3.68/1.68	YES
3.68/1.69	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.68/1.69	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.68/1.69	
3.68/1.69	
3.68/1.69	Quasi decreasingness of the given CTRS could be proven:
3.68/1.69	
3.68/1.69	(0) CTRS
3.68/1.69	(1) CTRSToQTRSProof [SOUND, 0 ms]
3.68/1.69	(2) QTRS
3.68/1.69	(3) QTRSRRRProof [EQUIVALENT, 24 ms]
3.68/1.69	(4) QTRS
3.68/1.69	(5) RisEmptyProof [EQUIVALENT, 0 ms]
3.68/1.69	(6) YES
3.68/1.69	
3.68/1.69	
3.68/1.69	----------------------------------------
3.68/1.69	
3.68/1.69	(0)
3.68/1.69	Obligation:
3.68/1.69	Conditional term rewrite system:
3.68/1.69	The TRS R consists of the following rules:
3.68/1.69	
3.68/1.69	   a -> c
3.68/1.69	   a -> d
3.68/1.69	   b -> c
3.68/1.69	   b -> d
3.68/1.69	   s(k) -> t(a)
3.68/1.69	   s(l) -> t(a)
3.68/1.69	   g(x, x) -> h(x, x)
3.68/1.69	
3.68/1.69	The conditional TRS C consists of the following conditional rules:
3.68/1.69	
3.68/1.69	   f(x, y) -> y <= s(x) -> t(y)
3.68/1.69	
3.68/1.69	
3.68/1.69	----------------------------------------
3.68/1.69	
3.68/1.69	(1) CTRSToQTRSProof (SOUND)
3.68/1.69	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
3.68/1.69	----------------------------------------
3.68/1.69	
3.68/1.69	(2)
3.68/1.69	Obligation:
3.68/1.69	Q restricted rewrite system:
3.68/1.69	The TRS R consists of the following rules:
3.68/1.69	
3.68/1.69	   f(x, y) -> U1(s(x), y)
3.68/1.69	   U1(t(y), y) -> y
3.68/1.69	   a -> c
3.68/1.69	   a -> d
3.68/1.69	   b -> c
3.68/1.69	   b -> d
3.68/1.69	   s(k) -> t(a)
3.68/1.69	   s(l) -> t(a)
3.68/1.69	   g(x, x) -> h(x, x)
3.68/1.69	
3.68/1.69	Q is empty.
3.68/1.69	
3.68/1.69	----------------------------------------
3.68/1.69	
3.68/1.69	(3) QTRSRRRProof (EQUIVALENT)
3.68/1.69	Used ordering:
3.68/1.69	Knuth-Bendix order [KBO] with precedence:g_2 > f_2 > U1_2 > h_2 > l > k > b > a > d > c > s_1 > t_1
3.68/1.69	
3.68/1.69	and weight map:
3.68/1.69	
3.68/1.69	   a=2
3.68/1.69	   c=2
3.68/1.69	   d=2
3.68/1.69	   b=2
3.68/1.69	   k=1
3.68/1.69	   l=1
3.68/1.69	   s_1=2
3.68/1.69	   t_1=1
3.68/1.69	   f_2=2
3.68/1.69	   U1_2=0
3.68/1.69	   g_2=0
3.68/1.69	   h_2=0
3.68/1.69	
3.68/1.69	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.68/1.69	
3.68/1.69	   f(x, y) -> U1(s(x), y)
3.68/1.69	   U1(t(y), y) -> y
3.68/1.69	   a -> c
3.68/1.69	   a -> d
3.68/1.69	   b -> c
3.68/1.69	   b -> d
3.68/1.69	   s(k) -> t(a)
3.68/1.69	   s(l) -> t(a)
3.68/1.69	   g(x, x) -> h(x, x)
3.68/1.69	
3.68/1.69	
3.68/1.69	
3.68/1.69	
3.68/1.69	----------------------------------------
3.68/1.69	
3.68/1.69	(4)
3.68/1.69	Obligation:
3.68/1.69	Q restricted rewrite system:
3.68/1.69	R is empty.
3.68/1.69	Q is empty.
3.68/1.69	
3.68/1.69	----------------------------------------
3.68/1.69	
3.68/1.69	(5) RisEmptyProof (EQUIVALENT)
3.68/1.69	The TRS R is empty. Hence, termination is trivially proven.
3.68/1.69	----------------------------------------
3.68/1.69	
3.68/1.69	(6)
3.68/1.69	YES
3.84/1.72	EOF