3.66/1.88	YES
3.66/1.89	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.66/1.89	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.66/1.89	
3.66/1.89	
3.66/1.89	Quasi decreasingness of the given CTRS could be proven:
3.66/1.89	
3.66/1.89	(0) CTRS
3.66/1.89	(1) CTRSToQTRSProof [SOUND, 0 ms]
3.66/1.89	(2) QTRS
3.66/1.89	(3) QTRSRRRProof [EQUIVALENT, 82 ms]
3.66/1.89	(4) QTRS
3.66/1.89	(5) RisEmptyProof [EQUIVALENT, 0 ms]
3.66/1.89	(6) YES
3.66/1.89	
3.66/1.89	
3.66/1.89	----------------------------------------
3.66/1.89	
3.66/1.89	(0)
3.66/1.89	Obligation:
3.66/1.89	Conditional term rewrite system:
3.66/1.89	The TRS R consists of the following rules:
3.66/1.89	
3.66/1.89	   h(a, a) -> i(b)
3.66/1.89	   h(a, b) -> i(c)
3.66/1.89	   h(b, b) -> i(d)
3.66/1.89	
3.66/1.89	The conditional TRS C consists of the following conditional rules:
3.66/1.89	
3.66/1.89	   f(x) -> g(x, y, z) <= h(a, x) -> i(y), h(a, y) -> i(z)
3.66/1.89	
3.66/1.89	
3.66/1.89	----------------------------------------
3.66/1.89	
3.66/1.89	(1) CTRSToQTRSProof (SOUND)
3.66/1.89	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
3.66/1.89	----------------------------------------
3.66/1.89	
3.66/1.89	(2)
3.66/1.89	Obligation:
3.66/1.89	Q restricted rewrite system:
3.66/1.89	The TRS R consists of the following rules:
3.66/1.89	
3.66/1.89	   f(x) -> U1(h(a, x), x)
3.66/1.89	   U1(i(y), x) -> U2(h(a, y), x, y)
3.66/1.89	   U2(i(z), x, y) -> g(x, y, z)
3.66/1.89	   h(a, a) -> i(b)
3.66/1.89	   h(a, b) -> i(c)
3.66/1.89	   h(b, b) -> i(d)
3.66/1.89	
3.66/1.89	Q is empty.
3.66/1.89	
3.66/1.89	----------------------------------------
3.66/1.89	
3.66/1.89	(3) QTRSRRRProof (EQUIVALENT)
3.66/1.89	Used ordering:
3.66/1.89	f/1(YES)
3.66/1.89	U1/2(YES,YES)
3.66/1.89	h/2(YES,YES)
3.66/1.89	a/0)
3.66/1.89	i/1)YES(
3.66/1.89	U2/3(YES,YES,YES)
3.66/1.89	g/3(YES,YES,YES)
3.66/1.89	b/0)
3.66/1.89	c/0)
3.66/1.89	d/0)
3.66/1.89	
3.66/1.89	Quasi precedence:
3.66/1.89	f_1 > U1_2 > [h_2, c, d] > b
3.66/1.89	f_1 > U1_2 > a
3.66/1.89	f_1 > U1_2 > U2_3 > g_3
3.66/1.89	
3.66/1.89	
3.66/1.89	Status:
3.66/1.89	f_1: multiset status
3.66/1.89	U1_2: multiset status
3.66/1.89	h_2: multiset status
3.66/1.89	a: multiset status
3.66/1.89	U2_3: multiset status
3.66/1.89	g_3: multiset status
3.66/1.89	b: multiset status
3.66/1.89	c: multiset status
3.66/1.89	d: multiset status
3.66/1.89	
3.66/1.89	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.66/1.89	
3.66/1.89	   f(x) -> U1(h(a, x), x)
3.66/1.89	   U1(i(y), x) -> U2(h(a, y), x, y)
3.66/1.89	   U2(i(z), x, y) -> g(x, y, z)
3.66/1.89	   h(a, a) -> i(b)
3.66/1.89	   h(a, b) -> i(c)
3.66/1.89	   h(b, b) -> i(d)
3.66/1.89	
3.66/1.89	
3.66/1.89	
3.66/1.89	
3.66/1.89	----------------------------------------
3.66/1.89	
3.66/1.89	(4)
3.66/1.89	Obligation:
3.66/1.89	Q restricted rewrite system:
3.66/1.89	R is empty.
3.66/1.89	Q is empty.
3.66/1.89	
3.66/1.89	----------------------------------------
3.66/1.89	
3.66/1.89	(5) RisEmptyProof (EQUIVALENT)
3.66/1.89	The TRS R is empty. Hence, termination is trivially proven.
3.66/1.89	----------------------------------------
3.66/1.89	
3.66/1.89	(6)
3.66/1.89	YES
3.74/1.92	EOF