3.35/1.62	YES
3.35/1.63	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.35/1.63	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.35/1.63	
3.35/1.63	
3.35/1.63	Quasi decreasingness of the given CTRS could be proven:
3.35/1.63	
3.35/1.63	(0) CTRS
3.35/1.63	(1) CTRSToQTRSProof [SOUND, 0 ms]
3.35/1.63	(2) QTRS
3.35/1.63	(3) QTRSRRRProof [EQUIVALENT, 39 ms]
3.35/1.63	(4) QTRS
3.35/1.63	(5) QTRSRRRProof [EQUIVALENT, 0 ms]
3.35/1.63	(6) QTRS
3.35/1.63	(7) RisEmptyProof [EQUIVALENT, 0 ms]
3.35/1.63	(8) YES
3.35/1.63	
3.35/1.63	
3.35/1.63	----------------------------------------
3.35/1.63	
3.35/1.63	(0)
3.35/1.63	Obligation:
3.35/1.63	Conditional term rewrite system:
3.35/1.63	The TRS R consists of the following rules:
3.35/1.63	
3.35/1.63	   a -> c
3.35/1.63	   a -> d
3.35/1.63	
3.35/1.63	The conditional TRS C consists of the following conditional rules:
3.35/1.63	
3.35/1.63	   f(x) -> x <= x -> c
3.35/1.63	
3.35/1.63	
3.35/1.63	----------------------------------------
3.35/1.63	
3.35/1.63	(1) CTRSToQTRSProof (SOUND)
3.35/1.63	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
3.35/1.63	----------------------------------------
3.35/1.63	
3.35/1.63	(2)
3.35/1.63	Obligation:
3.35/1.63	Q restricted rewrite system:
3.35/1.63	The TRS R consists of the following rules:
3.35/1.63	
3.35/1.63	   f(x) -> U1(x, x)
3.35/1.63	   U1(c, x) -> x
3.35/1.63	   a -> c
3.35/1.63	   a -> d
3.35/1.63	
3.35/1.63	Q is empty.
3.35/1.63	
3.35/1.63	----------------------------------------
3.35/1.63	
3.35/1.63	(3) QTRSRRRProof (EQUIVALENT)
3.35/1.63	Used ordering:
3.35/1.63	Quasi precedence:
3.35/1.63	f_1 > U1_2
3.35/1.63	[a, d] > c > U1_2
3.35/1.63	
3.35/1.63	
3.35/1.63	Status:
3.35/1.63	f_1: multiset status
3.35/1.63	U1_2: multiset status
3.35/1.63	c: multiset status
3.35/1.63	a: multiset status
3.35/1.63	d: multiset status
3.35/1.63	
3.35/1.63	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.35/1.63	
3.35/1.63	   f(x) -> U1(x, x)
3.35/1.63	   U1(c, x) -> x
3.35/1.63	   a -> c
3.35/1.63	
3.35/1.63	
3.35/1.63	
3.35/1.63	
3.35/1.63	----------------------------------------
3.35/1.63	
3.35/1.63	(4)
3.35/1.63	Obligation:
3.35/1.63	Q restricted rewrite system:
3.35/1.63	The TRS R consists of the following rules:
3.35/1.63	
3.35/1.63	   a -> d
3.35/1.63	
3.35/1.63	Q is empty.
3.35/1.63	
3.35/1.63	----------------------------------------
3.35/1.63	
3.35/1.63	(5) QTRSRRRProof (EQUIVALENT)
3.35/1.63	Used ordering:
3.35/1.63	Polynomial interpretation [POLO]:
3.35/1.63	
3.35/1.63	   POL(a) = 1
3.35/1.63	   POL(d) = 0
3.35/1.63	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.35/1.63	
3.35/1.63	   a -> d
3.35/1.63	
3.35/1.63	
3.35/1.63	
3.35/1.63	
3.35/1.63	----------------------------------------
3.35/1.63	
3.35/1.63	(6)
3.35/1.63	Obligation:
3.35/1.63	Q restricted rewrite system:
3.35/1.63	R is empty.
3.35/1.63	Q is empty.
3.35/1.63	
3.35/1.63	----------------------------------------
3.35/1.63	
3.35/1.63	(7) RisEmptyProof (EQUIVALENT)
3.35/1.63	The TRS R is empty. Hence, termination is trivially proven.
3.35/1.63	----------------------------------------
3.35/1.63	
3.35/1.63	(8)
3.35/1.63	YES
3.53/1.67	EOF