3.39/1.70	YES
3.65/1.71	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.65/1.71	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.65/1.71	
3.65/1.71	
3.65/1.71	Quasi decreasingness of the given CTRS could be proven:
3.65/1.71	
3.65/1.71	(0) CTRS
3.65/1.71	(1) CTRSToQTRSProof [SOUND, 0 ms]
3.65/1.71	(2) QTRS
3.65/1.71	(3) QTRSRRRProof [EQUIVALENT, 56 ms]
3.65/1.71	(4) QTRS
3.65/1.71	(5) QTRSRRRProof [EQUIVALENT, 0 ms]
3.65/1.71	(6) QTRS
3.65/1.71	(7) RisEmptyProof [EQUIVALENT, 0 ms]
3.65/1.71	(8) YES
3.65/1.71	
3.65/1.71	
3.65/1.71	----------------------------------------
3.65/1.71	
3.65/1.71	(0)
3.65/1.71	Obligation:
3.65/1.71	Conditional term rewrite system:
3.65/1.71	The TRS R is empty.
3.65/1.71	The conditional TRS C consists of the following conditional rules:
3.65/1.71	
3.65/1.71	   f(x, x') -> g(y, y) <= x -> y, x' -> y
3.65/1.71	   h(x, x', x'') -> c <= x -> y, x' -> y, x'' -> y
3.65/1.71	
3.65/1.71	
3.65/1.71	----------------------------------------
3.65/1.71	
3.65/1.71	(1) CTRSToQTRSProof (SOUND)
3.65/1.71	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
3.65/1.71	----------------------------------------
3.65/1.71	
3.65/1.71	(2)
3.65/1.71	Obligation:
3.65/1.71	Q restricted rewrite system:
3.65/1.71	The TRS R consists of the following rules:
3.65/1.71	
3.65/1.71	   f(x, x') -> U1(x, x')
3.65/1.71	   U1(y, x') -> U2(x', y)
3.65/1.71	   U2(y, y) -> g(y, y)
3.65/1.71	   h(x, x', x'') -> U3(x, x', x'')
3.65/1.71	   U3(y, x', x'') -> U4(x', x'')
3.65/1.71	   U4(y, x'') -> U5(x'')
3.65/1.71	   U5(y) -> c
3.65/1.71	
3.65/1.71	Q is empty.
3.65/1.71	
3.65/1.71	----------------------------------------
3.65/1.71	
3.65/1.71	(3) QTRSRRRProof (EQUIVALENT)
3.65/1.71	Used ordering:
3.65/1.71	Polynomial interpretation [POLO]:
3.65/1.71	
3.65/1.71	   POL(U1(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
3.65/1.71	   POL(U2(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
3.65/1.71	   POL(U3(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + 2*x_3
3.65/1.71	   POL(U4(x_1, x_2)) = 2*x_1 + 2*x_2
3.65/1.71	   POL(U5(x_1)) = 2*x_1
3.65/1.71	   POL(c) = 0
3.65/1.71	   POL(f(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
3.65/1.71	   POL(g(x_1, x_2)) = x_1 + x_2
3.65/1.71	   POL(h(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + 2*x_3
3.65/1.71	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.65/1.71	
3.65/1.71	   U2(y, y) -> g(y, y)
3.65/1.71	   U3(y, x', x'') -> U4(x', x'')
3.65/1.71	
3.65/1.71	
3.65/1.71	
3.65/1.71	
3.65/1.71	----------------------------------------
3.65/1.71	
3.65/1.71	(4)
3.65/1.71	Obligation:
3.65/1.71	Q restricted rewrite system:
3.65/1.71	The TRS R consists of the following rules:
3.65/1.71	
3.65/1.71	   f(x, x') -> U1(x, x')
3.65/1.71	   U1(y, x') -> U2(x', y)
3.65/1.71	   h(x, x', x'') -> U3(x, x', x'')
3.65/1.71	   U4(y, x'') -> U5(x'')
3.65/1.71	   U5(y) -> c
3.65/1.71	
3.65/1.71	Q is empty.
3.65/1.71	
3.65/1.71	----------------------------------------
3.65/1.71	
3.65/1.71	(5) QTRSRRRProof (EQUIVALENT)
3.65/1.71	Used ordering:
3.65/1.71	Knuth-Bendix order [KBO] with precedence:f_2 > U4_2 > U5_1 > c > h_3 > U3_3 > U1_2 > U2_2
3.65/1.71	
3.65/1.71	and weight map:
3.65/1.71	
3.65/1.71	   c=2
3.65/1.71	   U5_1=1
3.65/1.71	   f_2=0
3.65/1.71	   U1_2=0
3.65/1.71	   U2_2=0
3.65/1.71	   h_3=0
3.65/1.71	   U3_3=0
3.65/1.71	   U4_2=0
3.65/1.71	
3.65/1.71	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.65/1.71	
3.65/1.71	   f(x, x') -> U1(x, x')
3.65/1.71	   U1(y, x') -> U2(x', y)
3.65/1.71	   h(x, x', x'') -> U3(x, x', x'')
3.65/1.71	   U4(y, x'') -> U5(x'')
3.65/1.71	   U5(y) -> c
3.65/1.71	
3.65/1.71	
3.65/1.71	
3.65/1.71	
3.65/1.71	----------------------------------------
3.65/1.71	
3.65/1.71	(6)
3.65/1.71	Obligation:
3.65/1.71	Q restricted rewrite system:
3.65/1.71	R is empty.
3.65/1.71	Q is empty.
3.65/1.71	
3.65/1.71	----------------------------------------
3.65/1.71	
3.65/1.71	(7) RisEmptyProof (EQUIVALENT)
3.65/1.71	The TRS R is empty. Hence, termination is trivially proven.
3.65/1.71	----------------------------------------
3.65/1.71	
3.65/1.71	(8)
3.65/1.71	YES
3.65/1.74	EOF