4.10/1.83	YES
4.10/1.84	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
4.10/1.84	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.10/1.84	
4.10/1.84	
4.10/1.84	Termination of the given ETRS could be proven:
4.10/1.84	
4.10/1.84	(0) ETRS
4.10/1.84	(1) RRRPoloETRSProof [EQUIVALENT, 161 ms]
4.10/1.84	(2) ETRS
4.10/1.84	(3) RRRPoloETRSProof [EQUIVALENT, 41 ms]
4.10/1.84	(4) ETRS
4.10/1.84	(5) RisEmptyProof [EQUIVALENT, 0 ms]
4.10/1.84	(6) YES
4.10/1.84	
4.10/1.84	
4.10/1.84	----------------------------------------
4.10/1.84	
4.10/1.84	(0)
4.10/1.84	Obligation:
4.10/1.84	Equational rewrite system:
4.10/1.84	The TRS R consists of the following rules:
4.10/1.84	
4.10/1.84	   f(plus(a, a)) -> plus(f(a), f(a))
4.10/1.84	   plus(f(a), f(a)) -> plus(a, f(f(a)))
4.10/1.84	
4.10/1.84	The set E consists of the following equations:
4.10/1.84	
4.10/1.84	   plus(x, y) == plus(y, x)
4.10/1.84	   plus(plus(x, y), z) == plus(x, plus(y, z))
4.10/1.84	
4.10/1.84	
4.10/1.84	----------------------------------------
4.10/1.84	
4.10/1.84	(1) RRRPoloETRSProof (EQUIVALENT)
4.10/1.84	The following E TRS is given: Equational rewrite system:
4.10/1.84	The TRS R consists of the following rules:
4.10/1.84	
4.10/1.84	   f(plus(a, a)) -> plus(f(a), f(a))
4.10/1.84	   plus(f(a), f(a)) -> plus(a, f(f(a)))
4.10/1.84	
4.10/1.84	The set E consists of the following equations:
4.10/1.84	
4.10/1.84	   plus(x, y) == plus(y, x)
4.10/1.84	   plus(plus(x, y), z) == plus(x, plus(y, z))
4.10/1.84	
4.10/1.84	The following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering:
4.10/1.84	
4.10/1.84	   f(plus(a, a)) -> plus(f(a), f(a))
4.10/1.84	Used ordering:
4.10/1.84	Polynomial interpretation [POLO]:
4.10/1.84	
4.10/1.84	   POL(a) = 0
4.10/1.84	   POL(f(x_1)) = 3*x_1
4.10/1.84	   POL(plus(x_1, x_2)) = 3 + x_1 + x_2
4.10/1.84	
4.10/1.84	
4.10/1.84	
4.10/1.84	
4.10/1.84	----------------------------------------
4.10/1.84	
4.10/1.84	(2)
4.10/1.84	Obligation:
4.10/1.84	Equational rewrite system:
4.10/1.84	The TRS R consists of the following rules:
4.10/1.84	
4.10/1.84	   plus(f(a), f(a)) -> plus(a, f(f(a)))
4.10/1.84	
4.10/1.84	The set E consists of the following equations:
4.10/1.84	
4.10/1.84	   plus(x, y) == plus(y, x)
4.10/1.84	   plus(plus(x, y), z) == plus(x, plus(y, z))
4.10/1.84	
4.10/1.84	
4.10/1.84	----------------------------------------
4.10/1.84	
4.10/1.84	(3) RRRPoloETRSProof (EQUIVALENT)
4.10/1.84	The following E TRS is given: Equational rewrite system:
4.10/1.84	The TRS R consists of the following rules:
4.10/1.84	
4.10/1.84	   plus(f(a), f(a)) -> plus(a, f(f(a)))
4.10/1.84	
4.10/1.84	The set E consists of the following equations:
4.10/1.84	
4.10/1.84	   plus(x, y) == plus(y, x)
4.10/1.84	   plus(plus(x, y), z) == plus(x, plus(y, z))
4.10/1.84	
4.10/1.84	The following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering:
4.10/1.84	
4.10/1.84	   plus(f(a), f(a)) -> plus(a, f(f(a)))
4.10/1.84	Used ordering:
4.10/1.84	Polynomial interpretation [POLO]:
4.10/1.84	
4.10/1.84	   POL(a) = 0
4.10/1.84	   POL(f(x_1)) = 1 + x_1^2
4.10/1.84	   POL(plus(x_1, x_2)) = 1 + 2*x_1 + 2*x_1*x_2 + 2*x_2
4.10/1.84	
4.10/1.84	
4.10/1.84	
4.10/1.84	
4.10/1.84	----------------------------------------
4.10/1.84	
4.10/1.84	(4)
4.10/1.84	Obligation:
4.10/1.84	Equational rewrite system:
4.10/1.84	R is empty.
4.10/1.84	The set E consists of the following equations:
4.10/1.84	
4.10/1.84	   plus(x, y) == plus(y, x)
4.10/1.84	   plus(plus(x, y), z) == plus(x, plus(y, z))
4.10/1.84	
4.10/1.84	
4.10/1.84	----------------------------------------
4.10/1.84	
4.10/1.84	(5) RisEmptyProof (EQUIVALENT)
4.10/1.84	The TRS R is empty. Hence, termination is trivially proven.
4.10/1.84	----------------------------------------
4.10/1.84	
4.10/1.84	(6)
4.10/1.84	YES
4.19/1.88	EOF