3.44/1.59	YES
3.44/1.60	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.44/1.60	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.44/1.60	
3.44/1.60	
3.44/1.60	Quasi decreasingness of the given CTRS could be proven:
3.44/1.60	
3.44/1.60	(0) CTRS
3.44/1.60	(1) CTRSToQTRSProof [SOUND, 0 ms]
3.44/1.60	(2) QTRS
3.44/1.60	(3) QTRSRRRProof [EQUIVALENT, 23 ms]
3.44/1.60	(4) QTRS
3.44/1.60	(5) RisEmptyProof [EQUIVALENT, 0 ms]
3.44/1.60	(6) YES
3.44/1.60	
3.44/1.60	
3.44/1.60	----------------------------------------
3.44/1.60	
3.44/1.60	(0)
3.44/1.60	Obligation:
3.44/1.60	Conditional term rewrite system:
3.44/1.60	The TRS R consists of the following rules:
3.44/1.60	
3.44/1.60	   isnoc(cons(y, nil)) -> tp2(nil, y)
3.44/1.60	
3.44/1.60	The conditional TRS C consists of the following conditional rules:
3.44/1.60	
3.44/1.60	   isnoc(cons(x, ys)) -> tp2(cons(x, xs), y) <= isnoc(ys) -> tp2(xs, y)
3.44/1.60	
3.44/1.60	
3.44/1.60	----------------------------------------
3.44/1.60	
3.44/1.60	(1) CTRSToQTRSProof (SOUND)
3.44/1.60	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
3.44/1.60	----------------------------------------
3.44/1.60	
3.44/1.60	(2)
3.44/1.60	Obligation:
3.44/1.60	Q restricted rewrite system:
3.44/1.60	The TRS R consists of the following rules:
3.44/1.60	
3.44/1.60	   isnoc(cons(x, ys)) -> U1(isnoc(ys), x)
3.44/1.60	   U1(tp2(xs, y), x) -> tp2(cons(x, xs), y)
3.44/1.60	   isnoc(cons(y, nil)) -> tp2(nil, y)
3.44/1.60	
3.44/1.60	Q is empty.
3.44/1.60	
3.44/1.60	----------------------------------------
3.44/1.60	
3.44/1.60	(3) QTRSRRRProof (EQUIVALENT)
3.44/1.60	Used ordering:
3.44/1.60	Knuth-Bendix order [KBO] with precedence:nil > isnoc_1 > U1_2 > tp2_2 > cons_2
3.44/1.60	
3.44/1.60	and weight map:
3.44/1.60	
3.44/1.60	   nil=1
3.44/1.60	   isnoc_1=2
3.44/1.60	   cons_2=0
3.44/1.60	   U1_2=0
3.44/1.60	   tp2_2=2
3.44/1.60	
3.44/1.60	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.44/1.60	
3.44/1.60	   isnoc(cons(x, ys)) -> U1(isnoc(ys), x)
3.44/1.60	   U1(tp2(xs, y), x) -> tp2(cons(x, xs), y)
3.44/1.60	   isnoc(cons(y, nil)) -> tp2(nil, y)
3.44/1.60	
3.44/1.60	
3.44/1.60	
3.44/1.60	
3.44/1.60	----------------------------------------
3.44/1.60	
3.44/1.60	(4)
3.44/1.60	Obligation:
3.44/1.60	Q restricted rewrite system:
3.44/1.60	R is empty.
3.44/1.60	Q is empty.
3.44/1.60	
3.44/1.60	----------------------------------------
3.44/1.60	
3.44/1.60	(5) RisEmptyProof (EQUIVALENT)
3.44/1.60	The TRS R is empty. Hence, termination is trivially proven.
3.44/1.60	----------------------------------------
3.44/1.60	
3.44/1.60	(6)
3.44/1.60	YES
3.61/1.63	EOF