3.58/1.75	YES
3.58/1.75	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.58/1.75	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.58/1.75	
3.58/1.75	
3.58/1.75	Termination of the given ETRS could be proven:
3.58/1.75	
3.58/1.75	(0) ETRS
3.58/1.75	(1) RRRPoloETRSProof [EQUIVALENT, 138 ms]
3.58/1.75	(2) ETRS
3.58/1.75	(3) RRRPoloETRSProof [EQUIVALENT, 0 ms]
3.58/1.75	(4) ETRS
3.58/1.75	(5) RisEmptyProof [EQUIVALENT, 0 ms]
3.58/1.75	(6) YES
3.58/1.75	
3.58/1.75	
3.58/1.75	----------------------------------------
3.58/1.75	
3.58/1.75	(0)
3.58/1.75	Obligation:
3.58/1.75	Equational rewrite system:
3.58/1.75	The TRS R consists of the following rules:
3.58/1.75	
3.58/1.75	   plus(0, y) -> y
3.58/1.75	   plus(s(x), 0) -> s(x)
3.58/1.75	   plus(s(x), s(y)) -> s(plus(s(x), plus(y, 0)))
3.58/1.75	
3.58/1.75	The set E consists of the following equations:
3.58/1.75	
3.58/1.75	   plus(x, y) == plus(y, x)
3.58/1.75	   plus(plus(x, y), z) == plus(x, plus(y, z))
3.58/1.75	
3.58/1.75	
3.58/1.75	----------------------------------------
3.58/1.75	
3.58/1.75	(1) RRRPoloETRSProof (EQUIVALENT)
3.58/1.75	The following E TRS is given: Equational rewrite system:
3.58/1.75	The TRS R consists of the following rules:
3.58/1.75	
3.58/1.75	   plus(0, y) -> y
3.58/1.75	   plus(s(x), 0) -> s(x)
3.58/1.75	   plus(s(x), s(y)) -> s(plus(s(x), plus(y, 0)))
3.58/1.75	
3.58/1.75	The set E consists of the following equations:
3.58/1.76	
3.58/1.76	   plus(x, y) == plus(y, x)
3.58/1.76	   plus(plus(x, y), z) == plus(x, plus(y, z))
3.58/1.76	
3.58/1.76	The following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering:
3.58/1.76	
3.58/1.76	   plus(s(x), s(y)) -> s(plus(s(x), plus(y, 0)))
3.58/1.76	Used ordering:
3.58/1.76	Polynomial interpretation [POLO]:
3.58/1.76	
3.58/1.76	   POL(0) = 0
3.58/1.76	   POL(plus(x_1, x_2)) = x_1 + x_1*x_2 + x_2
3.58/1.76	   POL(s(x_1)) = 1 + x_1
3.58/1.76	
3.58/1.76	
3.58/1.76	
3.58/1.76	
3.58/1.76	----------------------------------------
3.58/1.76	
3.58/1.76	(2)
3.58/1.76	Obligation:
3.58/1.76	Equational rewrite system:
3.58/1.76	The TRS R consists of the following rules:
3.58/1.76	
3.58/1.76	   plus(0, y) -> y
3.58/1.76	   plus(s(x), 0) -> s(x)
3.58/1.76	
3.58/1.76	The set E consists of the following equations:
3.58/1.76	
3.58/1.76	   plus(x, y) == plus(y, x)
3.58/1.76	   plus(plus(x, y), z) == plus(x, plus(y, z))
3.58/1.76	
3.58/1.76	
3.58/1.76	----------------------------------------
3.58/1.76	
3.58/1.76	(3) RRRPoloETRSProof (EQUIVALENT)
3.58/1.76	The following E TRS is given: Equational rewrite system:
3.58/1.76	The TRS R consists of the following rules:
3.58/1.76	
3.58/1.76	   plus(0, y) -> y
3.58/1.76	   plus(s(x), 0) -> s(x)
3.58/1.76	
3.58/1.76	The set E consists of the following equations:
3.58/1.76	
3.58/1.76	   plus(x, y) == plus(y, x)
3.58/1.76	   plus(plus(x, y), z) == plus(x, plus(y, z))
3.58/1.76	
3.58/1.76	The following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering:
3.58/1.76	
3.58/1.76	   plus(0, y) -> y
3.58/1.76	   plus(s(x), 0) -> s(x)
3.58/1.76	Used ordering:
3.58/1.76	Polynomial interpretation [POLO]:
3.58/1.76	
3.58/1.76	   POL(0) = 0
3.58/1.76	   POL(plus(x_1, x_2)) = 1 + x_1 + x_2
3.58/1.76	   POL(s(x_1)) = 3 + x_1^2
3.58/1.76	
3.58/1.76	
3.58/1.76	
3.58/1.76	
3.58/1.76	----------------------------------------
3.58/1.76	
3.58/1.76	(4)
3.58/1.76	Obligation:
3.58/1.76	Equational rewrite system:
3.58/1.76	R is empty.
3.58/1.76	The set E consists of the following equations:
3.58/1.76	
3.58/1.76	   plus(x, y) == plus(y, x)
3.58/1.76	   plus(plus(x, y), z) == plus(x, plus(y, z))
3.58/1.76	
3.58/1.76	
3.58/1.76	----------------------------------------
3.58/1.76	
3.58/1.76	(5) RisEmptyProof (EQUIVALENT)
3.58/1.76	The TRS R is empty. Hence, termination is trivially proven.
3.58/1.76	----------------------------------------
3.58/1.76	
3.58/1.76	(6)
3.58/1.76	YES
3.77/1.78	EOF