3.43/1.67	YES
3.43/1.67	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.43/1.67	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.43/1.67	
3.43/1.67	
3.43/1.67	Quasi decreasingness of the given CTRS could be proven:
3.43/1.67	
3.43/1.67	(0) CTRS
3.43/1.67	(1) CTRSToQTRSProof [SOUND, 0 ms]
3.43/1.67	(2) QTRS
3.43/1.67	(3) QTRSRRRProof [EQUIVALENT, 48 ms]
3.43/1.67	(4) QTRS
3.43/1.67	(5) QTRSRRRProof [EQUIVALENT, 0 ms]
3.43/1.67	(6) QTRS
3.43/1.67	(7) RisEmptyProof [EQUIVALENT, 0 ms]
3.43/1.67	(8) YES
3.43/1.67	
3.43/1.67	
3.43/1.67	----------------------------------------
3.43/1.67	
3.43/1.67	(0)
3.43/1.67	Obligation:
3.43/1.67	Conditional term rewrite system:
3.43/1.67	The TRS R consists of the following rules:
3.43/1.67	
3.43/1.67	   a -> c
3.43/1.67	   a -> d
3.43/1.67	   s(c) -> t(k)
3.43/1.67	
3.43/1.67	The conditional TRS C consists of the following conditional rules:
3.43/1.67	
3.43/1.67	   f(x) -> z <= s(x) -> t(z)
3.43/1.67	
3.43/1.67	
3.43/1.67	----------------------------------------
3.43/1.67	
3.43/1.67	(1) CTRSToQTRSProof (SOUND)
3.43/1.67	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
3.43/1.67	----------------------------------------
3.43/1.67	
3.43/1.67	(2)
3.43/1.67	Obligation:
3.43/1.67	Q restricted rewrite system:
3.43/1.67	The TRS R consists of the following rules:
3.43/1.67	
3.43/1.67	   f(x) -> U1(s(x))
3.43/1.67	   U1(t(z)) -> z
3.43/1.67	   a -> c
3.43/1.67	   a -> d
3.43/1.67	   s(c) -> t(k)
3.43/1.67	
3.43/1.67	Q is empty.
3.43/1.67	
3.43/1.67	----------------------------------------
3.43/1.67	
3.43/1.67	(3) QTRSRRRProof (EQUIVALENT)
3.43/1.67	Used ordering:
3.43/1.67	Polynomial interpretation [POLO]:
3.43/1.67	
3.43/1.67	   POL(U1(x_1)) = 2 + x_1
3.43/1.67	   POL(a) = 1
3.43/1.67	   POL(c) = 0
3.43/1.67	   POL(d) = 1
3.43/1.67	   POL(f(x_1)) = 2 + 2*x_1
3.43/1.67	   POL(k) = 0
3.43/1.67	   POL(s(x_1)) = 2*x_1
3.43/1.67	   POL(t(x_1)) = x_1
3.43/1.67	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.43/1.67	
3.43/1.67	   U1(t(z)) -> z
3.43/1.67	   a -> c
3.43/1.67	
3.43/1.67	
3.43/1.67	
3.43/1.67	
3.43/1.67	----------------------------------------
3.43/1.67	
3.43/1.67	(4)
3.43/1.67	Obligation:
3.43/1.67	Q restricted rewrite system:
3.43/1.67	The TRS R consists of the following rules:
3.43/1.67	
3.43/1.67	   f(x) -> U1(s(x))
3.43/1.67	   a -> d
3.43/1.67	   s(c) -> t(k)
3.43/1.67	
3.43/1.67	Q is empty.
3.43/1.67	
3.43/1.67	----------------------------------------
3.43/1.67	
3.43/1.67	(5) QTRSRRRProof (EQUIVALENT)
3.43/1.67	Used ordering:
3.43/1.67	Knuth-Bendix order [KBO] with precedence:k > s_1 > f_1 > t_1 > c > a > d > U1_1
3.43/1.67	
3.43/1.67	and weight map:
3.43/1.67	
3.43/1.67	   a=1
3.43/1.67	   d=1
3.43/1.67	   c=1
3.43/1.67	   k=2
3.43/1.67	   f_1=3
3.43/1.67	   U1_1=1
3.43/1.67	   s_1=2
3.43/1.67	   t_1=1
3.43/1.67	
3.43/1.67	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.43/1.68	
3.43/1.68	   f(x) -> U1(s(x))
3.43/1.68	   a -> d
3.43/1.68	   s(c) -> t(k)
3.43/1.68	
3.43/1.68	
3.43/1.68	
3.43/1.68	
3.43/1.68	----------------------------------------
3.43/1.68	
3.43/1.68	(6)
3.43/1.68	Obligation:
3.43/1.68	Q restricted rewrite system:
3.43/1.68	R is empty.
3.43/1.68	Q is empty.
3.43/1.68	
3.43/1.68	----------------------------------------
3.43/1.68	
3.43/1.68	(7) RisEmptyProof (EQUIVALENT)
3.43/1.68	The TRS R is empty. Hence, termination is trivially proven.
3.43/1.68	----------------------------------------
3.43/1.68	
3.43/1.68	(8)
3.43/1.68	YES
3.59/1.71	EOF