3.43/1.63	YES
3.43/1.64	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.43/1.64	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.43/1.64	
3.43/1.64	
3.43/1.64	Quasi decreasingness of the given CTRS could be proven:
3.43/1.64	
3.43/1.64	(0) CTRS
3.43/1.64	(1) CTRSToQTRSProof [SOUND, 0 ms]
3.43/1.64	(2) QTRS
3.43/1.64	(3) QTRSRRRProof [EQUIVALENT, 59 ms]
3.43/1.64	(4) QTRS
3.43/1.64	(5) QTRSRRRProof [EQUIVALENT, 0 ms]
3.43/1.64	(6) QTRS
3.43/1.64	(7) RisEmptyProof [EQUIVALENT, 0 ms]
3.43/1.64	(8) YES
3.43/1.64	
3.43/1.64	
3.43/1.64	----------------------------------------
3.43/1.64	
3.43/1.64	(0)
3.43/1.64	Obligation:
3.43/1.64	Conditional term rewrite system:
3.43/1.64	The TRS R consists of the following rules:
3.43/1.64	
3.43/1.64	   g(d, x, x) -> A
3.43/1.64	   h(x, x) -> g(x, x, f(k))
3.43/1.64	   a -> c
3.43/1.64	   a -> d
3.43/1.64	   b -> c
3.43/1.64	   b -> d
3.43/1.64	   c -> e
3.43/1.64	   c -> l
3.43/1.64	   k -> l
3.43/1.64	   k -> m
3.43/1.64	   d -> m
3.43/1.64	
3.43/1.64	The conditional TRS C consists of the following conditional rules:
3.43/1.64	
3.43/1.64	   f(x) -> x <= x -> e
3.43/1.64	
3.43/1.64	
3.43/1.64	----------------------------------------
3.43/1.64	
3.43/1.64	(1) CTRSToQTRSProof (SOUND)
3.43/1.64	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
3.43/1.64	----------------------------------------
3.43/1.64	
3.43/1.64	(2)
3.43/1.64	Obligation:
3.43/1.64	Q restricted rewrite system:
3.43/1.64	The TRS R consists of the following rules:
3.43/1.64	
3.43/1.64	   f(x) -> U1(x, x)
3.43/1.64	   U1(e, x) -> x
3.43/1.64	   g(d, x, x) -> A
3.43/1.64	   h(x, x) -> g(x, x, f(k))
3.43/1.64	   a -> c
3.43/1.64	   a -> d
3.43/1.64	   b -> c
3.43/1.64	   b -> d
3.43/1.64	   c -> e
3.43/1.64	   c -> l
3.43/1.64	   k -> l
3.43/1.64	   k -> m
3.43/1.64	   d -> m
3.43/1.64	
3.43/1.64	Q is empty.
3.43/1.64	
3.43/1.64	----------------------------------------
3.43/1.64	
3.43/1.64	(3) QTRSRRRProof (EQUIVALENT)
3.43/1.64	Used ordering:
3.43/1.64	Polynomial interpretation [POLO]:
3.43/1.64	
3.43/1.64	   POL(A) = 2
3.43/1.64	   POL(U1(x_1, x_2)) = x_1 + x_2
3.43/1.64	   POL(a) = 2
3.43/1.64	   POL(b) = 2
3.43/1.64	   POL(c) = 2
3.43/1.64	   POL(d) = 2
3.43/1.64	   POL(e) = 2
3.43/1.64	   POL(f(x_1)) = 2 + 2*x_1
3.43/1.64	   POL(g(x_1, x_2, x_3)) = x_1 + x_2 + x_3
3.43/1.64	   POL(h(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
3.43/1.64	   POL(k) = 0
3.43/1.64	   POL(l) = 0
3.43/1.64	   POL(m) = 0
3.43/1.64	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.43/1.64	
3.43/1.64	   f(x) -> U1(x, x)
3.43/1.64	   U1(e, x) -> x
3.43/1.64	   c -> l
3.43/1.64	   d -> m
3.43/1.64	
3.43/1.64	
3.43/1.64	
3.43/1.64	
3.43/1.64	----------------------------------------
3.43/1.64	
3.43/1.64	(4)
3.43/1.64	Obligation:
3.43/1.64	Q restricted rewrite system:
3.43/1.64	The TRS R consists of the following rules:
3.43/1.64	
3.43/1.64	   g(d, x, x) -> A
3.43/1.64	   h(x, x) -> g(x, x, f(k))
3.43/1.64	   a -> c
3.43/1.64	   a -> d
3.43/1.64	   b -> c
3.43/1.64	   b -> d
3.43/1.64	   c -> e
3.43/1.64	   k -> l
3.43/1.64	   k -> m
3.43/1.64	
3.43/1.64	Q is empty.
3.43/1.64	
3.43/1.64	----------------------------------------
3.43/1.64	
3.43/1.64	(5) QTRSRRRProof (EQUIVALENT)
3.43/1.64	Used ordering:
3.43/1.64	Knuth-Bendix order [KBO] with precedence:k > h_2 > m > l > b > a > g_3 > c > e > f_1 > A > d
3.43/1.64	
3.43/1.64	and weight map:
3.43/1.64	
3.43/1.64	   d=1
3.43/1.64	   A=3
3.43/1.64	   k=1
3.43/1.64	   a=1
3.43/1.64	   c=1
3.43/1.64	   b=1
3.43/1.64	   e=1
3.43/1.64	   l=1
3.43/1.64	   m=1
3.43/1.64	   f_1=1
3.43/1.64	   g_3=0
3.43/1.64	   h_2=2
3.43/1.64	
3.43/1.64	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.43/1.64	
3.43/1.64	   g(d, x, x) -> A
3.43/1.64	   h(x, x) -> g(x, x, f(k))
3.43/1.64	   a -> c
3.43/1.64	   a -> d
3.43/1.64	   b -> c
3.43/1.64	   b -> d
3.43/1.64	   c -> e
3.43/1.64	   k -> l
3.43/1.64	   k -> m
3.43/1.64	
3.43/1.64	
3.43/1.64	
3.43/1.64	
3.43/1.64	----------------------------------------
3.43/1.64	
3.43/1.64	(6)
3.43/1.64	Obligation:
3.43/1.64	Q restricted rewrite system:
3.43/1.64	R is empty.
3.43/1.64	Q is empty.
3.43/1.64	
3.43/1.64	----------------------------------------
3.43/1.64	
3.43/1.64	(7) RisEmptyProof (EQUIVALENT)
3.43/1.64	The TRS R is empty. Hence, termination is trivially proven.
3.43/1.64	----------------------------------------
3.43/1.64	
3.43/1.64	(8)
3.43/1.64	YES
3.43/1.66	EOF