5.59/2.27	YES
5.59/2.28	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
5.59/2.28	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.59/2.28	
5.59/2.28	
5.59/2.28	Termination of the given RelTRS could be proven:
5.59/2.28	
5.59/2.28	(0) RelTRS
5.59/2.28	(1) RelTRS Reverse [EQUIVALENT, 0 ms]
5.59/2.28	(2) RelTRS
5.59/2.28	(3) RelTRSRRRProof [EQUIVALENT, 162 ms]
5.59/2.28	(4) RelTRS
5.59/2.28	(5) RIsEmptyProof [EQUIVALENT, 2 ms]
5.59/2.28	(6) YES
5.59/2.28	
5.59/2.28	
5.59/2.28	----------------------------------------
5.59/2.28	
5.59/2.28	(0)
5.59/2.28	Obligation:
5.59/2.28	Relative term rewrite system:
5.59/2.28	The relative TRS consists of the following R rules:
5.59/2.28	
5.59/2.28	   f(g(f(x))) -> f(g(g(g(f(x)))))
5.59/2.28	
5.59/2.28	The relative TRS consists of the following S rules:
5.59/2.28	
5.59/2.28	   g(x) -> g(g(x))
5.59/2.28	
5.59/2.28	
5.59/2.28	----------------------------------------
5.59/2.28	
5.59/2.28	(1) RelTRS Reverse (EQUIVALENT)
5.59/2.28	We have reversed the following relative TRS [REVERSE]:
5.59/2.28	The set of rules R is 
5.59/2.28	   f(g(f(x))) -> f(g(g(g(f(x)))))
5.59/2.28	
5.59/2.28	The set of rules S is 
5.59/2.28	   g(x) -> g(g(x))
5.59/2.28	
5.59/2.28	We have obtained the following relative TRS:
5.59/2.28	The set of rules R is 
5.59/2.28	   f(g(f(x))) -> f(g(g(g(f(x)))))
5.59/2.28	
5.59/2.28	The set of rules S is 
5.59/2.28	   g(x) -> g(g(x))
5.59/2.28	
5.59/2.28	
5.59/2.28	----------------------------------------
5.59/2.28	
5.59/2.28	(2)
5.59/2.28	Obligation:
5.59/2.28	Relative term rewrite system:
5.59/2.28	The relative TRS consists of the following R rules:
5.59/2.28	
5.59/2.28	   f(g(f(x))) -> f(g(g(g(f(x)))))
5.59/2.28	
5.59/2.28	The relative TRS consists of the following S rules:
5.59/2.28	
5.59/2.28	   g(x) -> g(g(x))
5.59/2.28	
5.59/2.28	
5.59/2.28	----------------------------------------
5.59/2.28	
5.59/2.28	(3) RelTRSRRRProof (EQUIVALENT)
5.59/2.28	We used the following monotonic ordering for rule removal:
5.59/2.28	Matrix interpretation [MATRO] to (N^3, +, *, >=, >) :
5.59/2.28	
5.59/2.28	   <<<
5.59/2.28	 POL(f(x_1)) =  	[[0], [0], [2]] 	 +  	[[2, 2, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
5.59/2.28	>>>
5.59/2.28	
5.59/2.28	   <<<
5.59/2.28	 POL(g(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 0, 2], [0, 0, 0]] 	* 	x_1
5.59/2.28	>>>
5.59/2.28	
5.59/2.28	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
5.59/2.28	Rules from R:
5.59/2.28	
5.59/2.28	   f(g(f(x))) -> f(g(g(g(f(x)))))
5.59/2.28	Rules from S:
5.59/2.28	none
5.59/2.28	
5.59/2.28	
5.59/2.28	
5.59/2.28	
5.59/2.28	----------------------------------------
5.59/2.28	
5.59/2.28	(4)
5.59/2.28	Obligation:
5.59/2.28	Relative term rewrite system:
5.59/2.28	R is empty.
5.59/2.28	The relative TRS consists of the following S rules:
5.59/2.28	
5.59/2.28	   g(x) -> g(g(x))
5.59/2.28	
5.59/2.28	
5.59/2.28	----------------------------------------
5.59/2.28	
5.59/2.28	(5) RIsEmptyProof (EQUIVALENT)
5.59/2.28	The TRS R is empty. Hence, termination is trivially proven.
5.59/2.28	----------------------------------------
5.59/2.28	
5.59/2.28	(6)
5.59/2.28	YES
5.90/2.33	EOF