3.81/1.77	YES
3.81/1.78	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.81/1.78	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.81/1.78	
3.81/1.78	
3.81/1.78	Termination of the given ETRS could be proven:
3.81/1.78	
3.81/1.78	(0) ETRS
3.81/1.78	(1) RRRPoloETRSProof [EQUIVALENT, 138 ms]
3.81/1.78	(2) ETRS
3.81/1.78	(3) RRRPoloETRSProof [EQUIVALENT, 0 ms]
3.81/1.78	(4) ETRS
3.81/1.78	(5) RRRPoloETRSProof [EQUIVALENT, 0 ms]
3.81/1.78	(6) ETRS
3.81/1.78	(7) RisEmptyProof [EQUIVALENT, 0 ms]
3.81/1.78	(8) YES
3.81/1.78	
3.81/1.78	
3.81/1.78	----------------------------------------
3.81/1.78	
3.81/1.78	(0)
3.81/1.78	Obligation:
3.81/1.78	Equational rewrite system:
3.81/1.78	The TRS R consists of the following rules:
3.81/1.78	
3.81/1.78	   0(S) -> S
3.81/1.78	   plus(S, x) -> x
3.81/1.78	   plus(0(x), 0(y)) -> 0(plus(x, y))
3.81/1.78	   plus(0(x), 1(y)) -> 1(plus(x, y))
3.81/1.78	   plus(1(x), 1(y)) -> 0(plus(x, plus(y, 1(S))))
3.81/1.78	   times(S, x) -> S
3.81/1.78	   times(0(x), y) -> 0(times(x, y))
3.81/1.78	   times(1(x), y) -> plus(0(times(x, y)), y)
3.81/1.78	
3.81/1.78	The set E consists of the following equations:
3.81/1.78	
3.81/1.78	   plus(x, y) == plus(y, x)
3.81/1.78	   times(x, y) == times(y, x)
3.81/1.78	   plus(plus(x, y), z) == plus(x, plus(y, z))
3.81/1.78	   times(times(x, y), z) == times(x, times(y, z))
3.81/1.78	
3.81/1.78	
3.81/1.78	----------------------------------------
3.81/1.78	
3.81/1.78	(1) RRRPoloETRSProof (EQUIVALENT)
3.81/1.78	The following E TRS is given: Equational rewrite system:
3.81/1.78	The TRS R consists of the following rules:
3.81/1.78	
3.81/1.78	   0(S) -> S
3.81/1.78	   plus(S, x) -> x
3.81/1.78	   plus(0(x), 0(y)) -> 0(plus(x, y))
3.81/1.78	   plus(0(x), 1(y)) -> 1(plus(x, y))
3.81/1.78	   plus(1(x), 1(y)) -> 0(plus(x, plus(y, 1(S))))
3.81/1.78	   times(S, x) -> S
3.81/1.78	   times(0(x), y) -> 0(times(x, y))
3.81/1.78	   times(1(x), y) -> plus(0(times(x, y)), y)
3.81/1.78	
3.81/1.78	The set E consists of the following equations:
3.81/1.78	
3.81/1.78	   plus(x, y) == plus(y, x)
3.81/1.78	   times(x, y) == times(y, x)
3.81/1.78	   plus(plus(x, y), z) == plus(x, plus(y, z))
3.81/1.78	   times(times(x, y), z) == times(x, times(y, z))
3.81/1.78	
3.81/1.78	The following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering:
3.81/1.78	
3.81/1.78	   plus(S, x) -> x
3.81/1.78	   times(S, x) -> S
3.81/1.78	   times(1(x), y) -> plus(0(times(x, y)), y)
3.81/1.78	Used ordering:
3.81/1.78	Polynomial interpretation [POLO]:
3.81/1.78	
3.81/1.78	   POL(0(x_1)) = x_1
3.81/1.78	   POL(1(x_1)) = 1 + x_1
3.81/1.78	   POL(S) = 1
3.81/1.78	   POL(plus(x_1, x_2)) = x_1 + x_2
3.81/1.78	   POL(times(x_1, x_2)) = 3 + 3*x_1 + 2*x_1*x_2 + 3*x_2
3.81/1.78	
3.81/1.78	
3.81/1.78	
3.81/1.78	
3.81/1.78	----------------------------------------
3.81/1.78	
3.81/1.78	(2)
3.81/1.78	Obligation:
3.81/1.78	Equational rewrite system:
3.81/1.78	The TRS R consists of the following rules:
3.81/1.78	
3.81/1.78	   0(S) -> S
3.81/1.78	   plus(0(x), 0(y)) -> 0(plus(x, y))
3.81/1.78	   plus(0(x), 1(y)) -> 1(plus(x, y))
3.81/1.78	   plus(1(x), 1(y)) -> 0(plus(x, plus(y, 1(S))))
3.81/1.78	   times(0(x), y) -> 0(times(x, y))
3.81/1.78	
3.81/1.78	The set E consists of the following equations:
3.81/1.78	
3.81/1.78	   plus(x, y) == plus(y, x)
3.81/1.78	   times(x, y) == times(y, x)
3.81/1.78	   plus(plus(x, y), z) == plus(x, plus(y, z))
3.81/1.78	   times(times(x, y), z) == times(x, times(y, z))
3.81/1.78	
3.81/1.78	
3.81/1.78	----------------------------------------
3.81/1.78	
3.81/1.78	(3) RRRPoloETRSProof (EQUIVALENT)
3.81/1.78	The following E TRS is given: Equational rewrite system:
3.81/1.78	The TRS R consists of the following rules:
3.81/1.78	
3.81/1.78	   0(S) -> S
3.81/1.78	   plus(0(x), 0(y)) -> 0(plus(x, y))
3.81/1.78	   plus(0(x), 1(y)) -> 1(plus(x, y))
3.81/1.78	   plus(1(x), 1(y)) -> 0(plus(x, plus(y, 1(S))))
3.81/1.78	   times(0(x), y) -> 0(times(x, y))
3.81/1.78	
3.81/1.78	The set E consists of the following equations:
3.81/1.78	
3.81/1.78	   plus(x, y) == plus(y, x)
3.81/1.78	   times(x, y) == times(y, x)
3.81/1.78	   plus(plus(x, y), z) == plus(x, plus(y, z))
3.81/1.78	   times(times(x, y), z) == times(x, times(y, z))
3.81/1.78	
3.81/1.78	The following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering:
3.81/1.78	
3.81/1.78	   0(S) -> S
3.81/1.78	   plus(0(x), 0(y)) -> 0(plus(x, y))
3.81/1.78	   plus(0(x), 1(y)) -> 1(plus(x, y))
3.81/1.78	   plus(1(x), 1(y)) -> 0(plus(x, plus(y, 1(S))))
3.81/1.78	Used ordering:
3.81/1.78	Polynomial interpretation [POLO]:
3.81/1.78	
3.81/1.78	   POL(0(x_1)) = 1 + x_1
3.81/1.78	   POL(1(x_1)) = 2 + x_1
3.81/1.78	   POL(S) = 0
3.81/1.78	   POL(plus(x_1, x_2)) = x_1 + x_2
3.81/1.78	   POL(times(x_1, x_2)) = x_1 + x_2
3.81/1.78	
3.81/1.78	
3.81/1.78	
3.81/1.78	
3.81/1.78	----------------------------------------
3.81/1.78	
3.81/1.78	(4)
3.81/1.78	Obligation:
3.81/1.78	Equational rewrite system:
3.81/1.78	The TRS R consists of the following rules:
3.81/1.78	
3.81/1.78	   times(0(x), y) -> 0(times(x, y))
3.81/1.78	
3.81/1.78	The set E consists of the following equations:
3.81/1.78	
3.81/1.78	   plus(x, y) == plus(y, x)
3.81/1.78	   times(x, y) == times(y, x)
3.81/1.78	   plus(plus(x, y), z) == plus(x, plus(y, z))
3.81/1.78	   times(times(x, y), z) == times(x, times(y, z))
3.81/1.78	
3.81/1.78	
3.81/1.78	----------------------------------------
3.81/1.78	
3.81/1.78	(5) RRRPoloETRSProof (EQUIVALENT)
3.81/1.78	The following E TRS is given: Equational rewrite system:
3.81/1.78	The TRS R consists of the following rules:
3.81/1.78	
3.81/1.78	   times(0(x), y) -> 0(times(x, y))
3.81/1.78	
3.81/1.78	The set E consists of the following equations:
3.81/1.78	
3.81/1.78	   plus(x, y) == plus(y, x)
3.81/1.78	   times(x, y) == times(y, x)
3.81/1.78	   plus(plus(x, y), z) == plus(x, plus(y, z))
3.81/1.78	   times(times(x, y), z) == times(x, times(y, z))
3.81/1.78	
3.81/1.78	The following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering:
3.81/1.78	
3.81/1.78	   times(0(x), y) -> 0(times(x, y))
3.81/1.78	Used ordering:
3.81/1.78	Polynomial interpretation [POLO]:
3.81/1.78	
3.81/1.78	   POL(0(x_1)) = 2 + x_1
3.81/1.78	   POL(plus(x_1, x_2)) = 1 + 2*x_1 + 2*x_1*x_2 + 2*x_2
3.81/1.78	   POL(times(x_1, x_2)) = 1 + 2*x_1 + 2*x_1*x_2 + 2*x_2
3.81/1.78	
3.81/1.78	
3.81/1.78	
3.81/1.78	
3.81/1.78	----------------------------------------
3.81/1.78	
3.81/1.78	(6)
3.81/1.78	Obligation:
3.81/1.78	Equational rewrite system:
3.81/1.78	R is empty.
3.81/1.78	The set E consists of the following equations:
3.81/1.78	
3.81/1.78	   plus(x, y) == plus(y, x)
3.81/1.78	   times(x, y) == times(y, x)
3.81/1.78	   plus(plus(x, y), z) == plus(x, plus(y, z))
3.81/1.78	   times(times(x, y), z) == times(x, times(y, z))
3.81/1.78	
3.81/1.78	
3.81/1.78	----------------------------------------
3.81/1.78	
3.81/1.78	(7) RisEmptyProof (EQUIVALENT)
3.81/1.78	The TRS R is empty. Hence, termination is trivially proven.
3.81/1.78	----------------------------------------
3.81/1.78	
3.81/1.78	(8)
3.81/1.78	YES
3.81/1.80	EOF