3.75/1.69	YES
3.75/1.69	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.75/1.69	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.75/1.69	
3.75/1.69	
3.75/1.69	Quasi decreasingness of the given CTRS could be proven:
3.75/1.69	
3.75/1.69	(0) CTRS
3.75/1.69	(1) CTRSToQTRSProof [SOUND, 0 ms]
3.75/1.69	(2) QTRS
3.75/1.69	(3) QTRSRRRProof [EQUIVALENT, 54 ms]
3.75/1.69	(4) QTRS
3.75/1.69	(5) QTRSRRRProof [EQUIVALENT, 0 ms]
3.75/1.69	(6) QTRS
3.75/1.69	(7) RisEmptyProof [EQUIVALENT, 0 ms]
3.75/1.69	(8) YES
3.75/1.69	
3.75/1.69	
3.75/1.69	----------------------------------------
3.75/1.69	
3.75/1.69	(0)
3.75/1.69	Obligation:
3.75/1.69	Conditional term rewrite system:
3.75/1.69	The TRS R consists of the following rules:
3.75/1.69	
3.75/1.69	   g(s(x)) -> x
3.75/1.69	   h(s(x)) -> x
3.75/1.69	
3.75/1.69	The conditional TRS C consists of the following conditional rules:
3.75/1.69	
3.75/1.69	   f(x, y) -> g(x) <= c(g(x)) -> c(a)
3.75/1.69	   f(x, y) -> h(x) <= c(h(x)) -> c(a)
3.75/1.69	
3.75/1.69	
3.75/1.69	----------------------------------------
3.75/1.69	
3.75/1.69	(1) CTRSToQTRSProof (SOUND)
3.75/1.69	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
3.75/1.69	----------------------------------------
3.75/1.69	
3.75/1.69	(2)
3.75/1.69	Obligation:
3.75/1.69	Q restricted rewrite system:
3.75/1.69	The TRS R consists of the following rules:
3.75/1.69	
3.75/1.69	   f(x, y) -> U1(c(g(x)), x)
3.75/1.70	   U1(c(a), x) -> g(x)
3.75/1.70	   f(x, y) -> U2(c(h(x)), x)
3.75/1.70	   U2(c(a), x) -> h(x)
3.75/1.70	   g(s(x)) -> x
3.75/1.70	   h(s(x)) -> x
3.75/1.70	
3.75/1.70	Q is empty.
3.75/1.70	
3.75/1.70	----------------------------------------
3.75/1.70	
3.75/1.70	(3) QTRSRRRProof (EQUIVALENT)
3.75/1.70	Used ordering:
3.75/1.70	Polynomial interpretation [POLO]:
3.75/1.70	
3.75/1.70	   POL(U1(x_1, x_2)) = x_1 + x_2
3.75/1.70	   POL(U2(x_1, x_2)) = x_1 + x_2
3.75/1.70	   POL(a) = 0
3.75/1.70	   POL(c(x_1)) = x_1
3.75/1.70	   POL(f(x_1, x_2)) = 1 + 2*x_1 + x_2
3.75/1.70	   POL(g(x_1)) = x_1
3.75/1.70	   POL(h(x_1)) = x_1
3.75/1.70	   POL(s(x_1)) = 2 + x_1
3.75/1.70	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.75/1.70	
3.75/1.70	   f(x, y) -> U1(c(g(x)), x)
3.75/1.70	   f(x, y) -> U2(c(h(x)), x)
3.75/1.70	   g(s(x)) -> x
3.75/1.70	   h(s(x)) -> x
3.75/1.70	
3.75/1.70	
3.75/1.70	
3.75/1.70	
3.75/1.70	----------------------------------------
3.75/1.70	
3.75/1.70	(4)
3.75/1.70	Obligation:
3.75/1.70	Q restricted rewrite system:
3.75/1.70	The TRS R consists of the following rules:
3.75/1.70	
3.75/1.70	   U1(c(a), x) -> g(x)
3.75/1.70	   U2(c(a), x) -> h(x)
3.75/1.70	
3.75/1.70	Q is empty.
3.75/1.70	
3.75/1.70	----------------------------------------
3.75/1.70	
3.75/1.70	(5) QTRSRRRProof (EQUIVALENT)
3.75/1.70	Used ordering:
3.75/1.70	Knuth-Bendix order [KBO] with precedence:U2_2 > h_1 > U1_2 > g_1 > a > c_1
3.75/1.70	
3.75/1.70	and weight map:
3.75/1.70	
3.75/1.70	   a=1
3.75/1.70	   c_1=1
3.75/1.70	   g_1=2
3.75/1.70	   h_1=2
3.75/1.70	   U1_2=0
3.75/1.70	   U2_2=0
3.75/1.70	
3.75/1.70	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.75/1.70	
3.75/1.70	   U1(c(a), x) -> g(x)
3.75/1.70	   U2(c(a), x) -> h(x)
3.75/1.70	
3.75/1.70	
3.75/1.70	
3.75/1.70	
3.75/1.70	----------------------------------------
3.75/1.70	
3.75/1.70	(6)
3.75/1.70	Obligation:
3.75/1.70	Q restricted rewrite system:
3.75/1.70	R is empty.
3.75/1.70	Q is empty.
3.75/1.70	
3.75/1.70	----------------------------------------
3.75/1.70	
3.75/1.70	(7) RisEmptyProof (EQUIVALENT)
3.75/1.70	The TRS R is empty. Hence, termination is trivially proven.
3.75/1.70	----------------------------------------
3.75/1.70	
3.75/1.70	(8)
3.75/1.70	YES
3.90/1.72	EOF