3.37/1.69	YES
3.37/1.70	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.37/1.70	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.37/1.70	
3.37/1.70	
3.37/1.70	Quasi decreasingness of the given CTRS could be proven:
3.37/1.70	
3.37/1.70	(0) CTRS
3.37/1.70	(1) CTRSToQTRSProof [SOUND, 0 ms]
3.37/1.70	(2) QTRS
3.37/1.70	(3) QTRSRRRProof [EQUIVALENT, 60 ms]
3.37/1.70	(4) QTRS
3.37/1.70	(5) QTRSRRRProof [EQUIVALENT, 0 ms]
3.37/1.70	(6) QTRS
3.37/1.70	(7) QTRSRRRProof [EQUIVALENT, 0 ms]
3.37/1.70	(8) QTRS
3.37/1.70	(9) RisEmptyProof [EQUIVALENT, 0 ms]
3.37/1.70	(10) YES
3.37/1.70	
3.37/1.70	
3.37/1.70	----------------------------------------
3.37/1.70	
3.37/1.70	(0)
3.37/1.70	Obligation:
3.37/1.70	Conditional term rewrite system:
3.37/1.70	The TRS R consists of the following rules:
3.37/1.70	
3.37/1.70	   add(x, 0) -> x
3.37/1.70	
3.37/1.70	The conditional TRS C consists of the following conditional rules:
3.37/1.70	
3.37/1.70	   add(x, s(y)) -> s(z) <= add(x, y) -> z
3.37/1.70	
3.37/1.70	
3.37/1.70	----------------------------------------
3.37/1.70	
3.37/1.70	(1) CTRSToQTRSProof (SOUND)
3.37/1.70	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
3.37/1.70	----------------------------------------
3.37/1.70	
3.37/1.70	(2)
3.37/1.70	Obligation:
3.37/1.70	Q restricted rewrite system:
3.37/1.70	The TRS R consists of the following rules:
3.37/1.70	
3.37/1.70	   add(x, s(y)) -> U1(add(x, y))
3.37/1.70	   U1(z) -> s(z)
3.37/1.70	   add(x, 0) -> x
3.37/1.70	
3.37/1.70	Q is empty.
3.37/1.70	
3.37/1.70	----------------------------------------
3.37/1.70	
3.37/1.70	(3) QTRSRRRProof (EQUIVALENT)
3.37/1.70	Used ordering:
3.37/1.70	Polynomial interpretation [POLO]:
3.37/1.70	
3.37/1.70	   POL(0) = 2
3.37/1.70	   POL(U1(x_1)) = x_1
3.37/1.70	   POL(add(x_1, x_2)) = 2*x_1 + 2*x_2
3.37/1.70	   POL(s(x_1)) = x_1
3.37/1.70	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.37/1.70	
3.37/1.70	   add(x, 0) -> x
3.37/1.70	
3.37/1.70	
3.37/1.70	
3.37/1.70	
3.37/1.70	----------------------------------------
3.37/1.70	
3.37/1.70	(4)
3.37/1.70	Obligation:
3.37/1.70	Q restricted rewrite system:
3.37/1.70	The TRS R consists of the following rules:
3.37/1.70	
3.37/1.70	   add(x, s(y)) -> U1(add(x, y))
3.37/1.70	   U1(z) -> s(z)
3.37/1.70	
3.37/1.70	Q is empty.
3.37/1.70	
3.37/1.70	----------------------------------------
3.37/1.70	
3.37/1.70	(5) QTRSRRRProof (EQUIVALENT)
3.37/1.70	Used ordering:
3.37/1.70	Polynomial interpretation [POLO]:
3.37/1.70	
3.37/1.70	   POL(U1(x_1)) = 2 + x_1
3.37/1.70	   POL(add(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
3.37/1.70	   POL(s(x_1)) = 1 + x_1
3.37/1.70	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.37/1.70	
3.37/1.70	   U1(z) -> s(z)
3.37/1.70	
3.37/1.70	
3.37/1.70	
3.37/1.70	
3.37/1.70	----------------------------------------
3.37/1.70	
3.37/1.70	(6)
3.37/1.70	Obligation:
3.37/1.70	Q restricted rewrite system:
3.37/1.70	The TRS R consists of the following rules:
3.37/1.70	
3.37/1.70	   add(x, s(y)) -> U1(add(x, y))
3.37/1.70	
3.37/1.70	Q is empty.
3.37/1.70	
3.37/1.70	----------------------------------------
3.37/1.70	
3.37/1.70	(7) QTRSRRRProof (EQUIVALENT)
3.37/1.70	Used ordering:
3.37/1.70	Knuth-Bendix order [KBO] with precedence:add_2 > U1_1 > s_1
3.37/1.70	
3.37/1.70	and weight map:
3.37/1.70	
3.37/1.70	   s_1=1
3.37/1.70	   U1_1=1
3.37/1.70	   add_2=0
3.37/1.70	
3.37/1.70	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.37/1.70	
3.37/1.70	   add(x, s(y)) -> U1(add(x, y))
3.37/1.70	
3.37/1.70	
3.37/1.70	
3.37/1.70	
3.37/1.70	----------------------------------------
3.37/1.70	
3.37/1.70	(8)
3.37/1.70	Obligation:
3.37/1.70	Q restricted rewrite system:
3.37/1.70	R is empty.
3.37/1.70	Q is empty.
3.37/1.70	
3.37/1.70	----------------------------------------
3.37/1.70	
3.37/1.70	(9) RisEmptyProof (EQUIVALENT)
3.37/1.70	The TRS R is empty. Hence, termination is trivially proven.
3.37/1.70	----------------------------------------
3.37/1.70	
3.37/1.70	(10)
3.37/1.70	YES
3.58/1.73	EOF