3.19/1.54	YES
3.19/1.55	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.19/1.55	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.19/1.55	
3.19/1.55	
3.19/1.55	Quasi decreasingness of the given CTRS could be proven:
3.19/1.55	
3.19/1.55	(0) CTRS
3.19/1.55	(1) CTRSToQTRSProof [SOUND, 0 ms]
3.19/1.55	(2) QTRS
3.19/1.55	(3) QTRSRRRProof [EQUIVALENT, 28 ms]
3.19/1.55	(4) QTRS
3.19/1.55	(5) RisEmptyProof [EQUIVALENT, 0 ms]
3.19/1.55	(6) YES
3.19/1.55	
3.19/1.55	
3.19/1.55	----------------------------------------
3.19/1.55	
3.19/1.55	(0)
3.19/1.55	Obligation:
3.19/1.55	Conditional term rewrite system:
3.19/1.55	The TRS R consists of the following rules:
3.19/1.55	
3.19/1.55	   even(0) -> true
3.19/1.55	   odd(0) -> false
3.19/1.55	
3.19/1.55	The conditional TRS C consists of the following conditional rules:
3.19/1.55	
3.19/1.55	   even(s(x)) -> false <= odd(x) -> false
3.19/1.55	   even(s(x)) -> true <= odd(x) -> true
3.19/1.55	   odd(s(x)) -> false <= even(x) -> false
3.19/1.55	   odd(s(x)) -> true <= even(x) -> true
3.19/1.55	
3.19/1.55	
3.19/1.55	----------------------------------------
3.19/1.55	
3.19/1.55	(1) CTRSToQTRSProof (SOUND)
3.19/1.55	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
3.19/1.55	----------------------------------------
3.19/1.55	
3.19/1.55	(2)
3.19/1.55	Obligation:
3.19/1.55	Q restricted rewrite system:
3.19/1.55	The TRS R consists of the following rules:
3.19/1.55	
3.19/1.55	   even(s(x)) -> U1(odd(x))
3.19/1.55	   U1(true) -> true
3.19/1.55	   U1(false) -> false
3.19/1.55	   odd(s(x)) -> U2(even(x))
3.19/1.55	   U2(true) -> true
3.19/1.55	   U2(false) -> false
3.19/1.55	   even(0) -> true
3.19/1.55	   odd(0) -> false
3.19/1.55	
3.19/1.55	Q is empty.
3.19/1.55	
3.19/1.55	----------------------------------------
3.19/1.55	
3.19/1.55	(3) QTRSRRRProof (EQUIVALENT)
3.19/1.55	Used ordering:
3.19/1.55	Knuth-Bendix order [KBO] with precedence:U2_1 > 0 > odd_1 > false > even_1 > true > s_1 > U1_1
3.19/1.55	
3.19/1.55	and weight map:
3.19/1.55	
3.19/1.55	   true=3
3.19/1.55	   false=3
3.19/1.55	   0=1
3.19/1.55	   even_1=2
3.19/1.55	   s_1=1
3.19/1.55	   U1_1=1
3.19/1.55	   odd_1=2
3.19/1.55	   U2_1=0
3.19/1.55	
3.19/1.55	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.19/1.55	
3.19/1.55	   even(s(x)) -> U1(odd(x))
3.19/1.55	   U1(true) -> true
3.19/1.55	   U1(false) -> false
3.19/1.55	   odd(s(x)) -> U2(even(x))
3.19/1.55	   U2(true) -> true
3.19/1.55	   U2(false) -> false
3.19/1.55	   even(0) -> true
3.19/1.55	   odd(0) -> false
3.19/1.55	
3.19/1.55	
3.19/1.55	
3.19/1.55	
3.19/1.55	----------------------------------------
3.19/1.55	
3.19/1.55	(4)
3.19/1.55	Obligation:
3.19/1.55	Q restricted rewrite system:
3.19/1.55	R is empty.
3.19/1.55	Q is empty.
3.19/1.55	
3.19/1.55	----------------------------------------
3.19/1.55	
3.19/1.55	(5) RisEmptyProof (EQUIVALENT)
3.19/1.55	The TRS R is empty. Hence, termination is trivially proven.
3.19/1.55	----------------------------------------
3.19/1.55	
3.19/1.55	(6)
3.19/1.55	YES
3.19/1.58	EOF