3.08/1.61	YES
3.08/1.62	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.08/1.62	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.08/1.62	
3.08/1.62	
3.08/1.62	Quasi decreasingness of the given CTRS could be proven:
3.08/1.62	
3.08/1.62	(0) CTRS
3.08/1.62	(1) CTRSToQTRSProof [SOUND, 0 ms]
3.08/1.62	(2) QTRS
3.08/1.62	(3) QTRSRRRProof [EQUIVALENT, 0 ms]
3.08/1.62	(4) QTRS
3.08/1.62	(5) RisEmptyProof [EQUIVALENT, 0 ms]
3.08/1.62	(6) YES
3.08/1.62	
3.08/1.62	
3.08/1.62	----------------------------------------
3.08/1.62	
3.08/1.62	(0)
3.08/1.62	Obligation:
3.08/1.62	Conditional term rewrite system:
3.08/1.62	The TRS R consists of the following rules:
3.08/1.62	
3.08/1.62	   e(0) -> true
3.08/1.62	
3.08/1.62	The conditional TRS C consists of the following conditional rules:
3.08/1.62	
3.08/1.62	   e(s(x)) -> true <= e(x) -> false
3.08/1.62	   e(s(x)) -> false <= e(x) -> true
3.08/1.62	
3.08/1.62	
3.08/1.62	----------------------------------------
3.08/1.62	
3.08/1.62	(1) CTRSToQTRSProof (SOUND)
3.08/1.62	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
3.08/1.62	----------------------------------------
3.08/1.62	
3.08/1.62	(2)
3.08/1.62	Obligation:
3.08/1.62	Q restricted rewrite system:
3.08/1.62	The TRS R consists of the following rules:
3.08/1.62	
3.08/1.62	   e(s(x)) -> U1(e(x))
3.08/1.62	   U1(false) -> true
3.08/1.62	   U1(true) -> false
3.08/1.62	   e(0) -> true
3.08/1.62	
3.08/1.62	Q is empty.
3.08/1.62	
3.08/1.62	----------------------------------------
3.08/1.62	
3.08/1.62	(3) QTRSRRRProof (EQUIVALENT)
3.08/1.62	Used ordering:
3.08/1.62	Knuth-Bendix order [KBO] with precedence:U1_1 > e_1 > true > s_1 > 0 > false
3.08/1.62	
3.08/1.62	and weight map:
3.08/1.62	
3.08/1.62	   false=3
3.08/1.62	   true=3
3.08/1.62	   0=1
3.08/1.62	   e_1=2
3.08/1.62	   s_1=1
3.08/1.62	   U1_1=0
3.08/1.62	
3.08/1.62	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.08/1.62	
3.08/1.62	   e(s(x)) -> U1(e(x))
3.08/1.62	   U1(false) -> true
3.08/1.62	   U1(true) -> false
3.08/1.62	   e(0) -> true
3.08/1.62	
3.08/1.62	
3.08/1.62	
3.08/1.62	
3.08/1.62	----------------------------------------
3.08/1.62	
3.08/1.62	(4)
3.08/1.62	Obligation:
3.08/1.62	Q restricted rewrite system:
3.08/1.62	R is empty.
3.08/1.62	Q is empty.
3.08/1.62	
3.08/1.62	----------------------------------------
3.08/1.62	
3.08/1.62	(5) RisEmptyProof (EQUIVALENT)
3.08/1.62	The TRS R is empty. Hence, termination is trivially proven.
3.08/1.62	----------------------------------------
3.08/1.62	
3.08/1.62	(6)
3.08/1.62	YES
3.37/1.64	EOF