3.17/1.63	YES
3.17/1.63	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.17/1.63	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.17/1.63	
3.17/1.63	
3.17/1.63	Termination of the given ETRS could be proven:
3.17/1.63	
3.17/1.63	(0) ETRS
3.17/1.63	(1) EDirectTerminationProof [EQUIVALENT, 0 ms]
3.17/1.63	(2) YES
3.17/1.63	
3.17/1.63	
3.17/1.63	----------------------------------------
3.17/1.63	
3.17/1.63	(0)
3.17/1.63	Obligation:
3.17/1.63	Equational rewrite system:
3.17/1.63	The TRS R consists of the following rules:
3.17/1.63	
3.17/1.63	   dx(X) -> 1
3.17/1.63	   dx(0) -> 0
3.17/1.63	   dx(1) -> 0
3.17/1.63	   dx(a) -> 0
3.17/1.63	   dx(+(f, g)) -> +(dx(f), dx(g))
3.17/1.63	   dx(*(f, g)) -> +(*(dx(f), g), *(dx(g), f))
3.17/1.63	   dx(-(f, g)) -> -(dx(f), dx(g))
3.17/1.63	   dx(neg(f)) -> neg(dx(f))
3.17/1.63	   dx(/(f, g)) -> -(/(dx(f), g), /(*(dx(g), f), exp(g, 2)))
3.17/1.63	   dx(ln(f)) -> /(dx(f), f)
3.17/1.63	   dx(exp(f, g)) -> +(*(dx(f), *(exp(f, -(g, 1)), g)), *(dx(g), *(exp(f, g), ln(f))))
3.17/1.63	
3.17/1.63	The set E consists of the following equations:
3.17/1.63	
3.17/1.63	   *(x, y) == *(y, x)
3.17/1.63	   +(x, y) == +(y, x)
3.17/1.63	   *(*(x, y), z) == *(x, *(y, z))
3.17/1.63	   +(+(x, y), z) == +(x, +(y, z))
3.17/1.63	
3.17/1.63	
3.17/1.63	----------------------------------------
3.17/1.63	
3.17/1.63	(1) EDirectTerminationProof (EQUIVALENT)
3.17/1.63	We use [DA_FALKE] with the following order to prove termination.
3.17/1.64	
3.17/1.64	Precedence:
3.17/1.64	dx_1 > 1
3.17/1.64	dx_1 > 0
3.17/1.64	dx_1 > +_2
3.17/1.64	dx_1 > *_2
3.17/1.64	dx_1 > -_2
3.17/1.64	dx_1 > neg_1
3.17/1.64	dx_1 > /_2
3.17/1.64	dx_1 > exp_2
3.17/1.64	dx_1 > 2
3.17/1.64	dx_1 > ln_1
3.17/1.64	
3.17/1.64	
3.17/1.64	Status:
3.17/1.64	*_2: flat status
3.17/1.64	+_2: flat status
3.17/1.64	
3.17/1.64	
3.17/1.64	
3.17/1.64	----------------------------------------
3.17/1.64	
3.17/1.64	(2)
3.17/1.64	YES
3.17/1.67	EOF