3.21/1.63	YES
3.21/1.63	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.21/1.63	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.21/1.63	
3.21/1.63	
3.21/1.63	Quasi decreasingness of the given CTRS could be proven:
3.21/1.63	
3.21/1.63	(0) CTRS
3.21/1.63	(1) CTRSToQTRSProof [SOUND, 0 ms]
3.21/1.63	(2) QTRS
3.21/1.63	(3) QTRSRRRProof [EQUIVALENT, 33 ms]
3.21/1.63	(4) QTRS
3.21/1.63	(5) RisEmptyProof [EQUIVALENT, 0 ms]
3.21/1.63	(6) YES
3.21/1.63	
3.21/1.63	
3.21/1.63	----------------------------------------
3.21/1.63	
3.21/1.63	(0)
3.21/1.63	Obligation:
3.21/1.63	Conditional term rewrite system:
3.21/1.63	The TRS R consists of the following rules:
3.21/1.63	
3.21/1.63	   a -> c
3.21/1.63	   b -> c
3.21/1.63	
3.21/1.63	The conditional TRS C consists of the following conditional rules:
3.21/1.63	
3.21/1.63	   f(x) -> x <= a -> x
3.21/1.63	
3.21/1.63	
3.21/1.63	----------------------------------------
3.21/1.63	
3.21/1.63	(1) CTRSToQTRSProof (SOUND)
3.21/1.63	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
3.21/1.63	----------------------------------------
3.21/1.63	
3.21/1.63	(2)
3.21/1.63	Obligation:
3.21/1.63	Q restricted rewrite system:
3.21/1.63	The TRS R consists of the following rules:
3.21/1.63	
3.21/1.63	   f(x) -> U1(a, x)
3.21/1.63	   U1(x, x) -> x
3.21/1.63	   a -> c
3.21/1.63	   b -> c
3.21/1.63	
3.21/1.63	Q is empty.
3.21/1.63	
3.21/1.63	----------------------------------------
3.21/1.63	
3.21/1.63	(3) QTRSRRRProof (EQUIVALENT)
3.21/1.63	Used ordering:
3.21/1.63	Knuth-Bendix order [KBO] with precedence:b > a > f_1 > c > U1_2
3.21/1.63	
3.21/1.63	and weight map:
3.21/1.63	
3.21/1.63	   a=2
3.21/1.63	   c=2
3.21/1.63	   b=2
3.21/1.63	   f_1=2
3.21/1.63	   U1_2=0
3.21/1.63	
3.21/1.63	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.21/1.63	
3.21/1.63	   f(x) -> U1(a, x)
3.21/1.63	   U1(x, x) -> x
3.21/1.63	   a -> c
3.21/1.63	   b -> c
3.21/1.63	
3.21/1.63	
3.21/1.63	
3.21/1.63	
3.21/1.63	----------------------------------------
3.21/1.63	
3.21/1.63	(4)
3.21/1.63	Obligation:
3.21/1.63	Q restricted rewrite system:
3.21/1.63	R is empty.
3.21/1.63	Q is empty.
3.21/1.63	
3.21/1.63	----------------------------------------
3.21/1.63	
3.21/1.63	(5) RisEmptyProof (EQUIVALENT)
3.21/1.63	The TRS R is empty. Hence, termination is trivially proven.
3.21/1.63	----------------------------------------
3.21/1.63	
3.21/1.63	(6)
3.21/1.63	YES
3.21/1.64	EOF