3.17/1.58	YES
3.17/1.59	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.17/1.59	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.17/1.59	
3.17/1.59	
3.17/1.59	Termination w.r.t. Q of the given QTRS could be proven:
3.17/1.59	
3.17/1.59	(0) QTRS
3.17/1.59	(1) RFCMatchBoundsTRSProof [EQUIVALENT, 0 ms]
3.17/1.59	(2) YES
3.17/1.59	
3.17/1.59	
3.17/1.59	----------------------------------------
3.17/1.59	
3.17/1.59	(0)
3.17/1.59	Obligation:
3.17/1.59	Q restricted rewrite system:
3.17/1.59	The TRS R consists of the following rules:
3.17/1.59	
3.17/1.59	   a__f(f(a)) -> a__f(g(f(a)))
3.17/1.59	   mark(f(X)) -> a__f(mark(X))
3.17/1.59	   mark(a) -> a
3.17/1.59	   mark(g(X)) -> g(X)
3.17/1.59	   a__f(X) -> f(X)
3.17/1.59	
3.17/1.59	The set Q consists of the following terms:
3.17/1.59	
3.17/1.59	   mark(f(x0))
3.17/1.59	   mark(a)
3.17/1.59	   mark(g(x0))
3.17/1.59	   a__f(x0)
3.17/1.59	
3.17/1.59	
3.17/1.59	----------------------------------------
3.17/1.59	
3.17/1.59	(1) RFCMatchBoundsTRSProof (EQUIVALENT)
3.17/1.59	Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 3. This implies Q-termination of R.
3.17/1.59	The following rules were used to construct the certificate:
3.17/1.59	
3.17/1.59	   a__f(f(a)) -> a__f(g(f(a)))
3.17/1.59	   mark(f(X)) -> a__f(mark(X))
3.17/1.59	   mark(a) -> a
3.17/1.59	   mark(g(X)) -> g(X)
3.17/1.59	   a__f(X) -> f(X)
3.17/1.59	
3.17/1.59	The certificate found is represented by the following graph.
3.17/1.59	The certificate consists of the following enumerated nodes:
3.17/1.59	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
3.17/1.59	
3.17/1.59	Node 1 is start node and node 2 is final node.
3.17/1.59	
3.17/1.59	Those nodes are connected through the following edges:
3.17/1.59	
3.17/1.59	* 1 to 3 labelled a__f_1(0), f_1(1)* 1 to 2 labelled a(0), g_1(0), f_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)* 3 to 2 labelled mark_1(0), a(1), g_1(1)* 3 to 6 labelled a__f_1(1), f_1(2)* 3 to 10 labelled a__f_1(2), f_1(3)* 4 to 5 labelled f_1(0)* 5 to 2 labelled a(0)* 6 to 2 labelled mark_1(1), a(1), g_1(1)* 6 to 6 labelled a__f_1(1), f_1(2)* 6 to 10 labelled a__f_1(2), f_1(3)* 7 to 8 labelled g_1(1)* 8 to 9 labelled f_1(1)* 9 to 2 labelled a(1)* 10 to 11 labelled g_1(2)* 11 to 12 labelled f_1(2)* 12 to 2 labelled a(2)
3.17/1.59	
3.17/1.59	
3.17/1.59	----------------------------------------
3.17/1.59	
3.17/1.59	(2)
3.17/1.59	YES
3.26/1.63	EOF