3.34/1.65	YES
3.34/1.66	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.34/1.66	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.34/1.66	
3.34/1.66	
3.34/1.66	Quasi decreasingness of the given CTRS could be proven:
3.34/1.66	
3.34/1.66	(0) CTRS
3.34/1.66	(1) CTRSToQTRSProof [SOUND, 0 ms]
3.34/1.66	(2) QTRS
3.34/1.66	(3) QTRSRRRProof [EQUIVALENT, 0 ms]
3.34/1.66	(4) QTRS
3.34/1.66	(5) QTRSRRRProof [EQUIVALENT, 1 ms]
3.34/1.66	(6) QTRS
3.34/1.66	(7) RisEmptyProof [EQUIVALENT, 0 ms]
3.34/1.66	(8) YES
3.34/1.66	
3.34/1.66	
3.34/1.66	----------------------------------------
3.34/1.66	
3.34/1.66	(0)
3.34/1.66	Obligation:
3.34/1.66	Conditional term rewrite system:
3.34/1.66	The TRS R consists of the following rules:
3.34/1.66	
3.34/1.66	   h(x, x) -> A
3.34/1.66	
3.34/1.66	The conditional TRS C consists of the following conditional rules:
3.34/1.66	
3.34/1.66	   f(x) -> e <= d -> l
3.34/1.66	
3.34/1.66	
3.34/1.66	----------------------------------------
3.34/1.66	
3.34/1.66	(1) CTRSToQTRSProof (SOUND)
3.34/1.66	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
3.34/1.66	----------------------------------------
3.34/1.66	
3.34/1.66	(2)
3.34/1.66	Obligation:
3.34/1.66	Q restricted rewrite system:
3.34/1.66	The TRS R consists of the following rules:
3.34/1.66	
3.34/1.66	   f(x) -> U1(d)
3.34/1.66	   U1(l) -> e
3.34/1.66	   h(x, x) -> A
3.34/1.66	
3.34/1.66	Q is empty.
3.34/1.66	
3.34/1.66	----------------------------------------
3.34/1.66	
3.34/1.66	(3) QTRSRRRProof (EQUIVALENT)
3.34/1.66	Used ordering:
3.34/1.66	Polynomial interpretation [POLO]:
3.34/1.66	
3.34/1.66	   POL(A) = 0
3.34/1.66	   POL(U1(x_1)) = 2*x_1
3.34/1.66	   POL(d) = 1
3.34/1.66	   POL(e) = 2
3.34/1.66	   POL(f(x_1)) = 2 + x_1
3.34/1.66	   POL(h(x_1, x_2)) = 1 + x_1 + x_2
3.34/1.66	   POL(l) = 1
3.34/1.66	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.34/1.66	
3.34/1.66	   h(x, x) -> A
3.34/1.66	
3.34/1.66	
3.34/1.66	
3.34/1.66	
3.34/1.66	----------------------------------------
3.34/1.66	
3.34/1.66	(4)
3.34/1.66	Obligation:
3.34/1.66	Q restricted rewrite system:
3.34/1.66	The TRS R consists of the following rules:
3.34/1.66	
3.34/1.66	   f(x) -> U1(d)
3.34/1.66	   U1(l) -> e
3.34/1.66	
3.34/1.66	Q is empty.
3.34/1.66	
3.34/1.66	----------------------------------------
3.34/1.66	
3.34/1.66	(5) QTRSRRRProof (EQUIVALENT)
3.34/1.66	Used ordering:
3.34/1.66	Knuth-Bendix order [KBO] with precedence:f_1 > U1_1 > e > l > d
3.34/1.66	
3.34/1.66	and weight map:
3.34/1.66	
3.34/1.66	   d=2
3.34/1.66	   l=1
3.34/1.66	   e=2
3.34/1.66	   f_1=2
3.34/1.66	   U1_1=1
3.34/1.66	
3.34/1.66	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.34/1.66	
3.34/1.66	   f(x) -> U1(d)
3.34/1.66	   U1(l) -> e
3.34/1.66	
3.34/1.66	
3.34/1.66	
3.34/1.66	
3.34/1.66	----------------------------------------
3.34/1.66	
3.34/1.66	(6)
3.34/1.66	Obligation:
3.34/1.66	Q restricted rewrite system:
3.34/1.66	R is empty.
3.34/1.66	Q is empty.
3.34/1.66	
3.34/1.66	----------------------------------------
3.34/1.66	
3.34/1.66	(7) RisEmptyProof (EQUIVALENT)
3.34/1.66	The TRS R is empty. Hence, termination is trivially proven.
3.34/1.66	----------------------------------------
3.34/1.66	
3.34/1.66	(8)
3.34/1.66	YES
3.50/1.69	EOF