3.04/1.78	YES
3.38/1.79	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.38/1.79	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.38/1.79	
3.38/1.79	
3.38/1.79	Quasi decreasingness of the given CTRS could be proven:
3.38/1.79	
3.38/1.79	(0) CTRS
3.38/1.79	(1) CTRSToQTRSProof [SOUND, 0 ms]
3.38/1.79	(2) QTRS
3.38/1.79	(3) QTRSRRRProof [EQUIVALENT, 29 ms]
3.38/1.79	(4) QTRS
3.38/1.79	(5) RisEmptyProof [EQUIVALENT, 0 ms]
3.38/1.79	(6) YES
3.38/1.79	
3.38/1.79	
3.38/1.79	----------------------------------------
3.38/1.79	
3.38/1.79	(0)
3.38/1.79	Obligation:
3.38/1.79	Conditional term rewrite system:
3.38/1.79	The TRS R consists of the following rules:
3.38/1.79	
3.38/1.79	   s(a) -> c
3.38/1.79	   s(b) -> c
3.38/1.79	   c -> t(k)
3.38/1.79	   c -> t(l)
3.38/1.79	   g(x, x) -> h(x, x)
3.38/1.79	
3.38/1.79	The conditional TRS C consists of the following conditional rules:
3.38/1.79	
3.38/1.79	   f(x) -> z <= s(x) -> z
3.38/1.79	
3.38/1.79	
3.38/1.79	----------------------------------------
3.38/1.79	
3.38/1.79	(1) CTRSToQTRSProof (SOUND)
3.38/1.79	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
3.38/1.79	----------------------------------------
3.38/1.79	
3.38/1.79	(2)
3.38/1.79	Obligation:
3.38/1.79	Q restricted rewrite system:
3.38/1.79	The TRS R consists of the following rules:
3.38/1.79	
3.38/1.79	   f(x) -> U1(s(x))
3.38/1.79	   U1(z) -> z
3.38/1.79	   s(a) -> c
3.38/1.79	   s(b) -> c
3.38/1.79	   c -> t(k)
3.38/1.79	   c -> t(l)
3.38/1.79	   g(x, x) -> h(x, x)
3.38/1.79	
3.38/1.79	Q is empty.
3.38/1.79	
3.38/1.79	----------------------------------------
3.38/1.79	
3.38/1.79	(3) QTRSRRRProof (EQUIVALENT)
3.38/1.79	Used ordering:
3.38/1.79	Knuth-Bendix order [KBO] with precedence:g_2 > h_2 > l > k > s_1 > c > t_1 > b > a > f_1 > U1_1
3.38/1.79	
3.38/1.79	and weight map:
3.38/1.79	
3.38/1.79	   a=1
3.38/1.79	   c=2
3.38/1.79	   b=1
3.38/1.79	   k=1
3.38/1.79	   l=1
3.38/1.79	   f_1=2
3.38/1.79	   U1_1=1
3.38/1.79	   s_1=1
3.38/1.79	   t_1=1
3.38/1.79	   g_2=0
3.38/1.79	   h_2=0
3.38/1.79	
3.38/1.79	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.38/1.79	
3.38/1.79	   f(x) -> U1(s(x))
3.38/1.79	   U1(z) -> z
3.38/1.79	   s(a) -> c
3.38/1.79	   s(b) -> c
3.38/1.79	   c -> t(k)
3.38/1.79	   c -> t(l)
3.38/1.79	   g(x, x) -> h(x, x)
3.38/1.79	
3.38/1.79	
3.38/1.79	
3.38/1.79	
3.38/1.79	----------------------------------------
3.38/1.79	
3.38/1.79	(4)
3.38/1.79	Obligation:
3.38/1.79	Q restricted rewrite system:
3.38/1.79	R is empty.
3.38/1.79	Q is empty.
3.38/1.79	
3.38/1.79	----------------------------------------
3.38/1.79	
3.38/1.79	(5) RisEmptyProof (EQUIVALENT)
3.38/1.79	The TRS R is empty. Hence, termination is trivially proven.
3.38/1.79	----------------------------------------
3.38/1.79	
3.38/1.79	(6)
3.38/1.79	YES
3.38/1.81	EOF