5.73/2.46	YES
5.73/2.47	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
5.73/2.47	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.73/2.47	
5.73/2.47	
5.73/2.47	Termination of the given ETRS could be proven:
5.73/2.47	
5.73/2.47	(0) ETRS
5.73/2.47	(1) RRRPoloETRSProof [EQUIVALENT, 162 ms]
5.73/2.47	(2) ETRS
5.73/2.47	(3) RRRPoloETRSProof [EQUIVALENT, 330 ms]
5.73/2.47	(4) ETRS
5.73/2.47	(5) RRRPoloETRSProof [EQUIVALENT, 269 ms]
5.73/2.47	(6) ETRS
5.73/2.47	(7) RisEmptyProof [EQUIVALENT, 0 ms]
5.73/2.47	(8) YES
5.73/2.47	
5.73/2.47	
5.73/2.47	----------------------------------------
5.73/2.47	
5.73/2.47	(0)
5.73/2.47	Obligation:
5.73/2.47	Equational rewrite system:
5.73/2.47	The TRS R consists of the following rules:
5.73/2.47	
5.73/2.47	   0(S) -> S
5.73/2.47	   plus(x, S) -> x
5.73/2.47	   plus(0(x), 0(y)) -> 0(plus(x, y))
5.73/2.47	   plus(0(x), 1(y)) -> 1(plus(x, y))
5.73/2.47	   plus(1(x), 1(y)) -> 0(1(plus(plus(x, y), S)))
5.73/2.47	   times(x, S) -> S
5.73/2.47	   times(x, 0(y)) -> 0(times(x, y))
5.73/2.47	   times(x, 1(y)) -> plus(x, 0(times(x, y)))
5.73/2.47	
5.73/2.47	The set E consists of the following equations:
5.73/2.47	
5.73/2.47	   plus(x, y) == plus(y, x)
5.73/2.47	   times(x, y) == times(y, x)
5.73/2.47	   plus(plus(x, y), z) == plus(x, plus(y, z))
5.73/2.47	   times(times(x, y), z) == times(x, times(y, z))
5.73/2.47	
5.73/2.47	
5.73/2.47	----------------------------------------
5.73/2.47	
5.73/2.47	(1) RRRPoloETRSProof (EQUIVALENT)
5.73/2.47	The following E TRS is given: Equational rewrite system:
5.73/2.47	The TRS R consists of the following rules:
5.73/2.47	
5.73/2.47	   0(S) -> S
5.73/2.47	   plus(x, S) -> x
5.73/2.47	   plus(0(x), 0(y)) -> 0(plus(x, y))
5.73/2.47	   plus(0(x), 1(y)) -> 1(plus(x, y))
5.73/2.47	   plus(1(x), 1(y)) -> 0(1(plus(plus(x, y), S)))
5.73/2.47	   times(x, S) -> S
5.73/2.47	   times(x, 0(y)) -> 0(times(x, y))
5.73/2.47	   times(x, 1(y)) -> plus(x, 0(times(x, y)))
5.73/2.47	
5.73/2.47	The set E consists of the following equations:
5.73/2.47	
5.73/2.47	   plus(x, y) == plus(y, x)
5.73/2.47	   times(x, y) == times(y, x)
5.73/2.47	   plus(plus(x, y), z) == plus(x, plus(y, z))
5.73/2.47	   times(times(x, y), z) == times(x, times(y, z))
5.73/2.47	
5.73/2.47	The following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering:
5.73/2.47	
5.73/2.47	   0(S) -> S
5.73/2.47	   plus(0(x), 0(y)) -> 0(plus(x, y))
5.73/2.47	   plus(0(x), 1(y)) -> 1(plus(x, y))
5.73/2.47	   times(x, S) -> S
5.73/2.47	   times(x, 0(y)) -> 0(times(x, y))
5.73/2.47	   times(x, 1(y)) -> plus(x, 0(times(x, y)))
5.73/2.47	Used ordering:
5.73/2.47	Polynomial interpretation [POLO]:
5.73/2.47	
5.73/2.47	   POL(0(x_1)) = 2 + x_1
5.73/2.47	   POL(1(x_1)) = 2 + x_1
5.73/2.47	   POL(S) = 0
5.73/2.47	   POL(plus(x_1, x_2)) = x_1 + x_2
5.73/2.47	   POL(times(x_1, x_2)) = 3 + 3*x_1 + 2*x_1*x_2 + 3*x_2
5.73/2.47	
5.73/2.47	
5.73/2.47	
5.73/2.47	
5.73/2.47	----------------------------------------
5.73/2.47	
5.73/2.47	(2)
5.73/2.47	Obligation:
5.73/2.47	Equational rewrite system:
5.73/2.47	The TRS R consists of the following rules:
5.73/2.47	
5.73/2.47	   plus(x, S) -> x
5.73/2.47	   plus(1(x), 1(y)) -> 0(1(plus(plus(x, y), S)))
5.73/2.47	
5.73/2.47	The set E consists of the following equations:
5.73/2.47	
5.73/2.47	   plus(x, y) == plus(y, x)
5.73/2.47	   times(x, y) == times(y, x)
5.73/2.47	   plus(plus(x, y), z) == plus(x, plus(y, z))
5.73/2.47	   times(times(x, y), z) == times(x, times(y, z))
5.73/2.47	
5.73/2.47	
5.73/2.47	----------------------------------------
5.73/2.47	
5.73/2.47	(3) RRRPoloETRSProof (EQUIVALENT)
5.73/2.47	The following E TRS is given: Equational rewrite system:
5.73/2.47	The TRS R consists of the following rules:
5.73/2.47	
5.73/2.47	   plus(x, S) -> x
5.73/2.47	   plus(1(x), 1(y)) -> 0(1(plus(plus(x, y), S)))
5.73/2.47	
5.73/2.47	The set E consists of the following equations:
5.73/2.47	
5.73/2.47	   plus(x, y) == plus(y, x)
5.73/2.47	   times(x, y) == times(y, x)
5.73/2.47	   plus(plus(x, y), z) == plus(x, plus(y, z))
5.73/2.47	   times(times(x, y), z) == times(x, times(y, z))
5.73/2.47	
5.73/2.47	The following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering:
5.73/2.47	
5.73/2.47	   plus(x, S) -> x
5.73/2.47	Used ordering:
5.73/2.47	Polynomial interpretation [POLO]:
5.73/2.47	
5.73/2.47	   POL(0(x_1)) = x_1
5.73/2.47	   POL(1(x_1)) = 2 + x_1
5.73/2.47	   POL(S) = 2
5.73/2.47	   POL(plus(x_1, x_2)) = x_1 + x_2
5.73/2.47	   POL(times(x_1, x_2)) = 3 + 3*x_1 + 2*x_1*x_2 + 3*x_2
5.73/2.47	
5.73/2.47	
5.73/2.47	
5.73/2.47	
5.73/2.47	----------------------------------------
5.73/2.47	
5.73/2.47	(4)
5.73/2.47	Obligation:
5.73/2.47	Equational rewrite system:
5.73/2.47	The TRS R consists of the following rules:
5.73/2.47	
5.73/2.47	   plus(1(x), 1(y)) -> 0(1(plus(plus(x, y), S)))
5.73/2.47	
5.73/2.47	The set E consists of the following equations:
5.73/2.47	
5.73/2.47	   plus(x, y) == plus(y, x)
5.73/2.47	   times(x, y) == times(y, x)
5.73/2.47	   plus(plus(x, y), z) == plus(x, plus(y, z))
5.73/2.47	   times(times(x, y), z) == times(x, times(y, z))
5.73/2.47	
5.73/2.47	
5.73/2.47	----------------------------------------
5.73/2.47	
5.73/2.47	(5) RRRPoloETRSProof (EQUIVALENT)
5.73/2.47	The following E TRS is given: Equational rewrite system:
5.73/2.47	The TRS R consists of the following rules:
5.73/2.47	
5.73/2.47	   plus(1(x), 1(y)) -> 0(1(plus(plus(x, y), S)))
5.73/2.47	
5.73/2.47	The set E consists of the following equations:
5.73/2.47	
5.73/2.47	   plus(x, y) == plus(y, x)
5.73/2.47	   times(x, y) == times(y, x)
5.73/2.47	   plus(plus(x, y), z) == plus(x, plus(y, z))
5.73/2.47	   times(times(x, y), z) == times(x, times(y, z))
5.73/2.47	
5.73/2.47	The following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering:
5.73/2.47	
5.73/2.47	   plus(1(x), 1(y)) -> 0(1(plus(plus(x, y), S)))
5.73/2.47	Used ordering:
5.73/2.47	Polynomial interpretation [POLO]:
5.73/2.47	
5.73/2.47	   POL(0(x_1)) = x_1
5.73/2.47	   POL(1(x_1)) = 2 + x_1
5.73/2.47	   POL(S) = 0
5.73/2.47	   POL(plus(x_1, x_2)) = x_1 + x_2
5.73/2.47	   POL(times(x_1, x_2)) = 3 + 3*x_1 + 2*x_1*x_2 + 3*x_2
5.73/2.47	
5.73/2.47	
5.73/2.47	
5.73/2.47	
5.73/2.47	----------------------------------------
5.73/2.47	
5.73/2.47	(6)
5.73/2.47	Obligation:
5.73/2.47	Equational rewrite system:
5.73/2.47	R is empty.
5.73/2.47	The set E consists of the following equations:
5.73/2.47	
5.73/2.47	   plus(x, y) == plus(y, x)
5.73/2.47	   times(x, y) == times(y, x)
5.73/2.47	   plus(plus(x, y), z) == plus(x, plus(y, z))
5.73/2.47	   times(times(x, y), z) == times(x, times(y, z))
5.73/2.47	
5.73/2.47	
5.73/2.47	----------------------------------------
5.73/2.47	
5.73/2.47	(7) RisEmptyProof (EQUIVALENT)
5.73/2.47	The TRS R is empty. Hence, termination is trivially proven.
5.73/2.47	----------------------------------------
5.73/2.47	
5.73/2.47	(8)
5.73/2.47	YES
5.79/2.53	EOF