3.86/1.81	YES
3.93/1.81	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.93/1.81	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.93/1.81	
3.93/1.81	
3.93/1.81	Termination of the given RelTRS could be proven:
3.93/1.81	
3.93/1.81	(0) RelTRS
3.93/1.81	(1) RelTRSRRRProof [EQUIVALENT, 82 ms]
3.93/1.81	(2) RelTRS
3.93/1.81	(3) RIsEmptyProof [EQUIVALENT, 0 ms]
3.93/1.81	(4) YES
3.93/1.81	
3.93/1.81	
3.93/1.81	----------------------------------------
3.93/1.81	
3.93/1.81	(0)
3.93/1.81	Obligation:
3.93/1.81	Relative term rewrite system:
3.93/1.81	The relative TRS consists of the following R rules:
3.93/1.81	
3.93/1.81	   f(f(x)) -> x
3.93/1.81	
3.93/1.81	The relative TRS consists of the following S rules:
3.93/1.81	
3.93/1.81	   f(x) -> g(f(g(f(x))))
3.93/1.81	
3.93/1.81	
3.93/1.81	----------------------------------------
3.93/1.81	
3.93/1.81	(1) RelTRSRRRProof (EQUIVALENT)
3.93/1.81	We used the following monotonic ordering for rule removal:
3.93/1.81	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
3.93/1.81	
3.93/1.81	   <<<
3.93/1.81	 POL(f(x_1)) =  	[[0], [1]] 	 +  	[[1, 1], [2, 0]] 	* 	x_1
3.93/1.81	>>>
3.93/1.81	
3.93/1.81	   <<<
3.93/1.81	 POL(g(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
3.93/1.81	>>>
3.93/1.81	
3.93/1.81	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
3.93/1.81	Rules from R:
3.93/1.81	
3.93/1.81	   f(f(x)) -> x
3.93/1.81	Rules from S:
3.93/1.81	none
3.93/1.81	
3.93/1.81	
3.93/1.81	
3.93/1.81	
3.93/1.81	----------------------------------------
3.93/1.81	
3.93/1.81	(2)
3.93/1.81	Obligation:
3.93/1.81	Relative term rewrite system:
3.93/1.81	R is empty.
3.93/1.81	The relative TRS consists of the following S rules:
3.93/1.81	
3.93/1.81	   f(x) -> g(f(g(f(x))))
3.93/1.81	
3.93/1.81	
3.93/1.81	----------------------------------------
3.93/1.81	
3.93/1.81	(3) RIsEmptyProof (EQUIVALENT)
3.93/1.81	The TRS R is empty. Hence, termination is trivially proven.
3.93/1.81	----------------------------------------
3.93/1.81	
3.93/1.81	(4)
3.93/1.81	YES
3.95/1.84	EOF