3.29/1.66	YES
3.29/1.67	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.29/1.67	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.29/1.67	
3.29/1.67	
3.29/1.67	Quasi decreasingness of the given CTRS could be proven:
3.29/1.67	
3.29/1.67	(0) CTRS
3.29/1.67	(1) CTRSToQTRSProof [SOUND, 0 ms]
3.29/1.67	(2) QTRS
3.29/1.67	(3) QTRSRRRProof [EQUIVALENT, 21 ms]
3.29/1.67	(4) QTRS
3.29/1.67	(5) RisEmptyProof [EQUIVALENT, 0 ms]
3.29/1.67	(6) YES
3.29/1.67	
3.29/1.67	
3.29/1.67	----------------------------------------
3.29/1.67	
3.29/1.67	(0)
3.29/1.67	Obligation:
3.29/1.67	Conditional term rewrite system:
3.29/1.67	The TRS R consists of the following rules:
3.29/1.67	
3.29/1.67	   e(0) -> true
3.29/1.67	   o(0) -> true
3.29/1.67	
3.29/1.67	The conditional TRS C consists of the following conditional rules:
3.29/1.67	
3.29/1.67	   e(s(x)) -> true <= o(x) -> true
3.29/1.67	   e(s(x)) -> false <= e(x) -> true
3.29/1.67	   o(s(x)) -> true <= e(x) -> true
3.29/1.67	   o(s(x)) -> false <= o(x) -> true
3.29/1.67	
3.29/1.67	
3.29/1.67	----------------------------------------
3.29/1.67	
3.29/1.67	(1) CTRSToQTRSProof (SOUND)
3.29/1.67	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
3.29/1.67	----------------------------------------
3.29/1.67	
3.29/1.67	(2)
3.29/1.67	Obligation:
3.29/1.67	Q restricted rewrite system:
3.29/1.67	The TRS R consists of the following rules:
3.29/1.67	
3.29/1.67	   e(s(x)) -> U1(o(x))
3.29/1.67	   U1(true) -> true
3.29/1.67	   e(s(x)) -> U2(e(x))
3.29/1.67	   U2(true) -> false
3.29/1.67	   o(s(x)) -> U3(e(x))
3.29/1.67	   U3(true) -> true
3.29/1.67	   o(s(x)) -> U4(o(x))
3.29/1.67	   U4(true) -> false
3.29/1.67	   e(0) -> true
3.29/1.67	   o(0) -> true
3.29/1.67	
3.29/1.67	Q is empty.
3.29/1.67	
3.29/1.67	----------------------------------------
3.29/1.67	
3.29/1.67	(3) QTRSRRRProof (EQUIVALENT)
3.29/1.67	Used ordering:
3.29/1.67	Knuth-Bendix order [KBO] with precedence:o_1 > 0 > s_1 > U3_1 > e_1 > U1_1 > true > U4_1 > false > U2_1
3.29/1.67	
3.29/1.67	and weight map:
3.29/1.67	
3.29/1.67	   true=5
3.29/1.67	   false=6
3.29/1.67	   0=1
3.29/1.67	   e_1=5
3.29/1.67	   s_1=3
3.29/1.67	   U1_1=4
3.29/1.67	   o_1=4
3.29/1.67	   U2_1=2
3.29/1.67	   U3_1=1
3.29/1.67	   U4_1=2
3.29/1.67	
3.29/1.67	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.29/1.67	
3.29/1.67	   e(s(x)) -> U1(o(x))
3.29/1.67	   U1(true) -> true
3.29/1.67	   e(s(x)) -> U2(e(x))
3.29/1.67	   U2(true) -> false
3.29/1.67	   o(s(x)) -> U3(e(x))
3.29/1.67	   U3(true) -> true
3.29/1.67	   o(s(x)) -> U4(o(x))
3.29/1.67	   U4(true) -> false
3.29/1.67	   e(0) -> true
3.29/1.67	   o(0) -> true
3.29/1.67	
3.29/1.67	
3.29/1.67	
3.29/1.67	
3.29/1.67	----------------------------------------
3.29/1.67	
3.29/1.67	(4)
3.29/1.67	Obligation:
3.29/1.67	Q restricted rewrite system:
3.29/1.67	R is empty.
3.29/1.67	Q is empty.
3.29/1.67	
3.29/1.67	----------------------------------------
3.29/1.67	
3.29/1.67	(5) RisEmptyProof (EQUIVALENT)
3.29/1.67	The TRS R is empty. Hence, termination is trivially proven.
3.29/1.67	----------------------------------------
3.29/1.67	
3.29/1.67	(6)
3.29/1.67	YES
3.29/1.69	EOF