6.32/2.76	YES
6.32/2.77	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
6.32/2.77	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.32/2.77	
6.32/2.77	
6.32/2.77	Termination of the given ETRS could be proven:
6.32/2.77	
6.32/2.77	(0) ETRS
6.32/2.77	(1) EquationalDependencyPairsProof [EQUIVALENT, 51 ms]
6.32/2.77	(2) EDP
6.32/2.77	(3) ERuleRemovalProof [EQUIVALENT, 0 ms]
6.32/2.77	(4) EDP
6.32/2.77	(5) EDPPoloProof [EQUIVALENT, 0 ms]
6.32/2.77	(6) EDP
6.32/2.77	(7) PisEmptyProof [EQUIVALENT, 0 ms]
6.32/2.77	(8) YES
6.32/2.77	
6.32/2.77	
6.32/2.77	----------------------------------------
6.32/2.77	
6.32/2.77	(0)
6.32/2.77	Obligation:
6.32/2.77	Equational rewrite system:
6.32/2.77	The TRS R consists of the following rules:
6.32/2.77	
6.32/2.77	   +(g(x), g(y)) -> g(+(g(a), +(x, y)))
6.32/2.77	
6.32/2.77	The set E consists of the following equations:
6.32/2.77	
6.32/2.77	   +(x, y) == +(y, x)
6.32/2.77	   +(+(x, y), z) == +(x, +(y, z))
6.32/2.77	
6.32/2.77	
6.32/2.77	----------------------------------------
6.32/2.77	
6.32/2.77	(1) EquationalDependencyPairsProof (EQUIVALENT)
6.32/2.77	Using Dependency Pairs [AG00,DA_STEIN] we result in the following initial EDP problem:
6.32/2.77	The TRS P consists of the following rules:
6.32/2.77	
6.32/2.77	   +^1(g(x), g(y)) -> +^1(g(a), +(x, y))
6.32/2.77	   +^1(g(x), g(y)) -> +^1(x, y)
6.32/2.77	   +^1(+(g(x), g(y)), ext) -> +^1(g(+(g(a), +(x, y))), ext)
6.32/2.77	   +^1(+(g(x), g(y)), ext) -> +^1(g(a), +(x, y))
6.32/2.77	   +^1(+(g(x), g(y)), ext) -> +^1(x, y)
6.32/2.77	
6.32/2.77	The TRS R consists of the following rules:
6.32/2.77	
6.32/2.77	   +(g(x), g(y)) -> g(+(g(a), +(x, y)))
6.32/2.77	   +(+(g(x), g(y)), ext) -> +(g(+(g(a), +(x, y))), ext)
6.32/2.77	
6.32/2.77	The set E consists of the following equations:
6.32/2.77	
6.32/2.77	   +(x, y) == +(y, x)
6.32/2.77	   +(+(x, y), z) == +(x, +(y, z))
6.32/2.77	
6.32/2.77	The set E# consists of the following equations:
6.32/2.77	
6.32/2.77	   +^1(x, y) == +^1(y, x)
6.32/2.77	   +^1(+(x, y), z) == +^1(x, +(y, z))
6.32/2.77	
6.32/2.77	We have to consider all minimal (P,E#,R,E)-chains
6.32/2.77	
6.32/2.77	----------------------------------------
6.32/2.77	
6.32/2.77	(2)
6.32/2.77	Obligation:
6.32/2.77	The TRS P consists of the following rules:
6.32/2.77	
6.32/2.77	   +^1(g(x), g(y)) -> +^1(g(a), +(x, y))
6.32/2.77	   +^1(g(x), g(y)) -> +^1(x, y)
6.32/2.77	   +^1(+(g(x), g(y)), ext) -> +^1(g(+(g(a), +(x, y))), ext)
6.32/2.77	   +^1(+(g(x), g(y)), ext) -> +^1(g(a), +(x, y))
6.32/2.77	   +^1(+(g(x), g(y)), ext) -> +^1(x, y)
6.32/2.77	
6.32/2.77	The TRS R consists of the following rules:
6.32/2.77	
6.32/2.77	   +(g(x), g(y)) -> g(+(g(a), +(x, y)))
6.32/2.77	   +(+(g(x), g(y)), ext) -> +(g(+(g(a), +(x, y))), ext)
6.32/2.77	
6.32/2.77	The set E consists of the following equations:
6.32/2.77	
6.32/2.77	   +(x, y) == +(y, x)
6.32/2.77	   +(+(x, y), z) == +(x, +(y, z))
6.32/2.77	
6.32/2.77	The set E# consists of the following equations:
6.32/2.77	
6.32/2.77	   +^1(x, y) == +^1(y, x)
6.32/2.77	   +^1(+(x, y), z) == +^1(x, +(y, z))
6.32/2.77	
6.32/2.77	We have to consider all minimal (P,E#,R,E)-chains
6.32/2.77	----------------------------------------
6.32/2.77	
6.32/2.77	(3) ERuleRemovalProof (EQUIVALENT)
6.32/2.77	By using the rule removal processor [DA_STEIN] with the following polynomial ordering [POLO], at least one Dependency Pair or term rewrite system rule of this EDP problem can be strictly oriented.
6.32/2.77	
6.32/2.77	Strictly oriented dependency pairs:
6.32/2.77	
6.32/2.77	   +^1(g(x), g(y)) -> +^1(g(a), +(x, y))
6.32/2.77	   +^1(g(x), g(y)) -> +^1(x, y)
6.32/2.77	   +^1(+(g(x), g(y)), ext) -> +^1(g(a), +(x, y))
6.32/2.77	   +^1(+(g(x), g(y)), ext) -> +^1(x, y)
6.32/2.77	
6.32/2.77	
6.32/2.77	Used ordering: POLO with Polynomial interpretation [POLO]:
6.32/2.77	
6.32/2.77	   POL(+(x_1, x_2)) = x_1 + x_2
6.32/2.77	   POL(+^1(x_1, x_2)) = x_1 + x_2
6.32/2.77	   POL(a) = 0
6.32/2.77	   POL(g(x_1)) = 1 + x_1
6.32/2.77	
6.32/2.77	
6.32/2.77	----------------------------------------
6.32/2.77	
6.32/2.77	(4)
6.32/2.77	Obligation:
6.32/2.77	The TRS P consists of the following rules:
6.32/2.77	
6.32/2.77	   +^1(+(g(x), g(y)), ext) -> +^1(g(+(g(a), +(x, y))), ext)
6.32/2.77	
6.32/2.77	The TRS R consists of the following rules:
6.32/2.77	
6.32/2.77	   +(g(x), g(y)) -> g(+(g(a), +(x, y)))
6.32/2.77	   +(+(g(x), g(y)), ext) -> +(g(+(g(a), +(x, y))), ext)
6.32/2.77	
6.32/2.77	The set E consists of the following equations:
6.32/2.77	
6.32/2.77	   +(x, y) == +(y, x)
6.32/2.77	   +(+(x, y), z) == +(x, +(y, z))
6.32/2.77	
6.32/2.77	The set E# consists of the following equations:
6.32/2.77	
6.32/2.77	   +^1(x, y) == +^1(y, x)
6.32/2.77	   +^1(+(x, y), z) == +^1(x, +(y, z))
6.32/2.77	
6.32/2.77	We have to consider all minimal (P,E#,R,E)-chains
6.32/2.77	----------------------------------------
6.32/2.77	
6.32/2.77	(5) EDPPoloProof (EQUIVALENT)
6.32/2.77	We use the reduction pair processor [DA_STEIN] with a polynomial ordering [POLO]. All Dependency Pairs of this DP problem can be strictly oriented.
6.32/2.77	
6.32/2.77	
6.32/2.77	   +^1(+(g(x), g(y)), ext) -> +^1(g(+(g(a), +(x, y))), ext)
6.32/2.77	With the implicit AFS we had to orient the following set of usable rules of R non-strictly.
6.32/2.77	
6.32/2.77	
6.32/2.77	   +(g(x), g(y)) -> g(+(g(a), +(x, y)))
6.32/2.77	   +(+(g(x), g(y)), ext) -> +(g(+(g(a), +(x, y))), ext)
6.32/2.77	We had to orient the following equations of E# equivalently.
6.32/2.77	
6.32/2.77	
6.32/2.77	   +^1(x, y) == +^1(y, x)
6.32/2.77	   +^1(+(x, y), z) == +^1(x, +(y, z))
6.32/2.77	With the implicit AFS we had to orient the following usable equations of E equivalently.
6.32/2.77	
6.32/2.77	
6.32/2.77	   +(+(x, y), z) == +(x, +(y, z))
6.32/2.77	   +(x, y) == +(y, x)
6.32/2.77	Used ordering: POLO with Polynomial interpretation [POLO]:
6.32/2.77	
6.32/2.77	   POL(+(x_1, x_2)) = x_1 + x_2
6.32/2.77	   POL(+^1(x_1, x_2)) = x_1 + x_2
6.32/2.77	   POL(a) = 0
6.32/2.77	   POL(g(x_1)) = 2
6.32/2.77	
6.32/2.77	
6.32/2.77	----------------------------------------
6.32/2.77	
6.32/2.77	(6)
6.32/2.77	Obligation:
6.32/2.77	P is empty.
6.32/2.77	The TRS R consists of the following rules:
6.32/2.77	
6.32/2.77	   +(g(x), g(y)) -> g(+(g(a), +(x, y)))
6.32/2.77	   +(+(g(x), g(y)), ext) -> +(g(+(g(a), +(x, y))), ext)
6.32/2.77	
6.32/2.77	The set E consists of the following equations:
6.32/2.77	
6.32/2.77	   +(x, y) == +(y, x)
6.32/2.77	   +(+(x, y), z) == +(x, +(y, z))
6.32/2.77	
6.32/2.77	The set E# consists of the following equations:
6.32/2.77	
6.32/2.77	   +^1(x, y) == +^1(y, x)
6.32/2.77	   +^1(+(x, y), z) == +^1(x, +(y, z))
6.32/2.77	
6.32/2.77	We have to consider all minimal (P,E#,R,E)-chains
6.32/2.77	----------------------------------------
6.32/2.77	
6.32/2.77	(7) PisEmptyProof (EQUIVALENT)
6.32/2.77	The TRS P is empty. Hence, there is no (P,E#,R,E) chain.
6.32/2.77	----------------------------------------
6.32/2.77	
6.32/2.77	(8)
6.32/2.77	YES
6.43/2.80	EOF