3.33/1.65	YES
3.64/1.66	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.64/1.66	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.64/1.66	
3.64/1.66	
3.64/1.66	Quasi decreasingness of the given CTRS could be proven:
3.64/1.66	
3.64/1.66	(0) CTRS
3.64/1.66	(1) CTRSToQTRSProof [SOUND, 0 ms]
3.64/1.66	(2) QTRS
3.64/1.66	(3) QTRSRRRProof [EQUIVALENT, 60 ms]
3.64/1.66	(4) QTRS
3.64/1.66	(5) QTRSRRRProof [EQUIVALENT, 0 ms]
3.64/1.66	(6) QTRS
3.64/1.66	(7) RisEmptyProof [EQUIVALENT, 0 ms]
3.64/1.66	(8) YES
3.64/1.66	
3.64/1.66	
3.64/1.66	----------------------------------------
3.64/1.66	
3.64/1.66	(0)
3.64/1.66	Obligation:
3.64/1.66	Conditional term rewrite system:
3.64/1.66	The TRS R consists of the following rules:
3.64/1.66	
3.64/1.66	   s(p(x)) -> x
3.64/1.66	   p(s(x)) -> x
3.64/1.66	   pos(0) -> false
3.64/1.66	   pos(s(0)) -> true
3.64/1.66	
3.64/1.66	The conditional TRS C consists of the following conditional rules:
3.64/1.66	
3.64/1.66	   pos(s(x)) -> true <= pos(x) -> true
3.64/1.66	   pos(p(x)) -> false <= pos(x) -> false
3.64/1.66	
3.64/1.66	
3.64/1.66	----------------------------------------
3.64/1.66	
3.64/1.66	(1) CTRSToQTRSProof (SOUND)
3.64/1.66	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
3.64/1.66	----------------------------------------
3.64/1.66	
3.64/1.66	(2)
3.64/1.66	Obligation:
3.64/1.66	Q restricted rewrite system:
3.64/1.66	The TRS R consists of the following rules:
3.64/1.66	
3.64/1.66	   pos(s(x)) -> U1(pos(x))
3.64/1.66	   U1(true) -> true
3.64/1.66	   pos(p(x)) -> U2(pos(x))
3.64/1.66	   U2(false) -> false
3.64/1.66	   s(p(x)) -> x
3.64/1.66	   p(s(x)) -> x
3.64/1.66	   pos(0) -> false
3.64/1.66	   pos(s(0)) -> true
3.64/1.66	
3.64/1.66	Q is empty.
3.64/1.66	
3.64/1.66	----------------------------------------
3.64/1.66	
3.64/1.66	(3) QTRSRRRProof (EQUIVALENT)
3.64/1.66	Used ordering:
3.64/1.66	Polynomial interpretation [POLO]:
3.64/1.66	
3.64/1.66	   POL(0) = 0
3.64/1.66	   POL(U1(x_1)) = x_1
3.64/1.66	   POL(U2(x_1)) = 2 + 2*x_1
3.64/1.66	   POL(false) = 1
3.64/1.66	   POL(p(x_1)) = 2 + 2*x_1
3.64/1.66	   POL(pos(x_1)) = 2 + 2*x_1
3.64/1.66	   POL(s(x_1)) = x_1
3.64/1.66	   POL(true) = 1
3.64/1.66	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.64/1.66	
3.64/1.66	   U2(false) -> false
3.64/1.66	   s(p(x)) -> x
3.64/1.66	   p(s(x)) -> x
3.64/1.66	   pos(0) -> false
3.64/1.66	   pos(s(0)) -> true
3.64/1.66	
3.64/1.66	
3.64/1.66	
3.64/1.66	
3.64/1.66	----------------------------------------
3.64/1.66	
3.64/1.66	(4)
3.64/1.66	Obligation:
3.64/1.66	Q restricted rewrite system:
3.64/1.66	The TRS R consists of the following rules:
3.64/1.66	
3.64/1.66	   pos(s(x)) -> U1(pos(x))
3.64/1.66	   U1(true) -> true
3.64/1.66	   pos(p(x)) -> U2(pos(x))
3.64/1.66	
3.64/1.66	Q is empty.
3.64/1.66	
3.64/1.66	----------------------------------------
3.64/1.66	
3.64/1.66	(5) QTRSRRRProof (EQUIVALENT)
3.64/1.66	Used ordering:
3.64/1.66	Knuth-Bendix order [KBO] with precedence:U1_1 > pos_1 > U2_1 > p_1 > s_1 > true
3.64/1.66	
3.64/1.66	and weight map:
3.64/1.66	
3.64/1.66	   true=1
3.64/1.66	   pos_1=2
3.64/1.66	   s_1=1
3.64/1.66	   U1_1=0
3.64/1.66	   p_1=1
3.64/1.66	   U2_1=1
3.64/1.66	
3.64/1.66	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.64/1.66	
3.64/1.66	   pos(s(x)) -> U1(pos(x))
3.64/1.66	   U1(true) -> true
3.64/1.66	   pos(p(x)) -> U2(pos(x))
3.64/1.66	
3.64/1.66	
3.64/1.66	
3.64/1.66	
3.64/1.66	----------------------------------------
3.64/1.66	
3.64/1.66	(6)
3.64/1.66	Obligation:
3.64/1.66	Q restricted rewrite system:
3.64/1.66	R is empty.
3.64/1.66	Q is empty.
3.64/1.66	
3.64/1.66	----------------------------------------
3.64/1.66	
3.64/1.66	(7) RisEmptyProof (EQUIVALENT)
3.64/1.66	The TRS R is empty. Hence, termination is trivially proven.
3.64/1.66	----------------------------------------
3.64/1.66	
3.64/1.66	(8)
3.64/1.66	YES
3.69/1.69	EOF