3.48/2.92	YES
3.48/2.93	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.48/2.93	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.48/2.93	
3.48/2.93	
3.48/2.93	Quasi decreasingness of the given CTRS could be proven:
3.48/2.93	
3.48/2.93	(0) CTRS
3.48/2.93	(1) CTRSToQTRSProof [SOUND, 0 ms]
3.48/2.93	(2) QTRS
3.48/2.93	(3) QTRSRRRProof [EQUIVALENT, 56 ms]
3.48/2.93	(4) QTRS
3.48/2.93	(5) QTRSRRRProof [EQUIVALENT, 0 ms]
3.48/2.93	(6) QTRS
3.48/2.93	(7) RisEmptyProof [EQUIVALENT, 0 ms]
3.48/2.93	(8) YES
3.48/2.93	
3.48/2.93	
3.48/2.93	----------------------------------------
3.48/2.93	
3.48/2.93	(0)
3.48/2.93	Obligation:
3.48/2.93	Conditional term rewrite system:
3.48/2.93	The TRS R is empty.
3.48/2.93	The conditional TRS C consists of the following conditional rules:
3.48/2.93	
3.48/2.93	   f(x, y) -> x <= g(x) -> z, g(y) -> z
3.48/2.93	   g(x) -> c <= d -> c
3.48/2.93	
3.48/2.93	
3.48/2.93	----------------------------------------
3.48/2.93	
3.48/2.93	(1) CTRSToQTRSProof (SOUND)
3.48/2.93	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
3.48/2.93	----------------------------------------
3.48/2.93	
3.48/2.93	(2)
3.48/2.93	Obligation:
3.48/2.93	Q restricted rewrite system:
3.48/2.93	The TRS R consists of the following rules:
3.48/2.93	
3.48/2.93	   f(x, y) -> U1(g(x), x, y)
3.48/2.93	   U1(z, x, y) -> U2(g(y), x)
3.48/2.93	   U2(z, x) -> x
3.48/2.93	   g(x) -> U3(d)
3.48/2.93	   U3(c) -> c
3.48/2.93	
3.48/2.93	Q is empty.
3.48/2.93	
3.48/2.93	----------------------------------------
3.48/2.93	
3.48/2.93	(3) QTRSRRRProof (EQUIVALENT)
3.48/2.93	Used ordering:
3.48/2.93	Polynomial interpretation [POLO]:
3.48/2.93	
3.48/2.93	   POL(U1(x_1, x_2, x_3)) = x_1 + x_2 + 2*x_3
3.48/2.93	   POL(U2(x_1, x_2)) = x_1 + x_2
3.48/2.93	   POL(U3(x_1)) = 2*x_1
3.48/2.93	   POL(c) = 2
3.48/2.93	   POL(d) = 0
3.48/2.93	   POL(f(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
3.48/2.93	   POL(g(x_1)) = x_1
3.48/2.93	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.48/2.93	
3.48/2.93	   f(x, y) -> U1(g(x), x, y)
3.48/2.93	   U3(c) -> c
3.48/2.93	
3.48/2.93	
3.48/2.93	
3.48/2.93	
3.48/2.93	----------------------------------------
3.48/2.93	
3.48/2.93	(4)
3.48/2.93	Obligation:
3.48/2.93	Q restricted rewrite system:
3.48/2.93	The TRS R consists of the following rules:
3.48/2.93	
3.48/2.93	   U1(z, x, y) -> U2(g(y), x)
3.48/2.93	   U2(z, x) -> x
3.48/2.93	   g(x) -> U3(d)
3.48/2.93	
3.48/2.93	Q is empty.
3.48/2.93	
3.48/2.93	----------------------------------------
3.48/2.93	
3.48/2.93	(5) QTRSRRRProof (EQUIVALENT)
3.48/2.93	Used ordering:
3.48/2.93	Knuth-Bendix order [KBO] with precedence:d > g_1 > U3_1 > U1_3 > U2_2
3.48/2.93	
3.48/2.93	and weight map:
3.48/2.93	
3.48/2.93	   d=2
3.48/2.93	   g_1=2
3.48/2.93	   U3_1=1
3.48/2.93	   U1_3=1
3.48/2.93	   U2_2=0
3.48/2.93	
3.48/2.93	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.48/2.93	
3.48/2.93	   U1(z, x, y) -> U2(g(y), x)
3.48/2.93	   U2(z, x) -> x
3.48/2.93	   g(x) -> U3(d)
3.48/2.93	
3.48/2.93	
3.48/2.93	
3.48/2.93	
3.48/2.93	----------------------------------------
3.48/2.93	
3.48/2.93	(6)
3.48/2.93	Obligation:
3.48/2.93	Q restricted rewrite system:
3.48/2.93	R is empty.
3.48/2.93	Q is empty.
3.48/2.93	
3.48/2.93	----------------------------------------
3.48/2.93	
3.48/2.93	(7) RisEmptyProof (EQUIVALENT)
3.48/2.93	The TRS R is empty. Hence, termination is trivially proven.
3.48/2.93	----------------------------------------
3.48/2.93	
3.48/2.93	(8)
3.48/2.93	YES
3.48/2.96	EOF