3.38/1.98	YES
3.38/1.98	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.38/1.98	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.38/1.98	
3.38/1.98	
3.38/1.98	Quasi decreasingness of the given CTRS could be proven:
3.38/1.98	
3.38/1.98	(0) CTRS
3.38/1.98	(1) CTRSToQTRSProof [SOUND, 0 ms]
3.38/1.98	(2) QTRS
3.38/1.98	(3) QTRSRRRProof [EQUIVALENT, 52 ms]
3.38/1.98	(4) QTRS
3.38/1.98	(5) QTRSRRRProof [EQUIVALENT, 0 ms]
3.38/1.98	(6) QTRS
3.38/1.98	(7) RisEmptyProof [EQUIVALENT, 0 ms]
3.38/1.98	(8) YES
3.38/1.98	
3.38/1.98	
3.38/1.98	----------------------------------------
3.38/1.98	
3.38/1.98	(0)
3.38/1.98	Obligation:
3.38/1.98	Conditional term rewrite system:
3.38/1.98	The TRS R is empty.
3.38/1.98	The conditional TRS C consists of the following conditional rules:
3.38/1.98	
3.38/1.98	   f(x) -> b <= g(x, x) -> a
3.38/1.98	
3.38/1.98	
3.38/1.98	----------------------------------------
3.38/1.98	
3.38/1.98	(1) CTRSToQTRSProof (SOUND)
3.38/1.98	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
3.38/1.98	----------------------------------------
3.38/1.98	
3.38/1.98	(2)
3.38/1.98	Obligation:
3.38/1.98	Q restricted rewrite system:
3.38/1.98	The TRS R consists of the following rules:
3.38/1.98	
3.38/1.98	   f(x) -> U1(g(x, x))
3.38/1.98	   U1(a) -> b
3.38/1.98	
3.38/1.98	Q is empty.
3.38/1.98	
3.38/1.98	----------------------------------------
3.38/1.98	
3.38/1.98	(3) QTRSRRRProof (EQUIVALENT)
3.38/1.98	Used ordering:
3.38/1.98	Polynomial interpretation [POLO]:
3.38/1.98	
3.38/1.98	   POL(U1(x_1)) = 1 + x_1
3.38/1.98	   POL(a) = 2
3.38/1.98	   POL(b) = 2
3.38/1.98	   POL(f(x_1)) = 1 + 2*x_1
3.38/1.98	   POL(g(x_1, x_2)) = x_1 + x_2
3.38/1.98	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.38/1.98	
3.38/1.98	   U1(a) -> b
3.38/1.98	
3.38/1.98	
3.38/1.98	
3.38/1.98	
3.38/1.98	----------------------------------------
3.38/1.98	
3.38/1.98	(4)
3.38/1.98	Obligation:
3.38/1.98	Q restricted rewrite system:
3.38/1.98	The TRS R consists of the following rules:
3.38/1.98	
3.38/1.98	   f(x) -> U1(g(x, x))
3.38/1.98	
3.38/1.98	Q is empty.
3.38/1.98	
3.38/1.98	----------------------------------------
3.38/1.98	
3.38/1.98	(5) QTRSRRRProof (EQUIVALENT)
3.38/1.98	Used ordering:
3.38/1.98	Polynomial interpretation [POLO]:
3.38/1.98	
3.38/1.98	   POL(U1(x_1)) = x_1
3.38/1.98	   POL(f(x_1)) = 2 + 2*x_1
3.38/1.98	   POL(g(x_1, x_2)) = 1 + x_1 + x_2
3.38/1.98	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.38/1.98	
3.38/1.98	   f(x) -> U1(g(x, x))
3.38/1.98	
3.38/1.98	
3.38/1.98	
3.38/1.98	
3.38/1.98	----------------------------------------
3.38/1.98	
3.38/1.98	(6)
3.38/1.98	Obligation:
3.38/1.98	Q restricted rewrite system:
3.38/1.98	R is empty.
3.38/1.98	Q is empty.
3.38/1.98	
3.38/1.98	----------------------------------------
3.38/1.98	
3.38/1.98	(7) RisEmptyProof (EQUIVALENT)
3.38/1.98	The TRS R is empty. Hence, termination is trivially proven.
3.38/1.98	----------------------------------------
3.38/1.98	
3.38/1.98	(8)
3.38/1.98	YES
3.38/2.00	EOF