbols get labeled
4.68/1.88	
4.68/1.88	
4.68/1.88	----------------------------------------
4.68/1.88	
4.68/1.88	(2)
4.68/1.88	Obligation:
4.68/1.88	Relative term rewrite system:
4.68/1.88	The relative TRS consists of the following R rules:
4.68/1.88	
4.68/1.88	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
4.68/1.88	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))
4.68/1.88	
4.68/1.88	The relative TRS consists of the following S rules:
4.68/1.88	
4.68/1.88	   b_{a_1}(x1) -> b_{b_1}(b_{a_1}(x1))
4.68/1.88	   b_{b_1}(x1) -> b_{b_1}(b_{b_1}(x1))
4.68/1.88	
4.68/1.88	
4.68/1.88	----------------------------------------
4.68/1.88	
4.68/1.88	(3) RelTRSRRRProof (EQUIVALENT)
4.68/1.88	We used the following monotonic ordering for rule removal:
4.68/1.88	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
4.68/1.88	
4.68/1.88	   <<<
4.68/1.88	 POL(a_{b_1}(x_1)) =  	[[0], [1]] 	 +  	[[1, 2], [0, 2]] 	* 	x_1
4.68/1.88	>>>
4.68/1.88	
4.68/1.88	   <<<
4.68/1.88	 POL(b_{a_1}(x_1)) =  	[[0], [2]] 	 +  	[[1, 0], [2, 2]] 	* 	x_1
4.68/1.88	>>>
4.68/1.88	
4.68/1.88	   <<<
4.68/1.88	 POL(a_{a_1}(x_1)) =  	[[0], [1]] 	 +  	[[2, 0], [2, 0]] 	* 	x_1
4.68/1.88	>>>
4.68/1.88	
4.68/1.88	   <<<
4.68/1.88	 POL(b_{b_1}(x_1)) =  	[[0], [2]] 	 +  	[[1, 0], [2, 0]] 	* 	x_1
4.68/1.88	>>>
4.68/1.88	
4.68/1.88	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
4.68/1.88	Rules from R:
4.68/1.88	
4.68/1.88	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
4.68/1.88	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))
4.68/1.88	Rules from S:
4.68/1.88	none
4.68/1.88	
4.68/1.88	
4.68/1.88	
4.68/1.88	
4.68/1.88	----------------------------------------
4.68/1.88	
4.68/1.88	(4)
4.68/1.88	Obligation:
4.68/1.88	Relative term rewrite system:
4.68/1.88	R is empty.
4.68/1.88	The relative TRS consists of the following S rules:
4.68/1.88	
4.68/1.88	   b_{a_1}(x1) -> b_{b_1}(b_{a_1}(x1))
4.68/1.88	   b_{b_1}(x1) -> b_{b_1}(b_{b_1}(x1))
4.68/1.88	
4.68/1.88	
4.68/1.88	----------------------------------------
4.68/1.88	
4.68/1.88	(5) RIsEmptyProof (EQUIVALENT)
4.68/1.88	The TRS R is empty. Hence, termination is trivially proven.
4.68/1.88	----------------------------------------
4.68/1.88	
4.68/1.88	(6)
4.68/1.88	YES
4.89/1.95	EOF