3.41/1.69	YES
3.41/1.74	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.41/1.74	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.41/1.74	
3.41/1.74	
3.41/1.74	Termination w.r.t. Q of the given QTRS could be proven:
3.41/1.74	
3.41/1.74	(0) QTRS
3.41/1.74	(1) RFCMatchBoundsTRSProof [EQUIVALENT, 23 ms]
3.41/1.74	(2) YES
3.41/1.74	
3.41/1.74	
3.41/1.74	----------------------------------------
3.41/1.74	
3.41/1.74	(0)
3.41/1.74	Obligation:
3.41/1.74	Q restricted rewrite system:
3.41/1.74	The TRS R consists of the following rules:
3.41/1.74	
3.41/1.74	   a(b(x1)) -> d(x1)
3.41/1.74	   b(a(x1)) -> a(b(x1))
3.41/1.74	   d(c(x1)) -> f(a(b(b(c(x1)))))
3.41/1.74	   d(f(x1)) -> f(a(b(x1)))
3.41/1.74	   a(f(x1)) -> a(x1)
3.41/1.74	
3.41/1.74	Q is empty.
3.41/1.74	
3.41/1.74	----------------------------------------
3.41/1.74	
3.41/1.74	(1) RFCMatchBoundsTRSProof (EQUIVALENT)
3.41/1.74	Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 2. This implies Q-termination of R.
3.41/1.74	The following rules were used to construct the certificate:
3.41/1.74	
3.41/1.74	   a(b(x1)) -> d(x1)
3.41/1.74	   b(a(x1)) -> a(b(x1))
3.41/1.74	   d(c(x1)) -> f(a(b(b(c(x1)))))
3.41/1.74	   d(f(x1)) -> f(a(b(x1)))
3.41/1.74	   a(f(x1)) -> a(x1)
3.41/1.74	
3.41/1.74	The certificate found is represented by the following graph.
3.41/1.74	The certificate consists of the following enumerated nodes:
3.41/1.74	1, 2, 3, 4, 5, 6, 7, 15, 16, 17, 18
3.41/1.74	
3.41/1.74	Node 1 is start node and node 2 is final node.
3.41/1.74	
3.41/1.74	Those nodes are connected through the following edges:
3.41/1.74	
3.41/1.74	* 1 to 2 labelled d_1(0), a_1(0), d_1(1), a_1(1)* 1 to 3 labelled a_1(0)* 1 to 4 labelled f_1(0)* 1 to 1 labelled f_1(0)* 1 to 15 labelled f_1(1), a_1(1), a_1(2)* 2 to 2 labelled #_1(0)* 3 to 2 labelled b_1(0), d_1(2)* 3 to 16 labelled a_1(1)* 3 to 17 labelled d_1(2)* 3 to 15 labelled f_1(1), a_1(2)* 4 to 5 labelled a_1(0)* 4 to 6 labelled d_1(1)* 5 to 6 labelled b_1(0)* 6 to 7 labelled b_1(0)* 7 to 2 labelled c_1(0)* 15 to 16 labelled a_1(1)* 15 to 17 labelled d_1(2)* 15 to 2 labelled d_1(2)* 15 to 15 labelled f_1(1), a_1(2)* 16 to 17 labelled b_1(1), d_1(2)* 16 to 2 labelled b_1(1), d_1(2)* 16 to 16 labelled a_1(1)* 16 to 15 labelled f_1(1), a_1(2)* 17 to 18 labelled b_1(1)* 18 to 2 labelled c_1(1)
3.41/1.74	
3.41/1.74	
3.41/1.74	----------------------------------------
3.41/1.74	
3.41/1.74	(2)
3.41/1.74	YES
3.69/1.77	EOF