3.31/1.61	YES
3.31/1.62	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.31/1.62	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.31/1.62	
3.31/1.62	
3.31/1.62	Quasi decreasingness of the given CTRS could be proven:
3.31/1.62	
3.31/1.62	(0) CTRS
3.31/1.62	(1) CTRSToQTRSProof [SOUND, 0 ms]
3.31/1.62	(2) QTRS
3.31/1.62	(3) QTRSRRRProof [EQUIVALENT, 41 ms]
3.31/1.62	(4) QTRS
3.31/1.62	(5) RisEmptyProof [EQUIVALENT, 0 ms]
3.31/1.62	(6) YES
3.31/1.62	
3.31/1.62	
3.31/1.62	----------------------------------------
3.31/1.62	
3.31/1.62	(0)
3.31/1.62	Obligation:
3.31/1.62	Conditional term rewrite system:
3.31/1.62	The TRS R consists of the following rules:
3.31/1.62	
3.31/1.62	   s(a) -> t
3.31/1.62	   s(b) -> t
3.31/1.62	   a -> c
3.31/1.62	   b -> c
3.31/1.62	   g(x, x) -> h(x, x)
3.31/1.62	
3.31/1.62	The conditional TRS C consists of the following conditional rules:
3.31/1.62	
3.31/1.62	   f(x) -> A <= s(x) -> t
3.31/1.62	   f(x) -> B <= s(x) -> t
3.31/1.62	
3.31/1.62	
3.31/1.62	----------------------------------------
3.31/1.62	
3.31/1.62	(1) CTRSToQTRSProof (SOUND)
3.31/1.62	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
3.31/1.62	----------------------------------------
3.31/1.62	
3.31/1.62	(2)
3.31/1.62	Obligation:
3.31/1.62	Q restricted rewrite system:
3.31/1.62	The TRS R consists of the following rules:
3.31/1.62	
3.31/1.62	   f(x) -> U1(s(x))
3.31/1.62	   U1(t) -> A
3.31/1.62	   U1(t) -> B
3.31/1.62	   s(a) -> t
3.31/1.62	   s(b) -> t
3.31/1.62	   a -> c
3.31/1.62	   b -> c
3.31/1.62	   g(x, x) -> h(x, x)
3.31/1.62	
3.31/1.62	Q is empty.
3.31/1.62	
3.31/1.62	----------------------------------------
3.31/1.62	
3.31/1.62	(3) QTRSRRRProof (EQUIVALENT)
3.31/1.62	Used ordering:
3.31/1.62	Knuth-Bendix order [KBO] with precedence:f_1 > U1_1 > s_1 > g_2 > t > h_2 > b > a > c > B > A
3.31/1.62	
3.31/1.62	and weight map:
3.31/1.62	
3.31/1.62	   t=2
3.31/1.62	   A=3
3.31/1.62	   B=3
3.31/1.62	   a=1
3.31/1.62	   b=1
3.31/1.62	   c=1
3.31/1.62	   f_1=2
3.31/1.62	   U1_1=1
3.31/1.62	   s_1=1
3.31/1.62	   g_2=0
3.31/1.62	   h_2=0
3.31/1.62	
3.31/1.62	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.31/1.62	
3.31/1.62	   f(x) -> U1(s(x))
3.31/1.62	   U1(t) -> A
3.31/1.62	   U1(t) -> B
3.31/1.62	   s(a) -> t
3.31/1.62	   s(b) -> t
3.31/1.62	   a -> c
3.31/1.62	   b -> c
3.31/1.62	   g(x, x) -> h(x, x)
3.31/1.62	
3.31/1.62	
3.31/1.62	
3.31/1.62	
3.31/1.62	----------------------------------------
3.31/1.62	
3.31/1.62	(4)
3.31/1.62	Obligation:
3.31/1.62	Q restricted rewrite system:
3.31/1.62	R is empty.
3.31/1.62	Q is empty.
3.31/1.62	
3.31/1.62	----------------------------------------
3.31/1.62	
3.31/1.62	(5) RisEmptyProof (EQUIVALENT)
3.31/1.62	The TRS R is empty. Hence, termination is trivially proven.
3.31/1.62	----------------------------------------
3.31/1.62	
3.31/1.62	(6)
3.31/1.62	YES
3.31/1.65	EOF