3.86/1.84	YES
3.86/1.85	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.86/1.85	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.86/1.85	
3.86/1.85	
3.86/1.85	Termination of the given ETRS could be proven:
3.86/1.85	
3.86/1.85	(0) ETRS
3.86/1.85	(1) RRRPoloETRSProof [EQUIVALENT, 156 ms]
3.86/1.85	(2) ETRS
3.86/1.85	(3) RRRPoloETRSProof [EQUIVALENT, 0 ms]
3.86/1.85	(4) ETRS
3.86/1.85	(5) RisEmptyProof [EQUIVALENT, 0 ms]
3.86/1.85	(6) YES
3.86/1.85	
3.86/1.85	
3.86/1.85	----------------------------------------
3.86/1.85	
3.86/1.85	(0)
3.86/1.85	Obligation:
3.86/1.85	Equational rewrite system:
3.86/1.85	The TRS R consists of the following rules:
3.86/1.85	
3.86/1.85	   zero(S) -> S
3.86/1.85	   plus(x, S) -> x
3.86/1.85	   plus(zero(x), zero(y)) -> zero(plus(x, y))
3.86/1.85	   plus(zero(x), un(y)) -> un(plus(x, y))
3.86/1.85	   plus(un(x), un(y)) -> zero(plus(x, plus(y, un(S))))
3.86/1.85	   times(x, S) -> S
3.86/1.85	   times(x, times(S, z)) -> times(S, z)
3.86/1.85	   times(x, zero(y)) -> zero(times(x, y))
3.86/1.85	   times(x, times(zero(y), z)) -> times(zero(times(x, y)), z)
3.86/1.85	   times(x, un(y)) -> plus(x, zero(times(x, y)))
3.86/1.85	   times(x, times(un(y), z)) -> times(plus(x, zero(times(x, y))), z)
3.86/1.85	
3.86/1.85	The set E consists of the following equations:
3.86/1.85	
3.86/1.85	   plus(x, y) == plus(y, x)
3.86/1.85	   times(x, y) == times(y, x)
3.86/1.85	   plus(plus(x, y), z) == plus(x, plus(y, z))
3.86/1.85	   times(times(x, y), z) == times(x, times(y, z))
3.86/1.85	
3.86/1.85	
3.86/1.85	----------------------------------------
3.86/1.85	
3.86/1.85	(1) RRRPoloETRSProof (EQUIVALENT)
3.86/1.85	The following E TRS is given: Equational rewrite system:
3.86/1.85	The TRS R consists of the following rules:
3.86/1.85	
3.86/1.85	   zero(S) -> S
3.86/1.85	   plus(x, S) -> x
3.86/1.85	   plus(zero(x), zero(y)) -> zero(plus(x, y))
3.86/1.85	   plus(zero(x), un(y)) -> un(plus(x, y))
3.86/1.85	   plus(un(x), un(y)) -> zero(plus(x, plus(y, un(S))))
3.86/1.85	   times(x, S) -> S
3.86/1.85	   times(x, times(S, z)) -> times(S, z)
3.86/1.85	   times(x, zero(y)) -> zero(times(x, y))
3.86/1.85	   times(x, times(zero(y), z)) -> times(zero(times(x, y)), z)
3.86/1.85	   times(x, un(y)) -> plus(x, zero(times(x, y)))
3.86/1.85	   times(x, times(un(y), z)) -> times(plus(x, zero(times(x, y))), z)
3.86/1.85	
3.86/1.85	The set E consists of the following equations:
3.86/1.85	
3.86/1.85	   plus(x, y) == plus(y, x)
3.86/1.85	   times(x, y) == times(y, x)
3.86/1.85	   plus(plus(x, y), z) == plus(x, plus(y, z))
3.86/1.85	   times(times(x, y), z) == times(x, times(y, z))
3.86/1.85	
3.86/1.85	The following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering:
3.86/1.85	
3.86/1.85	   plus(un(x), un(y)) -> zero(plus(x, plus(y, un(S))))
3.86/1.85	   times(x, un(y)) -> plus(x, zero(times(x, y)))
3.86/1.85	   times(x, times(un(y), z)) -> times(plus(x, zero(times(x, y))), z)
3.86/1.85	Used ordering:
3.86/1.85	Polynomial interpretation [POLO]:
3.86/1.85	
3.86/1.85	   POL(S) = 0
3.86/1.85	   POL(plus(x_1, x_2)) = x_1 + x_2
3.86/1.85	   POL(times(x_1, x_2)) = x_1 + x_1*x_2 + x_2
3.86/1.85	   POL(un(x_1)) = 2 + x_1
3.86/1.85	   POL(zero(x_1)) = x_1
3.86/1.85	
3.86/1.85	
3.86/1.85	
3.86/1.85	
3.86/1.85	----------------------------------------
3.86/1.85	
3.86/1.85	(2)
3.86/1.85	Obligation:
3.86/1.85	Equational rewrite system:
3.86/1.85	The TRS R consists of the following rules:
3.86/1.85	
3.86/1.85	   zero(S) -> S
3.86/1.85	   plus(x, S) -> x
3.86/1.85	   plus(zero(x), zero(y)) -> zero(plus(x, y))
3.86/1.85	   plus(zero(x), un(y)) -> un(plus(x, y))
3.86/1.85	   times(x, S) -> S
3.86/1.85	   times(x, times(S, z)) -> times(S, z)
3.86/1.85	   times(x, zero(y)) -> zero(times(x, y))
3.86/1.85	   times(x, times(zero(y), z)) -> times(zero(times(x, y)), z)
3.86/1.85	
3.86/1.85	The set E consists of the following equations:
3.86/1.85	
3.86/1.85	   plus(x, y) == plus(y, x)
3.86/1.85	   times(x, y) == times(y, x)
3.86/1.85	   plus(plus(x, y), z) == plus(x, plus(y, z))
3.86/1.85	   times(times(x, y), z) == times(x, times(y, z))
3.86/1.85	
3.86/1.85	
3.86/1.85	----------------------------------------
3.86/1.85	
3.86/1.85	(3) RRRPoloETRSProof (EQUIVALENT)
3.86/1.85	The following E TRS is given: Equational rewrite system:
3.86/1.85	The TRS R consists of the following rules:
3.86/1.85	
3.86/1.85	   zero(S) -> S
3.86/1.85	   plus(x, S) -> x
3.86/1.85	   plus(zero(x), zero(y)) -> zero(plus(x, y))
3.86/1.85	   plus(zero(x), un(y)) -> un(plus(x, y))
3.86/1.85	   times(x, S) -> S
3.86/1.85	   times(x, times(S, z)) -> times(S, z)
3.86/1.85	   times(x, zero(y)) -> zero(times(x, y))
3.86/1.85	   times(x, times(zero(y), z)) -> times(zero(times(x, y)), z)
3.86/1.85	
3.86/1.85	The set E consists of the following equations:
3.86/1.85	
3.86/1.85	   plus(x, y) == plus(y, x)
3.86/1.85	   times(x, y) == times(y, x)
3.86/1.85	   plus(plus(x, y), z) == plus(x, plus(y, z))
3.86/1.85	   times(times(x, y), z) == times(x, times(y, z))
3.86/1.85	
3.86/1.85	The following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering:
3.86/1.85	
3.86/1.85	   zero(S) -> S
3.86/1.85	   plus(x, S) -> x
3.86/1.85	   plus(zero(x), zero(y)) -> zero(plus(x, y))
3.86/1.85	   plus(zero(x), un(y)) -> un(plus(x, y))
3.86/1.85	   times(x, S) -> S
3.86/1.85	   times(x, times(S, z)) -> times(S, z)
3.86/1.85	   times(x, zero(y)) -> zero(times(x, y))
3.86/1.85	   times(x, times(zero(y), z)) -> times(zero(times(x, y)), z)
3.86/1.85	Used ordering:
3.86/1.85	Polynomial interpretation [POLO]:
3.86/1.85	
3.86/1.85	   POL(S) = 3
3.86/1.85	   POL(plus(x_1, x_2)) = 3 + 3*x_1 + 2*x_1*x_2 + 3*x_2
3.86/1.85	   POL(times(x_1, x_2)) = 2 + 2*x_1 + x_1*x_2 + 2*x_2
3.86/1.85	   POL(un(x_1)) = 2 + 2*x_1
3.86/1.85	   POL(zero(x_1)) = 2 + x_1
3.86/1.85	
3.86/1.85	
3.86/1.85	
3.86/1.85	
3.86/1.85	----------------------------------------
3.86/1.85	
3.86/1.85	(4)
3.86/1.85	Obligation:
3.86/1.85	Equational rewrite system:
3.86/1.85	R is empty.
3.86/1.85	The set E consists of the following equations:
3.86/1.85	
3.86/1.85	   plus(x, y) == plus(y, x)
3.86/1.85	   times(x, y) == times(y, x)
3.86/1.85	   plus(plus(x, y), z) == plus(x, plus(y, z))
3.86/1.85	   times(times(x, y), z) == times(x, times(y, z))
3.86/1.85	
3.86/1.85	
3.86/1.85	----------------------------------------
3.86/1.85	
3.86/1.85	(5) RisEmptyProof (EQUIVALENT)
3.86/1.85	The TRS R is empty. Hence, termination is trivially proven.
3.86/1.85	----------------------------------------
3.86/1.85	
3.86/1.85	(6)
3.86/1.85	YES
3.86/1.88	EOF