3.40/1.91	YES
3.40/1.92	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.40/1.92	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.40/1.92	
3.40/1.92	
3.40/1.92	Quasi decreasingness of the given CTRS could be proven:
3.40/1.92	
3.40/1.92	(0) CTRS
3.40/1.92	(1) CTRSToQTRSProof [SOUND, 0 ms]
3.40/1.92	(2) QTRS
3.40/1.92	(3) QTRSRRRProof [EQUIVALENT, 0 ms]
3.40/1.92	(4) QTRS
3.40/1.92	(5) QTRSRRRProof [EQUIVALENT, 2 ms]
3.40/1.92	(6) QTRS
3.40/1.92	(7) RisEmptyProof [EQUIVALENT, 0 ms]
3.40/1.92	(8) YES
3.40/1.92	
3.40/1.92	
3.40/1.92	----------------------------------------
3.40/1.92	
3.40/1.92	(0)
3.40/1.92	Obligation:
3.40/1.92	Conditional term rewrite system:
3.40/1.92	The TRS R is empty.
3.40/1.92	The conditional TRS C consists of the following conditional rules:
3.40/1.92	
3.40/1.92	   f(x, x) -> a <= g(x) -> b
3.40/1.92	
3.40/1.92	
3.40/1.92	----------------------------------------
3.40/1.92	
3.40/1.92	(1) CTRSToQTRSProof (SOUND)
3.40/1.92	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
3.40/1.92	----------------------------------------
3.40/1.92	
3.40/1.92	(2)
3.40/1.92	Obligation:
3.40/1.92	Q restricted rewrite system:
3.40/1.92	The TRS R consists of the following rules:
3.40/1.92	
3.40/1.92	   f(x, x) -> U1(g(x))
3.40/1.92	   U1(b) -> a
3.40/1.92	
3.40/1.92	Q is empty.
3.40/1.92	
3.40/1.92	----------------------------------------
3.40/1.92	
3.40/1.92	(3) QTRSRRRProof (EQUIVALENT)
3.40/1.92	Used ordering:
3.40/1.92	Polynomial interpretation [POLO]:
3.40/1.92	
3.40/1.92	   POL(U1(x_1)) = 2*x_1
3.40/1.92	   POL(a) = 0
3.40/1.92	   POL(b) = 1
3.40/1.92	   POL(f(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
3.40/1.92	   POL(g(x_1)) = 1 + x_1
3.40/1.92	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.40/1.92	
3.40/1.92	   U1(b) -> a
3.40/1.92	
3.40/1.92	
3.40/1.92	
3.40/1.92	
3.40/1.92	----------------------------------------
3.40/1.92	
3.40/1.92	(4)
3.40/1.92	Obligation:
3.40/1.92	Q restricted rewrite system:
3.40/1.92	The TRS R consists of the following rules:
3.40/1.92	
3.40/1.92	   f(x, x) -> U1(g(x))
3.40/1.92	
3.40/1.92	Q is empty.
3.40/1.92	
3.40/1.92	----------------------------------------
3.40/1.92	
3.40/1.92	(5) QTRSRRRProof (EQUIVALENT)
3.40/1.92	Used ordering:
3.40/1.92	Knuth-Bendix order [KBO] with precedence:g_1 > f_2 > U1_1
3.40/1.92	
3.40/1.92	and weight map:
3.40/1.92	
3.40/1.92	   U1_1=1
3.40/1.92	   g_1=1
3.40/1.92	   f_2=1
3.40/1.92	
3.40/1.92	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.40/1.92	
3.40/1.92	   f(x, x) -> U1(g(x))
3.40/1.92	
3.40/1.92	
3.40/1.92	
3.40/1.92	
3.40/1.92	----------------------------------------
3.40/1.92	
3.40/1.92	(6)
3.40/1.92	Obligation:
3.40/1.92	Q restricted rewrite system:
3.40/1.92	R is empty.
3.40/1.92	Q is empty.
3.40/1.92	
3.40/1.92	----------------------------------------
3.40/1.92	
3.40/1.92	(7) RisEmptyProof (EQUIVALENT)
3.40/1.92	The TRS R is empty. Hence, termination is trivially proven.
3.40/1.92	----------------------------------------
3.40/1.92	
3.40/1.92	(8)
3.40/1.92	YES
3.40/1.94	EOF