34.23/9.55	YES
34.23/9.57	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
34.23/9.57	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
34.23/9.57	
34.23/9.57	
34.23/9.57	Termination of the given RelTRS could be proven:
34.23/9.57	
34.23/9.57	(0) RelTRS
34.23/9.57	(1) FlatCCProof [EQUIVALENT, 0 ms]
34.23/9.57	(2) RelTRS
34.23/9.57	(3) RootLabelingProof [EQUIVALENT, 0 ms]
34.23/9.57	(4) RelTRS
34.23/9.57	(5) RelTRSRRRProof [EQUIVALENT, 282 ms]
34.23/9.57	(6) RelTRS
34.23/9.57	(7) RelTRSRRRProof [EQUIVALENT, 101 ms]
34.23/9.57	(8) RelTRS
34.23/9.57	(9) RelTRSRRRProof [EQUIVALENT, 8 ms]
34.23/9.57	(10) RelTRS
34.23/9.57	(11) RIsEmptyProof [EQUIVALENT, 0 ms]
34.23/9.57	(12) YES
34.23/9.57	
34.23/9.57	
34.23/9.57	----------------------------------------
34.23/9.57	
34.23/9.57	(0)
34.23/9.57	Obligation:
34.23/9.57	Relative term rewrite system:
34.23/9.57	The relative TRS consists of the following R rules:
34.23/9.57	
34.23/9.57	   a(a(x1)) -> x1
34.23/9.57	
34.23/9.57	The relative TRS consists of the following S rules:
34.23/9.57	
34.23/9.57	   a(a(x1)) -> b(a(a(a(b(x1)))))
34.23/9.57	
34.23/9.57	
34.23/9.57	----------------------------------------
34.23/9.57	
34.23/9.57	(1) FlatCCProof (EQUIVALENT)
34.23/9.57	We used flat context closure [ROOTLAB]
34.23/9.57	
34.23/9.57	----------------------------------------
34.23/9.57	
34.23/9.57	(2)
34.23/9.57	Obligation:
34.23/9.57	Relative term rewrite system:
34.23/9.57	The relative TRS consists of the following R rules:
34.23/9.57	
34.23/9.57	   a(a(a(x1))) -> a(x1)
34.23/9.57	   b(a(a(x1))) -> b(x1)
34.23/9.57	
34.23/9.57	The relative TRS consists of the following S rules:
34.23/9.57	
34.23/9.57	   a(a(a(x1))) -> a(b(a(a(a(b(x1))))))
34.23/9.57	   b(a(a(x1))) -> b(b(a(a(a(b(x1))))))
34.23/9.57	
34.23/9.57	
34.23/9.57	----------------------------------------
34.23/9.57	
34.23/9.57	(3) RootLabelingProof (EQUIVALENT)
34.23/9.57	We used plain root labeling [ROOTLAB] with the following heuristic:
34.23/9.57	LabelAll: All function symbols get labeled
34.23/9.57	
34.23/9.57	
34.23/9.57	----------------------------------------
34.23/9.57	
34.23/9.57	(4)
34.23/9.57	Obligation:
34.23/9.57	Relative term rewrite system:
34.23/9.57	The relative TRS consists of the following R rules:
34.23/9.57	
34.23/9.57	   a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{a_1}(x1)
34.23/9.57	   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(x1)
34.23/9.57	   b_{a_1}(a_{a_1}(a_{a_1}(x1))) -> b_{a_1}(x1)
34.23/9.57	   b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(x1)
34.23/9.57	
34.23/9.57	The relative TRS consists of the following S rules:
34.23/9.57	
34.23/9.57	   a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))
34.23/9.57	   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))
34.23/9.57	   b_{a_1}(a_{a_1}(a_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))
34.23/9.57	   b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))
34.23/9.57	
34.23/9.57	
34.23/9.57	----------------------------------------
34.23/9.57	
34.23/9.57	(5) RelTRSRRRProof (EQUIVALENT)
34.23/9.57	We used the following monotonic ordering for rule removal:
34.23/9.57	Matrix interpretation [MATRO] to (N^3, +, *, >=, >) :
34.23/9.57	
34.23/9.57	   <<<
34.23/9.57	 POL(a_{a_1}(x_1)) =  	[[0], [0], [1]] 	 +  	[[1, 1, 0], [0, 0, 1], [0, 0, 1]] 	* 	x_1
34.23/9.57	>>>
34.23/9.57	
34.23/9.57	   <<<
34.23/9.57	 POL(a_{b_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
34.23/9.57	>>>
34.23/9.57	
34.23/9.57	   <<<
34.23/9.57	 POL(b_{a_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
34.23/9.57	>>>
34.23/9.57	
34.23/9.57	   <<<
34.23/9.57	 POL(b_{b_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
34.23/9.57	>>>
34.23/9.57	
34.23/9.57	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
34.23/9.57	Rules from R:
34.23/9.57	
34.23/9.57	   a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{a_1}(x1)
34.23/9.57	Rules from S:
34.23/9.57	
34.23/9.57	   a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))
34.23/9.57	
34.23/9.57	
34.23/9.57	
34.23/9.57	
34.23/9.57	----------------------------------------
34.23/9.57	
34.23/9.57	(6)
34.23/9.57	Obligation:
34.23/9.57	Relative term rewrite system:
34.23/9.57	The relative TRS consists of the following R rules:
34.23/9.57	
34.23/9.57	   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(x1)
34.23/9.57	   b_{a_1}(a_{a_1}(a_{a_1}(x1))) -> b_{a_1}(x1)
34.23/9.57	   b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(x1)
34.23/9.57	
34.23/9.57	The relative TRS consists of the following S rules:
34.23/9.57	
34.23/9.57	   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))
34.23/9.57	   b_{a_1}(a_{a_1}(a_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))
34.23/9.57	   b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))
34.23/9.57	
34.23/9.57	
34.23/9.57	----------------------------------------
34.23/9.57	
34.23/9.57	(7) RelTRSRRRProof (EQUIVALENT)
34.23/9.57	We used the following monotonic ordering for rule removal:
34.23/9.57	Matrix interpretation [MATRO] to (N^3, +, *, >=, >) :
34.23/9.57	
34.23/9.57	   <<<
34.23/9.57	 POL(a_{a_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 0, 1], [0, 1, 0]] 	* 	x_1
34.23/9.57	>>>
34.23/9.57	
34.23/9.57	   <<<
34.23/9.57	 POL(a_{b_1}(x_1)) =  	[[0], [1], [0]] 	 +  	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
34.23/9.57	>>>
34.23/9.57	
34.23/9.57	   <<<
34.23/9.57	 POL(b_{a_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 1], [0, 0, 0], [0, 0, 0]] 	* 	x_1
34.23/9.57	>>>
34.23/9.57	
34.23/9.57	   <<<
34.23/9.57	 POL(b_{b_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
34.23/9.57	>>>
34.23/9.57	
34.23/9.57	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
34.23/9.57	Rules from R:
34.23/9.57	
34.23/9.57	   b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(x1)
34.23/9.57	Rules from S:
34.23/9.57	
34.23/9.57	   b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))
34.23/9.57	
34.23/9.57	
34.23/9.57	
34.23/9.57	
34.23/9.57	----------------------------------------
34.23/9.57	
34.23/9.57	(8)
34.23/9.57	Obligation:
34.23/9.57	Relative term rewrite system:
34.23/9.57	The relative TRS consists of the following R rules:
34.23/9.57	
34.23/9.57	   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(x1)
34.23/9.57	   b_{a_1}(a_{a_1}(a_{a_1}(x1))) -> b_{a_1}(x1)
34.23/9.57	
34.23/9.57	The relative TRS consists of the following S rules:
34.23/9.57	
34.23/9.57	   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))
34.23/9.57	   b_{a_1}(a_{a_1}(a_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))
34.23/9.57	
34.23/9.57	
34.23/9.57	----------------------------------------
34.23/9.57	
34.23/9.57	(9) RelTRSRRRProof (EQUIVALENT)
34.23/9.57	We used the following monotonic ordering for rule removal:
34.23/9.57	Polynomial interpretation [POLO]:
34.23/9.57	
34.23/9.57	   POL(a_{a_1}(x_1)) = 1 + x_1
34.23/9.57	   POL(a_{b_1}(x_1)) = x_1
34.23/9.57	   POL(b_{a_1}(x_1)) = x_1
34.23/9.57	   POL(b_{b_1}(x_1)) = x_1
34.23/9.57	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
34.23/9.57	Rules from R:
34.23/9.57	
34.23/9.57	   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(x1)
34.23/9.57	   b_{a_1}(a_{a_1}(a_{a_1}(x1))) -> b_{a_1}(x1)
34.23/9.57	Rules from S:
34.23/9.57	none
34.23/9.57	
34.23/9.57	
34.23/9.57	
34.23/9.57	
34.23/9.57	----------------------------------------
34.23/9.57	
34.23/9.57	(10)
34.23/9.57	Obligation:
34.23/9.57	Relative term rewrite system:
34.23/9.57	R is empty.
34.23/9.57	The relative TRS consists of the following S rules:
34.23/9.57	
34.23/9.57	   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))
34.23/9.57	   b_{a_1}(a_{a_1}(a_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))
34.23/9.57	
34.23/9.57	
34.23/9.57	----------------------------------------
34.23/9.57	
34.23/9.57	(11) RIsEmptyProof (EQUIVALENT)
34.23/9.57	The TRS R is empty. Hence, termination is trivially proven.
34.23/9.57	----------------------------------------
34.23/9.57	
34.23/9.57	(12)
34.23/9.57	YES
36.29/10.11	EOF