3.38/2.47	YES
3.43/2.48	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.43/2.48	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.43/2.48	
3.43/2.48	
3.43/2.48	Quasi decreasingness of the given CTRS could be proven:
3.43/2.48	
3.43/2.48	(0) CTRS
3.43/2.48	(1) CTRSToQTRSProof [SOUND, 0 ms]
3.43/2.48	(2) QTRS
3.43/2.48	(3) QTRSRRRProof [EQUIVALENT, 55 ms]
3.43/2.48	(4) QTRS
3.43/2.48	(5) RisEmptyProof [EQUIVALENT, 0 ms]
3.43/2.48	(6) YES
3.43/2.48	
3.43/2.48	
3.43/2.48	----------------------------------------
3.43/2.48	
3.43/2.48	(0)
3.43/2.48	Obligation:
3.43/2.48	Conditional term rewrite system:
3.43/2.48	The TRS R consists of the following rules:
3.43/2.48	
3.43/2.48	   even(0) -> true
3.43/2.48	
3.43/2.48	The conditional TRS C consists of the following conditional rules:
3.43/2.48	
3.43/2.48	   even(s(x)) -> true <= odd(x) -> true
3.43/2.48	   odd(s(x)) -> true <= even(x) -> true
3.43/2.48	
3.43/2.48	
3.43/2.48	----------------------------------------
3.43/2.48	
3.43/2.48	(1) CTRSToQTRSProof (SOUND)
3.43/2.48	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
3.43/2.48	----------------------------------------
3.43/2.48	
3.43/2.48	(2)
3.43/2.48	Obligation:
3.43/2.48	Q restricted rewrite system:
3.43/2.48	The TRS R consists of the following rules:
3.43/2.48	
3.43/2.48	   even(s(x)) -> U1(odd(x))
3.43/2.48	   U1(true) -> true
3.43/2.48	   odd(s(x)) -> U2(even(x))
3.43/2.48	   U2(true) -> true
3.43/2.48	   even(0) -> true
3.43/2.48	
3.43/2.48	Q is empty.
3.43/2.48	
3.43/2.48	----------------------------------------
3.43/2.48	
3.43/2.48	(3) QTRSRRRProof (EQUIVALENT)
3.43/2.48	Used ordering:
3.43/2.48	Polynomial interpretation [POLO]:
3.43/2.48	
3.43/2.48	   POL(0) = 0
3.43/2.48	   POL(U1(x_1)) = 2*x_1
3.43/2.48	   POL(U2(x_1)) = 2*x_1
3.43/2.48	   POL(even(x_1)) = 2 + 2*x_1
3.43/2.48	   POL(odd(x_1)) = 2 + 2*x_1
3.43/2.48	   POL(s(x_1)) = 2 + 2*x_1
3.43/2.48	   POL(true) = 1
3.43/2.48	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.43/2.48	
3.43/2.48	   even(s(x)) -> U1(odd(x))
3.43/2.48	   U1(true) -> true
3.43/2.48	   odd(s(x)) -> U2(even(x))
3.43/2.48	   U2(true) -> true
3.43/2.48	   even(0) -> true
3.43/2.48	
3.43/2.48	
3.43/2.48	
3.43/2.48	
3.43/2.48	----------------------------------------
3.43/2.48	
3.43/2.48	(4)
3.43/2.48	Obligation:
3.43/2.48	Q restricted rewrite system:
3.43/2.48	R is empty.
3.43/2.48	Q is empty.
3.43/2.48	
3.43/2.48	----------------------------------------
3.43/2.48	
3.43/2.48	(5) RisEmptyProof (EQUIVALENT)
3.43/2.48	The TRS R is empty. Hence, termination is trivially proven.
3.43/2.48	----------------------------------------
3.43/2.48	
3.43/2.48	(6)
3.43/2.48	YES
3.46/2.63	EOF