2.91/1.57	YES
2.91/1.58	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
2.91/1.58	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.91/1.58	
2.91/1.58	
2.91/1.58	Termination w.r.t. Q of the given QTRS could be proven:
2.91/1.58	
2.91/1.58	(0) QTRS
2.91/1.58	(1) QTRS Reverse [EQUIVALENT, 0 ms]
2.91/1.58	(2) QTRS
2.91/1.58	(3) RFCMatchBoundsTRSProof [EQUIVALENT, 0 ms]
2.91/1.58	(4) YES
2.91/1.58	
2.91/1.58	
2.91/1.58	----------------------------------------
2.91/1.58	
2.91/1.58	(0)
2.91/1.58	Obligation:
2.91/1.58	Q restricted rewrite system:
2.91/1.58	The TRS R consists of the following rules:
2.91/1.58	
2.91/1.58	   half(0) -> 0
2.91/1.58	   half(s(0)) -> 0
2.91/1.58	   half(s(s(x))) -> s(half(x))
2.91/1.58	   bits(0) -> 0
2.91/1.58	   bits(s(x)) -> s(bits(half(s(x))))
2.91/1.58	
2.91/1.58	The set Q consists of the following terms:
2.91/1.58	
2.91/1.58	   half(0)
2.91/1.58	   half(s(0))
2.91/1.58	   half(s(s(x0)))
2.91/1.58	   bits(0)
2.91/1.58	   bits(s(x0))
2.91/1.58	
2.91/1.58	
2.91/1.58	----------------------------------------
2.91/1.58	
2.91/1.58	(1) QTRS Reverse (EQUIVALENT)
2.91/1.58	We applied the QTRS Reverse Processor [REVERSE].
2.91/1.58	----------------------------------------
2.91/1.58	
2.91/1.58	(2)
2.91/1.58	Obligation:
2.91/1.58	Q restricted rewrite system:
2.91/1.58	The TRS R consists of the following rules:
2.91/1.58	
2.91/1.58	   0'(half(x)) -> 0'(x)
2.91/1.58	   0'(s(half(x))) -> 0'(x)
2.91/1.58	   s(s(half(x))) -> half(s(x))
2.91/1.58	   0'(bits(x)) -> 0'(x)
2.91/1.58	   s(bits(x)) -> s(half(bits(s(x))))
2.91/1.58	
2.91/1.58	Q is empty.
2.91/1.58	
2.91/1.58	----------------------------------------
2.91/1.58	
2.91/1.58	(3) RFCMatchBoundsTRSProof (EQUIVALENT)
2.91/1.58	Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1. This implies Q-termination of R.
2.91/1.58	The following rules were used to construct the certificate:
2.91/1.58	
2.91/1.58	   0'(half(x)) -> 0'(x)
2.91/1.58	   0'(s(half(x))) -> 0'(x)
2.91/1.58	   s(s(half(x))) -> half(s(x))
2.91/1.58	   0'(bits(x)) -> 0'(x)
2.91/1.58	   s(bits(x)) -> s(half(bits(s(x))))
2.91/1.58	
2.91/1.58	The certificate found is represented by the following graph.
2.91/1.58	The certificate consists of the following enumerated nodes:
2.91/1.58	3, 4, 9, 10, 11, 12, 17, 18, 19, 20
2.91/1.58	
2.91/1.58	Node 3 is start node and node 4 is final node.
2.91/1.58	
2.91/1.58	Those nodes are connected through the following edges:
2.91/1.58	
2.91/1.58	* 3 to 4 labelled 0'_1(0), 0'_1(1)* 3 to 9 labelled half_1(0)* 3 to 10 labelled s_1(0)* 4 to 4 labelled #_1(0)* 9 to 4 labelled s_1(0)* 9 to 17 labelled half_1(1)* 9 to 18 labelled s_1(1)* 10 to 11 labelled half_1(0)* 11 to 12 labelled bits_1(0)* 12 to 4 labelled s_1(0)* 12 to 17 labelled half_1(1)* 12 to 18 labelled s_1(1)* 17 to 4 labelled s_1(1)* 17 to 17 labelled half_1(1)* 17 to 18 labelled s_1(1)* 18 to 19 labelled half_1(1)* 19 to 20 labelled bits_1(1)* 20 to 4 labelled s_1(1)* 20 to 17 labelled half_1(1)* 20 to 18 labelled s_1(1)
2.91/1.58	
2.91/1.58	
2.91/1.58	----------------------------------------
2.91/1.58	
2.91/1.58	(4)
2.91/1.58	YES
3.26/1.60	EOF