3.55/1.69	YES
3.55/1.70	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.55/1.70	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.55/1.70	
3.55/1.70	
3.55/1.70	Quasi decreasingness of the given CTRS could be proven:
3.55/1.70	
3.55/1.70	(0) CTRS
3.55/1.70	(1) CTRSToQTRSProof [SOUND, 0 ms]
3.55/1.70	(2) QTRS
3.55/1.70	(3) QTRSRRRProof [EQUIVALENT, 46 ms]
3.55/1.70	(4) QTRS
3.55/1.70	(5) RisEmptyProof [EQUIVALENT, 0 ms]
3.55/1.70	(6) YES
3.55/1.70	
3.55/1.70	
3.55/1.70	----------------------------------------
3.55/1.70	
3.55/1.70	(0)
3.55/1.70	Obligation:
3.55/1.70	Conditional term rewrite system:
3.55/1.70	The TRS R consists of the following rules:
3.55/1.70	
3.55/1.70	   a -> b
3.55/1.70	   g(a) -> h(b)
3.55/1.70	
3.55/1.70	The conditional TRS C consists of the following conditional rules:
3.55/1.70	
3.55/1.70	   f(x) -> c(x, y) <= g(x) -> y, y -> h(x)
3.55/1.70	
3.55/1.70	
3.55/1.70	----------------------------------------
3.55/1.70	
3.55/1.70	(1) CTRSToQTRSProof (SOUND)
3.55/1.70	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
3.55/1.70	----------------------------------------
3.55/1.70	
3.55/1.70	(2)
3.55/1.70	Obligation:
3.55/1.70	Q restricted rewrite system:
3.55/1.70	The TRS R consists of the following rules:
3.55/1.70	
3.55/1.70	   f(x) -> U1(g(x), x)
3.55/1.70	   U1(y, x) -> U2(y, x, y)
3.55/1.70	   U2(h(x), x, y) -> c(x, y)
3.55/1.70	   a -> b
3.55/1.70	   g(a) -> h(b)
3.55/1.70	
3.55/1.70	Q is empty.
3.55/1.70	
3.55/1.70	----------------------------------------
3.55/1.70	
3.55/1.70	(3) QTRSRRRProof (EQUIVALENT)
3.55/1.70	Used ordering:
3.55/1.70	f/1(YES)
3.55/1.70	U1/2(YES,YES)
3.55/1.70	g/1)YES(
3.55/1.70	U2/3(YES,YES,YES)
3.55/1.70	h/1(YES)
3.55/1.70	c/2(YES,YES)
3.55/1.70	a/0)
3.55/1.70	b/0)
3.55/1.70	
3.55/1.70	Quasi precedence:
3.55/1.70	f_1 > U1_2 > U2_3 > c_2
3.55/1.70	a > h_1 > c_2
3.55/1.70	a > b > c_2
3.55/1.70	
3.55/1.70	
3.55/1.70	Status:
3.55/1.70	f_1: [1]
3.55/1.70	U1_2: [2,1]
3.55/1.70	U2_3: [2,1,3]
3.55/1.70	h_1: multiset status
3.55/1.70	c_2: [2,1]
3.55/1.70	a: multiset status
3.55/1.70	b: multiset status
3.55/1.70	
3.55/1.70	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.55/1.70	
3.55/1.70	   f(x) -> U1(g(x), x)
3.55/1.70	   U1(y, x) -> U2(y, x, y)
3.55/1.70	   U2(h(x), x, y) -> c(x, y)
3.55/1.70	   a -> b
3.55/1.70	   g(a) -> h(b)
3.55/1.70	
3.55/1.70	
3.55/1.70	
3.55/1.70	
3.55/1.70	----------------------------------------
3.55/1.70	
3.55/1.70	(4)
3.55/1.70	Obligation:
3.55/1.70	Q restricted rewrite system:
3.55/1.70	R is empty.
3.55/1.70	Q is empty.
3.55/1.70	
3.55/1.70	----------------------------------------
3.55/1.70	
3.55/1.70	(5) RisEmptyProof (EQUIVALENT)
3.55/1.70	The TRS R is empty. Hence, termination is trivially proven.
3.55/1.70	----------------------------------------
3.55/1.70	
3.55/1.70	(6)
3.55/1.70	YES
3.55/1.72	EOF