4.18/1.81	YES
4.18/1.82	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
4.18/1.82	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.18/1.82	
4.18/1.82	
4.18/1.82	Termination of the given RelTRS could be proven:
4.18/1.82	
4.18/1.82	(0) RelTRS
4.18/1.82	(1) RelTRSRRRProof [EQUIVALENT, 81 ms]
4.18/1.82	(2) RelTRS
4.18/1.82	(3) RIsEmptyProof [EQUIVALENT, 0 ms]
4.18/1.82	(4) YES
4.18/1.82	
4.18/1.82	
4.18/1.82	----------------------------------------
4.18/1.82	
4.18/1.82	(0)
4.18/1.82	Obligation:
4.18/1.82	Relative term rewrite system:
4.18/1.82	The relative TRS consists of the following R rules:
4.18/1.82	
4.18/1.82	   a(a(x1)) -> a(b(a(x1)))
4.18/1.82	
4.18/1.82	The relative TRS consists of the following S rules:
4.18/1.82	
4.18/1.82	   b(x1) -> b(b(x1))
4.18/1.82	
4.18/1.82	
4.18/1.82	----------------------------------------
4.18/1.82	
4.18/1.82	(1) RelTRSRRRProof (EQUIVALENT)
4.18/1.82	We used the following monotonic ordering for rule removal:
4.18/1.82	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
4.18/1.82	
4.18/1.82	   <<<
4.18/1.82	 POL(a(x_1)) =  	[[2], [2]] 	 +  	[[1, 2], [2, 2]] 	* 	x_1
4.18/1.82	>>>
4.18/1.82	
4.18/1.82	   <<<
4.18/1.82	 POL(b(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
4.18/1.82	>>>
4.18/1.82	
4.18/1.82	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
4.18/1.82	Rules from R:
4.18/1.82	
4.18/1.82	   a(a(x1)) -> a(b(a(x1)))
4.18/1.82	Rules from S:
4.18/1.82	none
4.18/1.82	
4.18/1.82	
4.18/1.82	
4.18/1.82	
4.18/1.82	----------------------------------------
4.18/1.82	
4.18/1.82	(2)
4.18/1.82	Obligation:
4.18/1.82	Relative term rewrite system:
4.18/1.82	R is empty.
4.18/1.82	The relative TRS consists of the following S rules:
4.18/1.82	
4.18/1.82	   b(x1) -> b(b(x1))
4.18/1.82	
4.18/1.82	
4.18/1.82	----------------------------------------
4.18/1.82	
4.18/1.82	(3) RIsEmptyProof (EQUIVALENT)
4.18/1.82	The TRS R is empty. Hence, termination is trivially proven.
4.18/1.82	----------------------------------------
4.18/1.82	
4.18/1.82	(4)
4.18/1.82	YES
4.18/1.85	EOF