4.50/2.06	YES
4.50/2.07	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
4.50/2.07	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.50/2.07	
4.50/2.07	
4.50/2.07	Termination of the given ETRS could be proven:
4.50/2.07	
4.50/2.07	(0) ETRS
4.50/2.07	(1) EquationalDependencyPairsProof [EQUIVALENT, 0 ms]
4.50/2.07	(2) EDP
4.50/2.07	(3) EDependencyGraphProof [EQUIVALENT, 0 ms]
4.50/2.07	(4) EDP
4.50/2.07	(5) EUsableRulesReductionPairsProof [EQUIVALENT, 0 ms]
4.50/2.07	(6) EDP
4.50/2.07	(7) EDPProblemToQDPProblemProof [EQUIVALENT, 0 ms]
4.50/2.07	(8) QDP
4.50/2.07	(9) MNOCProof [EQUIVALENT, 0 ms]
4.50/2.07	(10) QDP
4.50/2.07	(11) MRRProof [EQUIVALENT, 0 ms]
4.50/2.07	(12) QDP
4.50/2.07	(13) DependencyGraphProof [EQUIVALENT, 0 ms]
4.50/2.07	(14) TRUE
4.50/2.07	
4.50/2.07	
4.50/2.07	----------------------------------------
4.50/2.07	
4.50/2.07	(0)
4.50/2.07	Obligation:
4.50/2.07	Equational rewrite system:
4.50/2.07	The TRS R consists of the following rules:
4.50/2.07	
4.50/2.07	   p(s(x)) -> x
4.50/2.07	   fac(0) -> s(0)
4.50/2.07	   fac(s(x)) -> times(s(x), fac(p(s(x))))
4.50/2.07	
4.50/2.07	The set E consists of the following equations:
4.50/2.07	
4.50/2.07	   times(x, y) == times(y, x)
4.50/2.07	   times(times(x, y), z) == times(x, times(y, z))
4.50/2.07	
4.50/2.07	
4.50/2.07	----------------------------------------
4.50/2.07	
4.50/2.07	(1) EquationalDependencyPairsProof (EQUIVALENT)
4.50/2.07	Using Dependency Pairs [AG00,DA_STEIN] we result in the following initial EDP problem:
4.50/2.07	The TRS P consists of the following rules:
4.50/2.07	
4.50/2.07	   FAC(s(x)) -> FAC(p(s(x)))
4.50/2.07	   FAC(s(x)) -> P(s(x))
4.50/2.07	
4.50/2.07	The TRS R consists of the following rules:
4.50/2.07	
4.50/2.07	   p(s(x)) -> x
4.50/2.07	   fac(0) -> s(0)
4.50/2.07	   fac(s(x)) -> times(s(x), fac(p(s(x))))
4.50/2.07	
4.50/2.07	The set E consists of the following equations:
4.50/2.07	
4.50/2.07	   times(x, y) == times(y, x)
4.50/2.07	   times(times(x, y), z) == times(x, times(y, z))
4.50/2.07	
4.50/2.07	E# is empty.
4.50/2.07	We have to consider all minimal (P,E#,R,E)-chains
4.50/2.07	
4.50/2.07	----------------------------------------
4.50/2.07	
4.50/2.07	(2)
4.50/2.07	Obligation:
4.50/2.07	The TRS P consists of the following rules:
4.50/2.07	
4.50/2.07	   FAC(s(x)) -> FAC(p(s(x)))
4.50/2.07	   FAC(s(x)) -> P(s(x))
4.50/2.07	
4.50/2.07	The TRS R consists of the following rules:
4.50/2.07	
4.50/2.07	   p(s(x)) -> x
4.50/2.07	   fac(0) -> s(0)
4.50/2.07	   fac(s(x)) -> times(s(x), fac(p(s(x))))
4.50/2.07	
4.50/2.07	The set E consists of the following equations:
4.50/2.07	
4.50/2.07	   times(x, y) == times(y, x)
4.50/2.07	   times(times(x, y), z) == times(x, times(y, z))
4.50/2.07	
4.50/2.07	E# is empty.
4.50/2.07	We have to consider all minimal (P,E#,R,E)-chains
4.50/2.07	----------------------------------------
4.50/2.07	
4.50/2.07	(3) EDependencyGraphProof (EQUIVALENT)
4.50/2.07	The approximation of the Equational Dependency Graph [DA_STEIN] contains 1 SCC with 1 less node.
4.50/2.07	----------------------------------------
4.50/2.07	
4.50/2.07	(4)
4.50/2.07	Obligation:
4.50/2.07	The TRS P consists of the following rules:
4.50/2.07	
4.50/2.07	   FAC(s(x)) -> FAC(p(s(x)))
4.50/2.07	
4.50/2.07	The TRS R consists of the following rules:
4.50/2.07	
4.50/2.07	   p(s(x)) -> x
4.50/2.07	   fac(0) -> s(0)
4.50/2.07	   fac(s(x)) -> times(s(x), fac(p(s(x))))
4.50/2.07	
4.50/2.07	The set E consists of the following equations:
4.50/2.07	
4.50/2.07	   times(x, y) == times(y, x)
4.50/2.07	   times(times(x, y), z) == times(x, times(y, z))
4.50/2.07	
4.50/2.07	E# is empty.
4.50/2.07	We have to consider all minimal (P,E#,R,E)-chains
4.50/2.07	----------------------------------------
4.50/2.07	
4.50/2.07	(5) EUsableRulesReductionPairsProof (EQUIVALENT)
4.50/2.07	By using the usable rules and equations with reduction pair processor [DA_STEIN] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules can be oriented non-strictly, the  usable equations and the esharp equations can be oriented equivalently. All non-usable rules and equations are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
4.50/2.07	
4.50/2.07	No dependency pairs are removed.
4.50/2.07	
4.50/2.07	The following rules are removed from R:
4.50/2.07	
4.50/2.07	   fac(0) -> s(0)
4.50/2.07	   fac(s(x)) -> times(s(x), fac(p(s(x))))
4.50/2.07	The following equations are removed from E:
4.50/2.07	
4.50/2.07	   times(x, y) == times(y, x)
4.50/2.07	   times(times(x, y), z) == times(x, times(y, z))
4.50/2.07	Used ordering: POLO with Polynomial interpretation [POLO]:
4.50/2.07	
4.50/2.07	   POL(FAC(x_1)) = x_1
4.50/2.07	   POL(p(x_1)) = x_1
4.50/2.07	   POL(s(x_1)) = x_1
4.50/2.07	
4.50/2.07	
4.50/2.07	----------------------------------------
4.50/2.07	
4.50/2.07	(6)
4.50/2.07	Obligation:
4.50/2.07	The TRS P consists of the following rules:
4.50/2.07	
4.50/2.07	   FAC(s(x)) -> FAC(p(s(x)))
4.50/2.07	
4.50/2.07	The TRS R consists of the following rules:
4.50/2.07	
4.50/2.07	   p(s(x)) -> x
4.50/2.07	
4.50/2.07	E is empty.
4.50/2.07	E# is empty.
4.50/2.07	We have to consider all minimal (P,E#,R,E)-chains
4.50/2.07	----------------------------------------
4.50/2.07	
4.50/2.07	(7) EDPProblemToQDPProblemProof (EQUIVALENT)
4.50/2.07	The EDP problem does not contain equations anymore, so we can transform it with the EDP to QDP problem processor [DA_STEIN] into a QDP problem.
4.50/2.07	----------------------------------------
4.50/2.07	
4.50/2.07	(8)
4.50/2.07	Obligation:
4.50/2.07	Q DP problem:
4.50/2.07	The TRS P consists of the following rules:
4.50/2.07	
4.50/2.07	   FAC(s(x)) -> FAC(p(s(x)))
4.50/2.07	
4.50/2.07	The TRS R consists of the following rules:
4.50/2.07	
4.50/2.07	   p(s(x)) -> x
4.50/2.07	
4.50/2.07	Q is empty.
4.50/2.07	We have to consider all minimal (P,Q,R)-chains.
4.50/2.07	----------------------------------------
4.50/2.07	
4.50/2.07	(9) MNOCProof (EQUIVALENT)
4.50/2.07	We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.
4.50/2.07	----------------------------------------
4.50/2.07	
4.50/2.07	(10)
4.50/2.07	Obligation:
4.50/2.07	Q DP problem:
4.50/2.07	The TRS P consists of the following rules:
4.50/2.07	
4.50/2.07	   FAC(s(x)) -> FAC(p(s(x)))
4.50/2.07	
4.50/2.07	The TRS R consists of the following rules:
4.50/2.07	
4.50/2.07	   p(s(x)) -> x
4.50/2.07	
4.50/2.07	The set Q consists of the following terms:
4.50/2.07	
4.50/2.07	   p(s(x0))
4.50/2.07	
4.50/2.07	We have to consider all minimal (P,Q,R)-chains.
4.50/2.07	----------------------------------------
4.50/2.07	
4.50/2.07	(11) MRRProof (EQUIVALENT)
4.50/2.07	By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
4.50/2.07	
4.50/2.07	
4.50/2.07	Strictly oriented rules of the TRS R:
4.50/2.07	
4.50/2.07	   p(s(x)) -> x
4.50/2.07	
4.50/2.07	Used ordering: Polynomial interpretation [POLO]:
4.50/2.07	
4.50/2.07	   POL(FAC(x_1)) = 2*x_1
4.50/2.07	   POL(p(x_1)) = x_1
4.50/2.07	   POL(s(x_1)) = 2 + x_1
4.50/2.07	
4.50/2.07	
4.50/2.07	----------------------------------------
4.50/2.07	
4.50/2.07	(12)
4.50/2.07	Obligation:
4.50/2.07	Q DP problem:
4.50/2.07	The TRS P consists of the following rules:
4.50/2.07	
4.50/2.07	   FAC(s(x)) -> FAC(p(s(x)))
4.50/2.07	
4.50/2.07	R is empty.
4.50/2.07	The set Q consists of the following terms:
4.50/2.07	
4.50/2.07	   p(s(x0))
4.50/2.07	
4.50/2.07	We have to consider all minimal (P,Q,R)-chains.
4.50/2.07	----------------------------------------
4.50/2.07	
4.50/2.07	(13) DependencyGraphProof (EQUIVALENT)
4.50/2.07	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
4.50/2.07	----------------------------------------
4.50/2.07	
4.50/2.07	(14)
4.50/2.07	TRUE
4.50/2.09	EOF