3.26/1.63	YES
3.26/1.64	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.26/1.64	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.26/1.64	
3.26/1.64	
3.26/1.64	Quasi decreasingness of the given CTRS could be proven:
3.26/1.64	
3.26/1.64	(0) CTRS
3.26/1.64	(1) CTRSToQTRSProof [SOUND, 0 ms]
3.26/1.64	(2) QTRS
3.26/1.64	(3) QTRSRRRProof [EQUIVALENT, 50 ms]
3.26/1.64	(4) QTRS
3.26/1.64	(5) QTRSRRRProof [EQUIVALENT, 0 ms]
3.26/1.64	(6) QTRS
3.26/1.64	(7) RisEmptyProof [EQUIVALENT, 0 ms]
3.26/1.64	(8) YES
3.26/1.64	
3.26/1.64	
3.26/1.64	----------------------------------------
3.26/1.64	
3.26/1.64	(0)
3.26/1.64	Obligation:
3.26/1.64	Conditional term rewrite system:
3.26/1.64	The TRS R consists of the following rules:
3.26/1.64	
3.26/1.64	   g(s(x)) -> g(x)
3.26/1.64	
3.26/1.64	The conditional TRS C consists of the following conditional rules:
3.26/1.64	
3.26/1.64	   f(g(x)) -> x <= x -> s(0)
3.26/1.64	
3.26/1.64	
3.26/1.64	----------------------------------------
3.26/1.64	
3.26/1.64	(1) CTRSToQTRSProof (SOUND)
3.26/1.64	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
3.26/1.64	----------------------------------------
3.26/1.64	
3.26/1.64	(2)
3.26/1.64	Obligation:
3.26/1.64	Q restricted rewrite system:
3.26/1.64	The TRS R consists of the following rules:
3.26/1.64	
3.26/1.64	   f(g(x)) -> U1(x, x)
3.26/1.64	   U1(s(0), x) -> x
3.26/1.64	   g(s(x)) -> g(x)
3.26/1.64	
3.26/1.64	Q is empty.
3.26/1.64	
3.26/1.64	----------------------------------------
3.26/1.64	
3.26/1.64	(3) QTRSRRRProof (EQUIVALENT)
3.26/1.64	Used ordering:
3.26/1.64	Polynomial interpretation [POLO]:
3.26/1.64	
3.26/1.64	   POL(0) = 0
3.26/1.64	   POL(U1(x_1, x_2)) = 1 + x_1 + x_2
3.26/1.64	   POL(f(x_1)) = 2 + 2*x_1
3.26/1.64	   POL(g(x_1)) = 2 + 2*x_1
3.26/1.64	   POL(s(x_1)) = x_1
3.26/1.64	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.26/1.64	
3.26/1.64	   f(g(x)) -> U1(x, x)
3.26/1.64	   U1(s(0), x) -> x
3.26/1.64	
3.26/1.64	
3.26/1.64	
3.26/1.64	
3.26/1.64	----------------------------------------
3.26/1.64	
3.26/1.64	(4)
3.26/1.64	Obligation:
3.26/1.64	Q restricted rewrite system:
3.26/1.64	The TRS R consists of the following rules:
3.26/1.64	
3.26/1.64	   g(s(x)) -> g(x)
3.26/1.64	
3.26/1.64	Q is empty.
3.26/1.64	
3.26/1.64	----------------------------------------
3.26/1.64	
3.26/1.64	(5) QTRSRRRProof (EQUIVALENT)
3.26/1.64	Used ordering:
3.26/1.64	Knuth-Bendix order [KBO] with precedence:s_1 > g_1
3.26/1.64	
3.26/1.64	and weight map:
3.26/1.64	
3.26/1.64	   g_1=1
3.26/1.64	   s_1=0
3.26/1.64	
3.26/1.64	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.26/1.64	
3.26/1.64	   g(s(x)) -> g(x)
3.26/1.64	
3.26/1.64	
3.26/1.64	
3.26/1.64	
3.26/1.64	----------------------------------------
3.26/1.64	
3.26/1.64	(6)
3.26/1.64	Obligation:
3.26/1.64	Q restricted rewrite system:
3.26/1.64	R is empty.
3.26/1.64	Q is empty.
3.26/1.64	
3.26/1.64	----------------------------------------
3.26/1.64	
3.26/1.64	(7) RisEmptyProof (EQUIVALENT)
3.26/1.64	The TRS R is empty. Hence, termination is trivially proven.
3.26/1.64	----------------------------------------
3.26/1.64	
3.26/1.64	(8)
3.26/1.64	YES
3.26/1.65	EOF