2.88/1.67	YES
3.26/1.68	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.26/1.68	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.26/1.68	
3.26/1.68	
3.26/1.68	Quasi decreasingness of the given CTRS could be proven:
3.26/1.68	
3.26/1.68	(0) CTRS
3.26/1.68	(1) CTRSToQTRSProof [SOUND, 0 ms]
3.26/1.68	(2) QTRS
3.26/1.68	(3) QTRSRRRProof [EQUIVALENT, 33 ms]
3.26/1.68	(4) QTRS
3.26/1.68	(5) RisEmptyProof [EQUIVALENT, 0 ms]
3.26/1.68	(6) YES
3.26/1.68	
3.26/1.68	
3.26/1.68	----------------------------------------
3.26/1.68	
3.26/1.68	(0)
3.26/1.68	Obligation:
3.26/1.68	Conditional term rewrite system:
3.26/1.68	The TRS R is empty.
3.26/1.68	The conditional TRS C consists of the following conditional rules:
3.26/1.68	
3.26/1.68	   f(x) -> a <= x -> b
3.26/1.68	
3.26/1.68	
3.26/1.68	----------------------------------------
3.26/1.68	
3.26/1.68	(1) CTRSToQTRSProof (SOUND)
3.26/1.68	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
3.26/1.68	----------------------------------------
3.26/1.68	
3.26/1.68	(2)
3.26/1.68	Obligation:
3.26/1.68	Q restricted rewrite system:
3.26/1.68	The TRS R consists of the following rules:
3.26/1.68	
3.26/1.68	   f(x) -> U1(x)
3.26/1.68	   U1(b) -> a
3.26/1.68	
3.26/1.68	Q is empty.
3.26/1.68	
3.26/1.68	----------------------------------------
3.26/1.68	
3.26/1.68	(3) QTRSRRRProof (EQUIVALENT)
3.26/1.68	Used ordering:
3.26/1.68	Knuth-Bendix order [KBO] with precedence:f_1 > U1_1 > a > b
3.26/1.68	
3.26/1.68	and weight map:
3.26/1.68	
3.26/1.68	   b=1
3.26/1.68	   a=2
3.26/1.68	   f_1=1
3.26/1.68	   U1_1=1
3.26/1.68	
3.26/1.68	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.26/1.68	
3.26/1.68	   f(x) -> U1(x)
3.26/1.68	   U1(b) -> a
3.26/1.68	
3.26/1.68	
3.26/1.68	
3.26/1.68	
3.26/1.68	----------------------------------------
3.26/1.68	
3.26/1.68	(4)
3.26/1.68	Obligation:
3.26/1.68	Q restricted rewrite system:
3.26/1.68	R is empty.
3.26/1.68	Q is empty.
3.26/1.68	
3.26/1.68	----------------------------------------
3.26/1.68	
3.26/1.68	(5) RisEmptyProof (EQUIVALENT)
3.26/1.68	The TRS R is empty. Hence, termination is trivially proven.
3.26/1.68	----------------------------------------
3.26/1.68	
3.26/1.68	(6)
3.26/1.68	YES
3.26/1.70	EOF