3.11/1.70	YES
3.34/1.71	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.34/1.71	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.34/1.71	
3.34/1.71	
3.34/1.71	Termination w.r.t. Q of the given QTRS could be proven:
3.34/1.71	
3.34/1.71	(0) QTRS
3.34/1.71	(1) RFCMatchBoundsTRSProof [EQUIVALENT, 0 ms]
3.34/1.71	(2) YES
3.34/1.71	
3.34/1.71	
3.34/1.71	----------------------------------------
3.34/1.71	
3.34/1.71	(0)
3.34/1.71	Obligation:
3.34/1.71	Q restricted rewrite system:
3.34/1.71	The TRS R consists of the following rules:
3.34/1.71	
3.34/1.71	   active(f(f(a))) -> mark(f(g(f(a))))
3.34/1.71	   mark(f(X)) -> active(f(X))
3.34/1.71	   mark(a) -> active(a)
3.34/1.71	   mark(g(X)) -> active(g(mark(X)))
3.34/1.71	   f(mark(X)) -> f(X)
3.34/1.71	   f(active(X)) -> f(X)
3.34/1.71	   g(mark(X)) -> g(X)
3.34/1.71	   g(active(X)) -> g(X)
3.34/1.71	
3.34/1.71	The set Q consists of the following terms:
3.34/1.71	
3.34/1.71	   active(f(f(a)))
3.34/1.71	   mark(f(x0))
3.34/1.71	   mark(a)
3.34/1.71	   mark(g(x0))
3.34/1.71	   f(mark(x0))
3.34/1.71	   f(active(x0))
3.34/1.71	   g(mark(x0))
3.34/1.71	   g(active(x0))
3.34/1.71	
3.34/1.71	
3.34/1.71	----------------------------------------
3.34/1.71	
3.34/1.71	(1) RFCMatchBoundsTRSProof (EQUIVALENT)
3.34/1.71	Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 2. This implies Q-termination of R.
3.34/1.71	The following rules were used to construct the certificate:
3.34/1.71	
3.34/1.71	   active(f(f(a))) -> mark(f(g(f(a))))
3.34/1.71	   mark(f(X)) -> active(f(X))
3.34/1.71	   mark(a) -> active(a)
3.34/1.71	   mark(g(X)) -> active(g(mark(X)))
3.34/1.71	   f(mark(X)) -> f(X)
3.34/1.71	   f(active(X)) -> f(X)
3.34/1.71	   g(mark(X)) -> g(X)
3.34/1.71	   g(active(X)) -> g(X)
3.34/1.71	
3.34/1.71	The certificate found is represented by the following graph.
3.34/1.71	The certificate consists of the following enumerated nodes:
3.34/1.71	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17
3.34/1.71	
3.34/1.71	Node 1 is start node and node 2 is final node.
3.34/1.71	
3.34/1.71	Those nodes are connected through the following edges:
3.34/1.71	
3.34/1.71	* 1 to 3 labelled mark_1(0)* 1 to 7 labelled active_1(0)* 1 to 6 labelled active_1(0)* 1 to 8 labelled active_1(0)* 1 to 2 labelled f_1(0), g_1(0), f_1(1), g_1(1)* 1 to 10 labelled active_1(1)* 1 to 13 labelled mark_1(1)* 1 to 17 labelled active_1(2)* 2 to 2 labelled #_1(0)* 3 to 4 labelled f_1(0)* 4 to 5 labelled g_1(0)* 5 to 6 labelled f_1(0)* 6 to 2 labelled a(0)* 7 to 2 labelled f_1(0), f_1(1), a(1)* 8 to 9 labelled g_1(0)* 8 to 2 labelled g_1(1)* 8 to 7 labelled g_1(1)* 8 to 11 labelled g_1(1)* 8 to 13 labelled g_1(1)* 8 to 17 labelled g_1(1)* 9 to 2 labelled mark_1(0)* 9 to 7 labelled active_1(1)* 9 to 11 labelled active_1(1)* 9 to 13 labelled mark_1(1)* 9 to 17 labelled active_1(2)* 10 to 4 labelled f_1(1)* 11 to 12 labelled g_1(1)* 11 to 2 labelled g_1(2), g_1(1)* 11 to 7 labelled g_1(2)* 11 to 11 labelled g_1(2)* 11 to 13 labelled g_1(2)* 11 to 17 labelled g_1(2)* 12 to 2 labelled mark_1(1)* 12 to 7 labelled active_1(1)* 12 to 11 labelled active_1(1)* 12 to 13 labelled mark_1(1)* 12 to 17 labelled active_1(2)* 13 to 14 labelled f_1(1)* 14 to 15 labelled g_1(1)* 15 to 16 labelled f_1(1)* 16 to 2 labelled a(1)* 17 to 14 labelled f_1(2)
3.34/1.71	
3.34/1.71	
3.34/1.71	----------------------------------------
3.34/1.71	
3.34/1.71	(2)
3.34/1.71	YES
3.34/1.74	EOF