3.42/1.68	YES
3.42/1.69	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.42/1.69	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.42/1.69	
3.42/1.69	
3.42/1.69	Quasi decreasingness of the given CTRS could be proven:
3.42/1.69	
3.42/1.69	(0) CTRS
3.42/1.69	(1) CTRSToQTRSProof [SOUND, 0 ms]
3.42/1.69	(2) QTRS
3.42/1.69	(3) QTRSRRRProof [EQUIVALENT, 46 ms]
3.42/1.69	(4) QTRS
3.42/1.69	(5) QTRSRRRProof [EQUIVALENT, 0 ms]
3.42/1.69	(6) QTRS
3.42/1.69	(7) RisEmptyProof [EQUIVALENT, 0 ms]
3.42/1.69	(8) YES
3.42/1.69	
3.42/1.69	
3.42/1.69	----------------------------------------
3.42/1.69	
3.42/1.69	(0)
3.42/1.69	Obligation:
3.42/1.69	Conditional term rewrite system:
3.42/1.69	The TRS R consists of the following rules:
3.42/1.69	
3.42/1.69	   a -> c
3.42/1.69	   g(a) -> h(b)
3.42/1.69	   h(b) -> g(c)
3.42/1.69	
3.42/1.69	The conditional TRS C consists of the following conditional rules:
3.42/1.69	
3.42/1.69	   f(x) -> z <= g(x) -> h(z)
3.42/1.69	
3.42/1.69	
3.42/1.69	----------------------------------------
3.42/1.69	
3.42/1.69	(1) CTRSToQTRSProof (SOUND)
3.42/1.69	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
3.42/1.69	----------------------------------------
3.42/1.69	
3.42/1.69	(2)
3.42/1.69	Obligation:
3.42/1.69	Q restricted rewrite system:
3.42/1.69	The TRS R consists of the following rules:
3.42/1.69	
3.42/1.69	   f(x) -> U1(g(x))
3.42/1.69	   U1(h(z)) -> z
3.42/1.69	   a -> c
3.42/1.69	   g(a) -> h(b)
3.42/1.69	   h(b) -> g(c)
3.42/1.69	
3.42/1.69	Q is empty.
3.42/1.69	
3.42/1.69	----------------------------------------
3.42/1.69	
3.42/1.69	(3) QTRSRRRProof (EQUIVALENT)
3.42/1.69	Used ordering:
3.42/1.69	Polynomial interpretation [POLO]:
3.42/1.69	
3.42/1.69	   POL(U1(x_1)) = 2*x_1
3.42/1.69	   POL(a) = 0
3.42/1.69	   POL(b) = 0
3.42/1.69	   POL(c) = 0
3.42/1.69	   POL(f(x_1)) = 2 + 2*x_1
3.42/1.69	   POL(g(x_1)) = x_1
3.42/1.69	   POL(h(x_1)) = 2*x_1
3.42/1.69	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.42/1.69	
3.42/1.69	   f(x) -> U1(g(x))
3.42/1.69	
3.42/1.69	
3.42/1.69	
3.42/1.69	
3.42/1.69	----------------------------------------
3.42/1.69	
3.42/1.69	(4)
3.42/1.69	Obligation:
3.42/1.69	Q restricted rewrite system:
3.42/1.69	The TRS R consists of the following rules:
3.42/1.69	
3.42/1.69	   U1(h(z)) -> z
3.42/1.69	   a -> c
3.42/1.69	   g(a) -> h(b)
3.42/1.69	   h(b) -> g(c)
3.42/1.69	
3.42/1.69	Q is empty.
3.42/1.69	
3.42/1.69	----------------------------------------
3.42/1.69	
3.42/1.69	(5) QTRSRRRProof (EQUIVALENT)
3.42/1.69	Used ordering:
3.42/1.69	Knuth-Bendix order [KBO] with precedence:b > h_1 > g_1 > c > a > U1_1
3.42/1.69	
3.42/1.69	and weight map:
3.42/1.69	
3.42/1.69	   a=3
3.42/1.69	   c=2
3.42/1.69	   b=1
3.42/1.69	   U1_1=1
3.42/1.69	   h_1=2
3.42/1.69	   g_1=1
3.42/1.69	
3.42/1.69	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.42/1.69	
3.42/1.69	   U1(h(z)) -> z
3.42/1.69	   a -> c
3.42/1.69	   g(a) -> h(b)
3.42/1.69	   h(b) -> g(c)
3.42/1.69	
3.42/1.69	
3.42/1.69	
3.42/1.69	
3.42/1.69	----------------------------------------
3.42/1.69	
3.42/1.69	(6)
3.42/1.69	Obligation:
3.42/1.69	Q restricted rewrite system:
3.42/1.69	R is empty.
3.42/1.69	Q is empty.
3.42/1.69	
3.42/1.69	----------------------------------------
3.42/1.69	
3.42/1.69	(7) RisEmptyProof (EQUIVALENT)
3.42/1.69	The TRS R is empty. Hence, termination is trivially proven.
3.42/1.69	----------------------------------------
3.42/1.69	
3.42/1.69	(8)
3.42/1.69	YES
3.42/1.71	EOF