3.51/1.65	YES
3.51/1.66	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.51/1.66	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.51/1.66	
3.51/1.66	
3.51/1.66	Quasi decreasingness of the given CTRS could be proven:
3.51/1.66	
3.51/1.66	(0) CTRS
3.51/1.66	(1) CTRSToQTRSProof [SOUND, 0 ms]
3.51/1.66	(2) QTRS
3.51/1.66	(3) QTRSRRRProof [EQUIVALENT, 67 ms]
3.51/1.66	(4) QTRS
3.51/1.66	(5) QTRSRRRProof [EQUIVALENT, 0 ms]
3.51/1.66	(6) QTRS
3.51/1.66	(7) RisEmptyProof [EQUIVALENT, 0 ms]
3.51/1.66	(8) YES
3.51/1.66	
3.51/1.66	
3.51/1.66	----------------------------------------
3.51/1.66	
3.51/1.66	(0)
3.51/1.66	Obligation:
3.51/1.66	Conditional term rewrite system:
3.51/1.66	The TRS R consists of the following rules:
3.51/1.66	
3.51/1.66	   plus(0, y) -> y
3.51/1.66	   plus(s(x), y) -> plus(x, s(y))
3.51/1.66	
3.51/1.66	The conditional TRS C consists of the following conditional rules:
3.51/1.66	
3.51/1.66	   f(x, y) -> plus(y, z) <= plus(x, y) -> plus(z, z)
3.51/1.66	
3.51/1.66	
3.51/1.66	----------------------------------------
3.51/1.66	
3.51/1.66	(1) CTRSToQTRSProof (SOUND)
3.51/1.66	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
3.51/1.66	----------------------------------------
3.51/1.66	
3.51/1.66	(2)
3.51/1.66	Obligation:
3.51/1.66	Q restricted rewrite system:
3.51/1.66	The TRS R consists of the following rules:
3.51/1.66	
3.51/1.66	   f(x, y) -> U1(plus(x, y), y)
3.51/1.66	   U1(plus(z, z), y) -> plus(y, z)
3.51/1.66	   plus(0, y) -> y
3.51/1.66	   plus(s(x), y) -> plus(x, s(y))
3.51/1.66	
3.51/1.66	Q is empty.
3.51/1.66	
3.51/1.66	----------------------------------------
3.51/1.66	
3.51/1.66	(3) QTRSRRRProof (EQUIVALENT)
3.51/1.66	Used ordering:
3.51/1.66	Polynomial interpretation [POLO]:
3.51/1.66	
3.51/1.66	   POL(0) = 2
3.51/1.66	   POL(U1(x_1, x_2)) = x_1 + x_2
3.51/1.66	   POL(f(x_1, x_2)) = 2 + x_1 + 2*x_2
3.51/1.66	   POL(plus(x_1, x_2)) = x_1 + x_2
3.51/1.66	   POL(s(x_1)) = 2 + x_1
3.51/1.66	   POL(z) = 0
3.51/1.66	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.51/1.66	
3.51/1.66	   f(x, y) -> U1(plus(x, y), y)
3.51/1.66	   plus(0, y) -> y
3.51/1.66	
3.51/1.66	
3.51/1.66	
3.51/1.66	
3.51/1.66	----------------------------------------
3.51/1.66	
3.51/1.66	(4)
3.51/1.66	Obligation:
3.51/1.66	Q restricted rewrite system:
3.51/1.66	The TRS R consists of the following rules:
3.51/1.66	
3.51/1.66	   U1(plus(z, z), y) -> plus(y, z)
3.51/1.66	   plus(s(x), y) -> plus(x, s(y))
3.51/1.66	
3.51/1.66	Q is empty.
3.51/1.66	
3.51/1.66	----------------------------------------
3.51/1.66	
3.51/1.66	(5) QTRSRRRProof (EQUIVALENT)
3.51/1.66	Used ordering:
3.51/1.66	Polynomial interpretation [POLO]:
3.51/1.66	
3.51/1.66	   POL(U1(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
3.51/1.66	   POL(plus(x_1, x_2)) = 2 + 2*x_1 + x_2
3.51/1.66	   POL(s(x_1)) = 1 + x_1
3.51/1.66	   POL(z) = 2
3.51/1.66	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.51/1.66	
3.51/1.66	   U1(plus(z, z), y) -> plus(y, z)
3.51/1.66	   plus(s(x), y) -> plus(x, s(y))
3.51/1.66	
3.51/1.66	
3.51/1.66	
3.51/1.66	
3.51/1.66	----------------------------------------
3.51/1.66	
3.51/1.66	(6)
3.51/1.66	Obligation:
3.51/1.66	Q restricted rewrite system:
3.51/1.66	R is empty.
3.51/1.66	Q is empty.
3.51/1.66	
3.51/1.66	----------------------------------------
3.51/1.66	
3.51/1.66	(7) RisEmptyProof (EQUIVALENT)
3.51/1.66	The TRS R is empty. Hence, termination is trivially proven.
3.51/1.66	----------------------------------------
3.51/1.66	
3.51/1.66	(8)
3.51/1.66	YES
3.51/1.69	EOF