3.42/1.58	YES
3.42/1.59	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.42/1.59	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.42/1.59	
3.42/1.59	
3.42/1.59	Quasi decreasingness of the given CTRS could be proven:
3.42/1.59	
3.42/1.59	(0) CTRS
3.42/1.59	(1) CTRSToQTRSProof [SOUND, 0 ms]
3.42/1.59	(2) QTRS
3.42/1.59	(3) QTRSRRRProof [EQUIVALENT, 0 ms]
3.42/1.59	(4) QTRS
3.42/1.59	(5) RisEmptyProof [EQUIVALENT, 0 ms]
3.42/1.59	(6) YES
3.42/1.59	
3.42/1.59	
3.42/1.59	----------------------------------------
3.42/1.59	
3.42/1.59	(0)
3.42/1.59	Obligation:
3.42/1.59	Conditional term rewrite system:
3.42/1.59	The TRS R consists of the following rules:
3.42/1.59	
3.42/1.59	   minus(x, 0) -> x
3.42/1.59	
3.42/1.59	The conditional TRS C consists of the following conditional rules:
3.42/1.59	
3.42/1.59	   minus(x, s(y)) -> z <= minus(x, y) -> s(z)
3.42/1.59	
3.42/1.59	
3.42/1.59	----------------------------------------
3.42/1.59	
3.42/1.59	(1) CTRSToQTRSProof (SOUND)
3.42/1.59	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
3.42/1.59	----------------------------------------
3.42/1.59	
3.42/1.59	(2)
3.42/1.59	Obligation:
3.42/1.59	Q restricted rewrite system:
3.42/1.59	The TRS R consists of the following rules:
3.42/1.59	
3.42/1.59	   minus(x, s(y)) -> U1(minus(x, y))
3.42/1.59	   U1(s(z)) -> z
3.42/1.59	   minus(x, 0) -> x
3.42/1.59	
3.42/1.59	Q is empty.
3.42/1.59	
3.42/1.59	----------------------------------------
3.42/1.59	
3.42/1.59	(3) QTRSRRRProof (EQUIVALENT)
3.42/1.59	Used ordering:
3.42/1.59	Knuth-Bendix order [KBO] with precedence:minus_2 > U1_1 > s_1 > 0
3.42/1.59	
3.42/1.59	and weight map:
3.42/1.59	
3.42/1.59	   0=1
3.42/1.59	   s_1=2
3.42/1.59	   U1_1=1
3.42/1.59	   minus_2=0
3.42/1.59	
3.42/1.59	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.42/1.59	
3.42/1.59	   minus(x, s(y)) -> U1(minus(x, y))
3.42/1.59	   U1(s(z)) -> z
3.42/1.59	   minus(x, 0) -> x
3.42/1.59	
3.42/1.59	
3.42/1.59	
3.42/1.59	
3.42/1.59	----------------------------------------
3.42/1.59	
3.42/1.59	(4)
3.42/1.59	Obligation:
3.42/1.59	Q restricted rewrite system:
3.42/1.59	R is empty.
3.42/1.59	Q is empty.
3.42/1.59	
3.42/1.59	----------------------------------------
3.42/1.59	
3.42/1.59	(5) RisEmptyProof (EQUIVALENT)
3.42/1.59	The TRS R is empty. Hence, termination is trivially proven.
3.42/1.59	----------------------------------------
3.42/1.59	
3.42/1.59	(6)
3.42/1.59	YES
3.42/1.61	EOF