3.63/1.72	YES
3.63/1.73	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.63/1.73	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.63/1.73	
3.63/1.73	
3.63/1.73	Quasi decreasingness of the given CTRS could be proven:
3.63/1.73	
3.63/1.73	(0) CTRS
3.63/1.73	(1) CTRSToQTRSProof [SOUND, 0 ms]
3.63/1.73	(2) QTRS
3.63/1.73	(3) QTRSRRRProof [EQUIVALENT, 56 ms]
3.63/1.73	(4) QTRS
3.63/1.73	(5) QTRSRRRProof [EQUIVALENT, 0 ms]
3.63/1.73	(6) QTRS
3.63/1.73	(7) RisEmptyProof [EQUIVALENT, 0 ms]
3.63/1.73	(8) YES
3.63/1.73	
3.63/1.73	
3.63/1.73	----------------------------------------
3.63/1.73	
3.63/1.73	(0)
3.63/1.73	Obligation:
3.63/1.73	Conditional term rewrite system:
3.63/1.73	The TRS R consists of the following rules:
3.63/1.73	
3.63/1.73	   a -> c
3.63/1.73	   a -> d
3.63/1.73	   b -> c
3.63/1.73	   b -> d
3.63/1.73	   g(x, x) -> h(x, x)
3.63/1.73	   h(x, f(x)) -> x
3.63/1.73	
3.63/1.73	The conditional TRS C consists of the following conditional rules:
3.63/1.73	
3.63/1.73	   f(x) -> x <= x -> c
3.63/1.73	
3.63/1.73	
3.63/1.73	----------------------------------------
3.63/1.73	
3.63/1.73	(1) CTRSToQTRSProof (SOUND)
3.63/1.73	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
3.63/1.73	----------------------------------------
3.63/1.73	
3.63/1.73	(2)
3.63/1.73	Obligation:
3.63/1.73	Q restricted rewrite system:
3.63/1.73	The TRS R consists of the following rules:
3.63/1.73	
3.63/1.73	   f(x) -> U1(x, x)
3.63/1.73	   U1(c, x) -> x
3.63/1.73	   a -> c
3.63/1.73	   a -> d
3.63/1.73	   b -> c
3.63/1.73	   b -> d
3.63/1.73	   g(x, x) -> h(x, x)
3.63/1.73	   h(x, f(x)) -> x
3.63/1.73	
3.63/1.73	Q is empty.
3.63/1.73	
3.63/1.73	----------------------------------------
3.63/1.73	
3.63/1.73	(3) QTRSRRRProof (EQUIVALENT)
3.63/1.73	Used ordering:
3.63/1.73	Polynomial interpretation [POLO]:
3.63/1.73	
3.63/1.73	   POL(U1(x_1, x_2)) = x_1 + x_2
3.63/1.73	   POL(a) = 2
3.63/1.73	   POL(b) = 2
3.63/1.73	   POL(c) = 2
3.63/1.73	   POL(d) = 2
3.63/1.73	   POL(f(x_1)) = 2 + 2*x_1
3.63/1.73	   POL(g(x_1, x_2)) = 2*x_1 + 2*x_2
3.63/1.73	   POL(h(x_1, x_2)) = x_1 + x_2
3.63/1.73	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.63/1.73	
3.63/1.73	   f(x) -> U1(x, x)
3.63/1.73	   U1(c, x) -> x
3.63/1.73	   h(x, f(x)) -> x
3.63/1.73	
3.63/1.73	
3.63/1.73	
3.63/1.73	
3.63/1.73	----------------------------------------
3.63/1.73	
3.63/1.73	(4)
3.63/1.73	Obligation:
3.63/1.73	Q restricted rewrite system:
3.63/1.73	The TRS R consists of the following rules:
3.63/1.73	
3.63/1.73	   a -> c
3.63/1.73	   a -> d
3.63/1.73	   b -> c
3.63/1.73	   b -> d
3.63/1.73	   g(x, x) -> h(x, x)
3.63/1.73	
3.63/1.73	Q is empty.
3.63/1.73	
3.63/1.73	----------------------------------------
3.63/1.73	
3.63/1.73	(5) QTRSRRRProof (EQUIVALENT)
3.63/1.73	Used ordering:
3.63/1.73	Knuth-Bendix order [KBO] with precedence:g_2 > h_2 > b > a > d > c
3.63/1.73	
3.63/1.73	and weight map:
3.63/1.73	
3.63/1.73	   a=1
3.63/1.73	   c=1
3.63/1.73	   d=1
3.63/1.73	   b=1
3.63/1.74	   g_2=0
3.63/1.74	   h_2=0
3.63/1.74	
3.63/1.74	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.63/1.74	
3.63/1.74	   a -> c
3.63/1.74	   a -> d
3.63/1.74	   b -> c
3.63/1.74	   b -> d
3.63/1.74	   g(x, x) -> h(x, x)
3.63/1.74	
3.63/1.74	
3.63/1.74	
3.63/1.74	
3.63/1.74	----------------------------------------
3.63/1.74	
3.63/1.74	(6)
3.63/1.74	Obligation:
3.63/1.74	Q restricted rewrite system:
3.63/1.74	R is empty.
3.63/1.74	Q is empty.
3.63/1.74	
3.63/1.74	----------------------------------------
3.63/1.74	
3.63/1.74	(7) RisEmptyProof (EQUIVALENT)
3.63/1.74	The TRS R is empty. Hence, termination is trivially proven.
3.63/1.74	----------------------------------------
3.63/1.74	
3.63/1.74	(8)
3.63/1.74	YES
3.63/1.76	EOF