2.59/1.44	YES
2.59/1.45	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
2.59/1.45	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.59/1.45	
2.59/1.45	
2.59/1.45	Termination of the given CSR could be proven:
2.59/1.45	
2.59/1.45	(0) CSR
2.59/1.45	(1) CSRRRRProof [EQUIVALENT, 74 ms]
2.59/1.45	(2) CSR
2.59/1.45	(3) RisEmptyProof [EQUIVALENT, 0 ms]
2.59/1.45	(4) YES
2.59/1.45	
2.59/1.45	
2.59/1.45	----------------------------------------
2.59/1.45	
2.59/1.45	(0)
2.59/1.45	Obligation:
2.59/1.45	Context-sensitive rewrite system:
2.59/1.45	The TRS R consists of the following rules:
2.59/1.45	
2.59/1.45	   f(f(a)) -> f(g(f(a)))
2.59/1.45	
2.59/1.45	The replacement map contains the following entries:
2.59/1.45	
2.59/1.45	f: empty set
2.59/1.45	a: empty set
2.59/1.45	g: {1}
2.59/1.45	
2.59/1.45	----------------------------------------
2.59/1.45	
2.59/1.45	(1) CSRRRRProof (EQUIVALENT)
2.59/1.45	The following CSR is given: Context-sensitive rewrite system:
2.59/1.45	The TRS R consists of the following rules:
2.59/1.45	
2.59/1.45	   f(f(a)) -> f(g(f(a)))
2.59/1.45	
2.59/1.45	The replacement map contains the following entries:
2.59/1.45	
2.59/1.45	f: empty set
2.59/1.45	a: empty set
2.59/1.45	g: {1}
2.59/1.45	Used ordering:
2.59/1.45	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
2.59/1.45	
2.59/1.45	   <<<
2.59/1.45	 POL(f(x_1)) =  	[[0], [1]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
2.59/1.45	>>>
2.59/1.45	
2.59/1.45	   <<<
2.59/1.45	 POL(a) =  	[[0], [0]]
2.59/1.45	>>>
2.59/1.45	
2.59/1.45	   <<<
2.59/1.45	 POL(g(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [1, 0]] 	* 	x_1
2.59/1.45	>>>
2.59/1.45	
2.59/1.45	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
2.59/1.45	
2.59/1.45	   f(f(a)) -> f(g(f(a)))
2.59/1.45	
2.59/1.45	
2.59/1.45	
2.59/1.45	
2.59/1.45	----------------------------------------
2.59/1.45	
2.59/1.45	(2)
2.59/1.45	Obligation:
2.59/1.45	Context-sensitive rewrite system:
2.59/1.45	R is empty.
2.59/1.45	
2.59/1.45	----------------------------------------
2.59/1.45	
2.59/1.45	(3) RisEmptyProof (EQUIVALENT)
2.59/1.45	The CSR R is empty. Hence, termination is trivially proven.
2.59/1.45	----------------------------------------
2.59/1.45	
2.59/1.45	(4)
2.59/1.45	YES
2.80/1.47	EOF