3.82/1.86	YES
4.23/1.87	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
4.23/1.87	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.23/1.87	
4.23/1.87	
4.23/1.87	Termination of the given RelTRS could be proven:
4.23/1.87	
4.23/1.87	(0) RelTRS
4.23/1.87	(1) RelTRSRRRProof [EQUIVALENT, 36 ms]
4.23/1.87	(2) RelTRS
4.23/1.87	(3) RIsEmptyProof [EQUIVALENT, 0 ms]
4.23/1.87	(4) YES
4.23/1.87	
4.23/1.87	
4.23/1.87	----------------------------------------
4.23/1.87	
4.23/1.87	(0)
4.23/1.87	Obligation:
4.23/1.87	Relative term rewrite system:
4.23/1.87	The relative TRS consists of the following R rules:
4.23/1.87	
4.23/1.87	   f(g(x)) -> x
4.23/1.87	
4.23/1.87	The relative TRS consists of the following S rules:
4.23/1.87	
4.23/1.87	   a -> h(g(f(a)))
4.23/1.87	
4.23/1.87	
4.23/1.87	----------------------------------------
4.23/1.87	
4.23/1.87	(1) RelTRSRRRProof (EQUIVALENT)
4.23/1.87	We used the following monotonic ordering for rule removal:
4.23/1.87	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
4.23/1.87	
4.23/1.87	   <<<
4.23/1.87	 POL(f(x_1)) =  	[[0], [0]] 	 +  	[[1, 2], [2, 0]] 	* 	x_1
4.23/1.87	>>>
4.23/1.87	
4.23/1.87	   <<<
4.23/1.87	 POL(g(x_1)) =  	[[0], [2]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
4.23/1.87	>>>
4.23/1.87	
4.23/1.87	   <<<
4.23/1.87	 POL(a) =  	[[0], [0]]
4.23/1.87	>>>
4.23/1.87	
4.23/1.87	   <<<
4.23/1.87	 POL(h(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
4.23/1.87	>>>
4.23/1.87	
4.23/1.87	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
4.23/1.87	Rules from R:
4.23/1.87	
4.23/1.87	   f(g(x)) -> x
4.23/1.87	Rules from S:
4.23/1.87	none
4.23/1.87	
4.23/1.87	
4.23/1.87	
4.23/1.87	
4.23/1.87	----------------------------------------
4.23/1.87	
4.23/1.87	(2)
4.23/1.87	Obligation:
4.23/1.87	Relative term rewrite system:
4.23/1.87	R is empty.
4.23/1.87	The relative TRS consists of the following S rules:
4.23/1.87	
4.23/1.87	   a -> h(g(f(a)))
4.23/1.87	
4.23/1.87	
4.23/1.87	----------------------------------------
4.23/1.87	
4.23/1.87	(3) RIsEmptyProof (EQUIVALENT)
4.23/1.87	The TRS R is empty. Hence, termination is trivially proven.
4.23/1.87	----------------------------------------
4.23/1.87	
4.23/1.87	(4)
4.23/1.87	YES
4.32/1.92	EOF