2.79/1.62	YES
2.79/1.63	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
2.79/1.63	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.79/1.63	
2.79/1.63	
2.79/1.63	Termination w.r.t. Q of the given QTRS could be proven:
2.79/1.63	
2.79/1.63	(0) QTRS
2.79/1.63	(1) RFCMatchBoundsTRSProof [EQUIVALENT, 0 ms]
2.79/1.63	(2) YES
2.79/1.63	
2.79/1.63	
2.79/1.63	----------------------------------------
2.79/1.63	
2.79/1.63	(0)
2.79/1.63	Obligation:
2.79/1.63	Q restricted rewrite system:
2.79/1.63	The TRS R consists of the following rules:
2.79/1.63	
2.79/1.63	   a__f(f(a)) -> a__f(g(f(a)))
2.79/1.63	   mark(f(X)) -> a__f(X)
2.79/1.63	   mark(a) -> a
2.79/1.63	   mark(g(X)) -> g(mark(X))
2.79/1.63	   a__f(X) -> f(X)
2.79/1.63	
2.79/1.63	The set Q consists of the following terms:
2.79/1.63	
2.79/1.63	   mark(f(x0))
2.79/1.63	   mark(a)
2.79/1.63	   mark(g(x0))
2.79/1.63	   a__f(x0)
2.79/1.63	
2.79/1.63	
2.79/1.63	----------------------------------------
2.79/1.63	
2.79/1.63	(1) RFCMatchBoundsTRSProof (EQUIVALENT)
2.79/1.63	Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 2. This implies Q-termination of R.
2.79/1.63	The following rules were used to construct the certificate:
2.79/1.63	
2.79/1.63	   a__f(f(a)) -> a__f(g(f(a)))
2.79/1.63	   mark(f(X)) -> a__f(X)
2.79/1.63	   mark(a) -> a
2.79/1.63	   mark(g(X)) -> g(mark(X))
2.79/1.63	   a__f(X) -> f(X)
2.79/1.63	
2.79/1.63	The certificate found is represented by the following graph.
2.79/1.64	The certificate consists of the following enumerated nodes:
2.79/1.64	1, 2, 3, 4, 5, 6, 7, 8, 9, 10
2.79/1.64	
2.79/1.64	Node 1 is start node and node 2 is final node.
2.79/1.64	
2.79/1.64	Those nodes are connected through the following edges:
2.79/1.64	
2.79/1.64	* 1 to 3 labelled a__f_1(0), f_1(1)* 1 to 2 labelled a__f_1(0), a(0), f_1(0), f_1(1)* 1 to 6 labelled g_1(0)* 1 to 7 labelled a__f_1(1), f_1(2)* 2 to 2 labelled #_1(0)* 3 to 4 labelled g_1(0)* 4 to 5 labelled f_1(0)* 5 to 2 labelled a(0)* 6 to 2 labelled mark_1(0), a__f_1(1), a(1), f_1(2)* 6 to 10 labelled g_1(1)* 6 to 7 labelled a__f_1(1), f_1(2)* 7 to 8 labelled g_1(1)* 8 to 9 labelled f_1(1)* 9 to 2 labelled a(1)* 10 to 2 labelled mark_1(1), a__f_1(1), a(1), f_1(2)* 10 to 10 labelled g_1(1)* 10 to 7 labelled a__f_1(1), f_1(2)
2.79/1.64	
2.79/1.64	
2.79/1.64	----------------------------------------
2.79/1.64	
2.79/1.64	(2)
2.79/1.64	YES
3.03/1.68	EOF