4.14/1.92	YES
4.14/1.93	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
4.14/1.93	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.14/1.93	
4.14/1.93	
4.14/1.93	Termination of the given ETRS could be proven:
4.14/1.93	
4.14/1.93	(0) ETRS
4.14/1.93	(1) RRRPoloETRSProof [EQUIVALENT, 130 ms]
4.14/1.93	(2) ETRS
4.14/1.93	(3) RRRPoloETRSProof [EQUIVALENT, 27 ms]
4.14/1.93	(4) ETRS
4.14/1.93	(5) RRRPoloETRSProof [EQUIVALENT, 41 ms]
4.14/1.93	(6) ETRS
4.14/1.93	(7) RRRPoloETRSProof [EQUIVALENT, 0 ms]
4.14/1.93	(8) ETRS
4.14/1.93	(9) RisEmptyProof [EQUIVALENT, 0 ms]
4.14/1.93	(10) YES
4.14/1.93	
4.14/1.93	
4.14/1.93	----------------------------------------
4.14/1.93	
4.14/1.93	(0)
4.14/1.93	Obligation:
4.14/1.93	Equational rewrite system:
4.14/1.93	The TRS R consists of the following rules:
4.14/1.93	
4.14/1.93	   f(plus(x, y)) -> plus(f(x), y)
4.14/1.93	   plus(g(x), y) -> g(plus(x, y))
4.14/1.93	   plus(f(a), g(b)) -> plus(f(b), g(a))
4.14/1.93	   h(a, b) -> h(b, a)
4.14/1.93	   h(a, g(g(a))) -> h(g(a), f(a))
4.14/1.93	   h(g(a), a) -> h(a, g(b))
4.14/1.93	   h(g(a), b) -> h(a, g(a))
4.14/1.93	
4.14/1.93	The set E consists of the following equations:
4.14/1.93	
4.14/1.93	   plus(x, y) == plus(y, x)
4.14/1.93	   plus(plus(x, y), z) == plus(x, plus(y, z))
4.14/1.93	
4.14/1.93	
4.14/1.93	----------------------------------------
4.14/1.93	
4.14/1.93	(1) RRRPoloETRSProof (EQUIVALENT)
4.14/1.93	The following E TRS is given: Equational rewrite system:
4.14/1.93	The TRS R consists of the following rules:
4.14/1.93	
4.14/1.93	   f(plus(x, y)) -> plus(f(x), y)
4.14/1.93	   plus(g(x), y) -> g(plus(x, y))
4.14/1.93	   plus(f(a), g(b)) -> plus(f(b), g(a))
4.14/1.93	   h(a, b) -> h(b, a)
4.14/1.93	   h(a, g(g(a))) -> h(g(a), f(a))
4.14/1.93	   h(g(a), a) -> h(a, g(b))
4.14/1.93	   h(g(a), b) -> h(a, g(a))
4.14/1.93	
4.14/1.93	The set E consists of the following equations:
4.14/1.93	
4.14/1.93	   plus(x, y) == plus(y, x)
4.14/1.93	   plus(plus(x, y), z) == plus(x, plus(y, z))
4.14/1.93	
4.14/1.93	The following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering:
4.14/1.93	
4.14/1.93	   f(plus(x, y)) -> plus(f(x), y)
4.14/1.93	   plus(g(x), y) -> g(plus(x, y))
4.14/1.93	   h(a, g(g(a))) -> h(g(a), f(a))
4.14/1.93	Used ordering:
4.14/1.93	Polynomial interpretation [POLO]:
4.14/1.93	
4.14/1.93	   POL(a) = 0
4.14/1.93	   POL(b) = 0
4.14/1.93	   POL(f(x_1)) = 2 + 3*x_1
4.14/1.93	   POL(g(x_1)) = 3 + x_1
4.14/1.93	   POL(h(x_1, x_2)) = x_1 + x_2
4.14/1.93	   POL(plus(x_1, x_2)) = 2 + 2*x_1 + x_1*x_2 + 2*x_2
4.14/1.93	
4.14/1.93	
4.14/1.93	
4.14/1.93	
4.14/1.93	----------------------------------------
4.14/1.93	
4.14/1.93	(2)
4.14/1.93	Obligation:
4.14/1.93	Equational rewrite system:
4.14/1.93	The TRS R consists of the following rules:
4.14/1.93	
4.14/1.93	   plus(f(a), g(b)) -> plus(f(b), g(a))
4.14/1.93	   h(a, b) -> h(b, a)
4.14/1.93	   h(g(a), a) -> h(a, g(b))
4.14/1.93	   h(g(a), b) -> h(a, g(a))
4.14/1.93	
4.14/1.93	The set E consists of the following equations:
4.14/1.93	
4.14/1.93	   plus(x, y) == plus(y, x)
4.14/1.93	   plus(plus(x, y), z) == plus(x, plus(y, z))
4.14/1.93	
4.14/1.93	
4.14/1.93	----------------------------------------
4.14/1.93	
4.14/1.93	(3) RRRPoloETRSProof (EQUIVALENT)
4.14/1.93	The following E TRS is given: Equational rewrite system:
4.14/1.93	The TRS R consists of the following rules:
4.14/1.93	
4.14/1.93	   plus(f(a), g(b)) -> plus(f(b), g(a))
4.14/1.93	   h(a, b) -> h(b, a)
4.14/1.93	   h(g(a), a) -> h(a, g(b))
4.14/1.93	   h(g(a), b) -> h(a, g(a))
4.14/1.93	
4.14/1.93	The set E consists of the following equations:
4.14/1.93	
4.14/1.93	   plus(x, y) == plus(y, x)
4.14/1.93	   plus(plus(x, y), z) == plus(x, plus(y, z))
4.14/1.93	
4.14/1.93	The following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering:
4.14/1.93	
4.14/1.93	   h(a, b) -> h(b, a)
4.14/1.93	   h(g(a), a) -> h(a, g(b))
4.14/1.93	Used ordering:
4.14/1.93	Polynomial interpretation [POLO]:
4.14/1.93	
4.14/1.93	   POL(a) = 2
4.14/1.93	   POL(b) = 0
4.14/1.93	   POL(f(x_1)) = 2*x_1
4.14/1.93	   POL(g(x_1)) = x_1^2
4.14/1.93	   POL(h(x_1, x_2)) = 2*x_1 + x_2
4.14/1.93	   POL(plus(x_1, x_2)) = 3 + x_1 + x_2
4.14/1.93	
4.14/1.93	
4.14/1.93	
4.14/1.93	
4.14/1.93	----------------------------------------
4.14/1.93	
4.14/1.93	(4)
4.14/1.93	Obligation:
4.14/1.93	Equational rewrite system:
4.14/1.93	The TRS R consists of the following rules:
4.14/1.93	
4.14/1.93	   plus(f(a), g(b)) -> plus(f(b), g(a))
4.14/1.93	   h(g(a), b) -> h(a, g(a))
4.14/1.93	
4.14/1.93	The set E consists of the following equations:
4.14/1.93	
4.14/1.93	   plus(x, y) == plus(y, x)
4.14/1.93	   plus(plus(x, y), z) == plus(x, plus(y, z))
4.14/1.93	
4.14/1.93	
4.14/1.93	----------------------------------------
4.14/1.93	
4.14/1.93	(5) RRRPoloETRSProof (EQUIVALENT)
4.14/1.93	The following E TRS is given: Equational rewrite system:
4.14/1.93	The TRS R consists of the following rules:
4.14/1.93	
4.14/1.93	   plus(f(a), g(b)) -> plus(f(b), g(a))
4.14/1.93	   h(g(a), b) -> h(a, g(a))
4.14/1.93	
4.14/1.93	The set E consists of the following equations:
4.14/1.93	
4.14/1.93	   plus(x, y) == plus(y, x)
4.14/1.93	   plus(plus(x, y), z) == plus(x, plus(y, z))
4.14/1.93	
4.14/1.93	The following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering:
4.14/1.93	
4.14/1.93	   plus(f(a), g(b)) -> plus(f(b), g(a))
4.14/1.93	Used ordering:
4.14/1.93	Polynomial interpretation [POLO]:
4.14/1.93	
4.14/1.93	   POL(a) = 2
4.14/1.93	   POL(b) = 0
4.14/1.93	   POL(f(x_1)) = 3*x_1
4.14/1.93	   POL(g(x_1)) = 2*x_1
4.14/1.93	   POL(h(x_1, x_2)) = 2*x_1 + x_2
4.14/1.93	   POL(plus(x_1, x_2)) = 3 + x_1 + x_2
4.14/1.93	
4.14/1.93	
4.14/1.93	
4.14/1.93	
4.14/1.93	----------------------------------------
4.14/1.93	
4.14/1.93	(6)
4.14/1.93	Obligation:
4.14/1.93	Equational rewrite system:
4.14/1.93	The TRS R consists of the following rules:
4.14/1.93	
4.14/1.93	   h(g(a), b) -> h(a, g(a))
4.14/1.93	
4.14/1.93	The set E consists of the following equations:
4.14/1.93	
4.14/1.93	   plus(x, y) == plus(y, x)
4.14/1.93	   plus(plus(x, y), z) == plus(x, plus(y, z))
4.14/1.93	
4.14/1.93	
4.14/1.93	----------------------------------------
4.14/1.93	
4.14/1.93	(7) RRRPoloETRSProof (EQUIVALENT)
4.14/1.93	The following E TRS is given: Equational rewrite system:
4.14/1.93	The TRS R consists of the following rules:
4.14/1.93	
4.14/1.93	   h(g(a), b) -> h(a, g(a))
4.14/1.93	
4.14/1.93	The set E consists of the following equations:
4.14/1.93	
4.14/1.93	   plus(x, y) == plus(y, x)
4.14/1.93	   plus(plus(x, y), z) == plus(x, plus(y, z))
4.14/1.93	
4.14/1.93	The following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering:
4.14/1.93	
4.14/1.93	   h(g(a), b) -> h(a, g(a))
4.14/1.93	Used ordering:
4.14/1.93	Polynomial interpretation [POLO]:
4.14/1.93	
4.14/1.93	   POL(a) = 0
4.14/1.93	   POL(b) = 1
4.14/1.93	   POL(g(x_1)) = x_1^2
4.14/1.93	   POL(h(x_1, x_2)) = 2*x_1 + x_2
4.14/1.93	   POL(plus(x_1, x_2)) = 3 + 3*x_1 + 2*x_1*x_2 + 3*x_2
4.14/1.93	
4.14/1.93	
4.14/1.93	
4.14/1.93	
4.14/1.93	----------------------------------------
4.14/1.93	
4.14/1.93	(8)
4.14/1.93	Obligation:
4.14/1.93	Equational rewrite system:
4.14/1.93	R is empty.
4.14/1.93	The set E consists of the following equations:
4.14/1.93	
4.14/1.93	   plus(x, y) == plus(y, x)
4.14/1.93	   plus(plus(x, y), z) == plus(x, plus(y, z))
4.14/1.93	
4.14/1.93	
4.14/1.93	----------------------------------------
4.14/1.93	
4.14/1.93	(9) RisEmptyProof (EQUIVALENT)
4.14/1.93	The TRS R is empty. Hence, termination is trivially proven.
4.14/1.93	----------------------------------------
4.14/1.93	
4.14/1.93	(10)
4.14/1.93	YES
4.35/1.97	EOF