3.65/1.67	YES
3.65/1.68	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.65/1.68	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.65/1.68	
3.65/1.68	
3.65/1.68	Quasi decreasingness of the given CTRS could be proven:
3.65/1.68	
3.65/1.68	(0) CTRS
3.65/1.68	(1) CTRSToQTRSProof [SOUND, 0 ms]
3.65/1.68	(2) QTRS
3.65/1.68	(3) QTRSRRRProof [EQUIVALENT, 54 ms]
3.65/1.68	(4) QTRS
3.65/1.68	(5) QTRSRRRProof [EQUIVALENT, 0 ms]
3.65/1.68	(6) QTRS
3.65/1.68	(7) QTRSRRRProof [EQUIVALENT, 0 ms]
3.65/1.68	(8) QTRS
3.65/1.68	(9) RisEmptyProof [EQUIVALENT, 0 ms]
3.65/1.68	(10) YES
3.65/1.68	
3.65/1.68	
3.65/1.68	----------------------------------------
3.65/1.68	
3.65/1.68	(0)
3.65/1.68	Obligation:
3.65/1.68	Conditional term rewrite system:
3.65/1.68	The TRS R consists of the following rules:
3.65/1.68	
3.65/1.68	   a -> c
3.65/1.68	   a -> d
3.65/1.68	
3.65/1.68	The conditional TRS C consists of the following conditional rules:
3.65/1.68	
3.65/1.68	   f(x) -> x <= x -> c, d -> x
3.65/1.68	
3.65/1.68	
3.65/1.68	----------------------------------------
3.65/1.68	
3.65/1.68	(1) CTRSToQTRSProof (SOUND)
3.65/1.68	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
3.65/1.68	----------------------------------------
3.65/1.68	
3.65/1.68	(2)
3.65/1.68	Obligation:
3.65/1.68	Q restricted rewrite system:
3.65/1.68	The TRS R consists of the following rules:
3.65/1.68	
3.65/1.68	   f(x) -> U1(x, x)
3.65/1.68	   U1(c, x) -> U2(d, x)
3.65/1.68	   U2(x, x) -> x
3.65/1.68	   a -> c
3.65/1.68	   a -> d
3.65/1.68	
3.65/1.68	Q is empty.
3.65/1.68	
3.65/1.68	----------------------------------------
3.65/1.68	
3.65/1.68	(3) QTRSRRRProof (EQUIVALENT)
3.65/1.68	Used ordering:
3.65/1.68	Polynomial interpretation [POLO]:
3.65/1.68	
3.65/1.68	   POL(U1(x_1, x_2)) = 2 + x_1 + x_2
3.65/1.68	   POL(U2(x_1, x_2)) = 2*x_1 + x_2
3.65/1.68	   POL(a) = 2
3.65/1.68	   POL(c) = 1
3.65/1.68	   POL(d) = 1
3.65/1.68	   POL(f(x_1)) = 2 + 2*x_1
3.65/1.68	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.65/1.68	
3.65/1.68	   U1(c, x) -> U2(d, x)
3.65/1.68	   a -> c
3.65/1.68	   a -> d
3.65/1.68	
3.65/1.68	
3.65/1.68	
3.65/1.68	
3.65/1.68	----------------------------------------
3.65/1.68	
3.65/1.68	(4)
3.65/1.68	Obligation:
3.65/1.68	Q restricted rewrite system:
3.65/1.68	The TRS R consists of the following rules:
3.65/1.68	
3.65/1.68	   f(x) -> U1(x, x)
3.65/1.68	   U2(x, x) -> x
3.65/1.68	
3.65/1.68	Q is empty.
3.65/1.68	
3.65/1.68	----------------------------------------
3.65/1.68	
3.65/1.68	(5) QTRSRRRProof (EQUIVALENT)
3.65/1.68	Used ordering:
3.65/1.68	Polynomial interpretation [POLO]:
3.65/1.68	
3.65/1.68	   POL(U1(x_1, x_2)) = 2 + x_1 + x_2
3.65/1.68	   POL(U2(x_1, x_2)) = 1 + x_1 + 2*x_2
3.65/1.68	   POL(f(x_1)) = 2 + 2*x_1
3.65/1.68	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.65/1.68	
3.65/1.68	   U2(x, x) -> x
3.65/1.68	
3.65/1.68	
3.65/1.68	
3.65/1.68	
3.65/1.68	----------------------------------------
3.65/1.68	
3.65/1.68	(6)
3.65/1.68	Obligation:
3.65/1.68	Q restricted rewrite system:
3.65/1.68	The TRS R consists of the following rules:
3.65/1.68	
3.65/1.68	   f(x) -> U1(x, x)
3.65/1.68	
3.65/1.68	Q is empty.
3.65/1.68	
3.65/1.68	----------------------------------------
3.65/1.68	
3.65/1.68	(7) QTRSRRRProof (EQUIVALENT)
3.65/1.68	Used ordering:
3.65/1.68	Quasi precedence:
3.65/1.68	f_1 > U1_2
3.65/1.68	
3.65/1.68	
3.65/1.68	Status:
3.65/1.68	f_1: multiset status
3.65/1.68	U1_2: multiset status
3.65/1.68	
3.65/1.68	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.65/1.68	
3.65/1.68	   f(x) -> U1(x, x)
3.65/1.68	
3.65/1.68	
3.65/1.68	
3.65/1.68	
3.65/1.68	----------------------------------------
3.65/1.68	
3.65/1.68	(8)
3.65/1.68	Obligation:
3.65/1.68	Q restricted rewrite system:
3.65/1.68	R is empty.
3.65/1.68	Q is empty.
3.65/1.68	
3.65/1.68	----------------------------------------
3.65/1.68	
3.65/1.68	(9) RisEmptyProof (EQUIVALENT)
3.65/1.68	The TRS R is empty. Hence, termination is trivially proven.
3.65/1.68	----------------------------------------
3.65/1.68	
3.65/1.68	(10)
3.65/1.68	YES
3.68/1.71	EOF