3.43/1.77	YES
3.43/1.77	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.43/1.77	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.43/1.77	
3.43/1.77	
3.43/1.77	Quasi decreasingness of the given CTRS could be proven:
3.43/1.77	
3.43/1.77	(0) CTRS
3.43/1.77	(1) CTRSToQTRSProof [SOUND, 0 ms]
3.43/1.77	(2) QTRS
3.43/1.77	(3) QTRSRRRProof [EQUIVALENT, 141 ms]
3.43/1.77	(4) QTRS
3.43/1.77	(5) RisEmptyProof [EQUIVALENT, 0 ms]
3.43/1.77	(6) YES
3.43/1.77	
3.43/1.77	
3.43/1.77	----------------------------------------
3.43/1.77	
3.43/1.77	(0)
3.43/1.77	Obligation:
3.43/1.77	Conditional term rewrite system:
3.43/1.77	The TRS R consists of the following rules:
3.43/1.77	
3.43/1.77	   split(x, nil) -> tp2(nil, nil)
3.43/1.77	   le(0, y) -> true
3.43/1.77	   le(s(x), 0) -> false
3.43/1.77	   le(s(x), s(y)) -> le(x, y)
3.43/1.77	
3.43/1.77	The conditional TRS C consists of the following conditional rules:
3.43/1.77	
3.43/1.77	   split(x, cons(y, ys)) -> tp2(zs1, cons(y, zs2)) <= split(x, ys) -> tp2(zs1, zs2), le(x, y) -> true
3.43/1.77	   split(x, cons(y, ys)) -> tp2(cons(y, zs1), zs2) <= split(x, ys) -> tp2(zs1, zs2), le(x, y) -> false
3.43/1.77	
3.43/1.77	
3.43/1.77	----------------------------------------
3.43/1.77	
3.43/1.77	(1) CTRSToQTRSProof (SOUND)
3.43/1.77	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
3.43/1.77	----------------------------------------
3.43/1.77	
3.43/1.77	(2)
3.43/1.77	Obligation:
3.43/1.77	Q restricted rewrite system:
3.43/1.77	The TRS R consists of the following rules:
3.43/1.77	
3.43/1.77	   split(x, cons(y, ys)) -> U1(split(x, ys), x, y)
3.43/1.77	   U1(tp2(zs1, zs2), x, y) -> U2(le(x, y), y, zs1, zs2)
3.43/1.77	   U2(false, y, zs1, zs2) -> tp2(cons(y, zs1), zs2)
3.43/1.77	   U2(true, y, zs1, zs2) -> tp2(zs1, cons(y, zs2))
3.43/1.77	   split(x, nil) -> tp2(nil, nil)
3.43/1.77	   le(0, y) -> true
3.43/1.77	   le(s(x), 0) -> false
3.43/1.77	   le(s(x), s(y)) -> le(x, y)
3.43/1.77	
3.43/1.77	Q is empty.
3.43/1.77	
3.43/1.77	----------------------------------------
3.43/1.77	
3.43/1.77	(3) QTRSRRRProof (EQUIVALENT)
3.43/1.77	Used ordering:
3.43/1.77	Quasi precedence:
3.43/1.77	split_2 > U1_3 > U2_4 > [cons_2, tp2_2, le_2]
3.43/1.77	split_2 > nil > [cons_2, tp2_2, le_2]
3.43/1.77	0 > [false, s_1] > [cons_2, tp2_2, le_2]
3.43/1.77	0 > true > [cons_2, tp2_2, le_2]
3.43/1.77	
3.43/1.77	
3.43/1.77	Status:
3.43/1.77	split_2: multiset status
3.43/1.77	cons_2: [1,2]
3.43/1.77	U1_3: multiset status
3.43/1.77	tp2_2: multiset status
3.43/1.77	U2_4: multiset status
3.43/1.77	le_2: multiset status
3.43/1.77	false: multiset status
3.43/1.77	true: multiset status
3.43/1.77	nil: multiset status
3.43/1.77	0: multiset status
3.43/1.77	s_1: [1]
3.43/1.77	
3.43/1.77	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.43/1.77	
3.43/1.77	   split(x, cons(y, ys)) -> U1(split(x, ys), x, y)
3.43/1.77	   U1(tp2(zs1, zs2), x, y) -> U2(le(x, y), y, zs1, zs2)
3.43/1.77	   U2(false, y, zs1, zs2) -> tp2(cons(y, zs1), zs2)
3.43/1.77	   U2(true, y, zs1, zs2) -> tp2(zs1, cons(y, zs2))
3.43/1.77	   split(x, nil) -> tp2(nil, nil)
3.43/1.77	   le(0, y) -> true
3.43/1.77	   le(s(x), 0) -> false
3.43/1.77	   le(s(x), s(y)) -> le(x, y)
3.43/1.77	
3.43/1.77	
3.43/1.77	
3.43/1.77	
3.43/1.77	----------------------------------------
3.43/1.77	
3.43/1.77	(4)
3.43/1.77	Obligation:
3.43/1.77	Q restricted rewrite system:
3.43/1.77	R is empty.
3.43/1.77	Q is empty.
3.43/1.77	
3.43/1.77	----------------------------------------
3.43/1.77	
3.43/1.77	(5) RisEmptyProof (EQUIVALENT)
3.43/1.77	The TRS R is empty. Hence, termination is trivially proven.
3.43/1.77	----------------------------------------
3.43/1.77	
3.43/1.77	(6)
3.43/1.77	YES
3.55/1.80	EOF