4.78/2.08	MAYBE
4.78/2.09	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
4.78/2.09	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.78/2.09	
4.78/2.09	
4.78/2.09	Quasi decreasingness of the given CTRS could not be shown:
4.78/2.09	
4.78/2.09	(0) CTRS
4.78/2.09	(1) CTRSToQTRSProof [SOUND, 0 ms]
4.78/2.09	(2) QTRS
4.78/2.09	(3) QTRSRRRProof [EQUIVALENT, 44 ms]
4.78/2.09	(4) QTRS
4.78/2.09	(5) Overlay + Local Confluence [EQUIVALENT, 0 ms]
4.78/2.09	(6) QTRS
4.78/2.09	(7) DependencyPairsProof [EQUIVALENT, 0 ms]
4.78/2.09	(8) QDP
4.78/2.09	(9) UsableRulesProof [EQUIVALENT, 0 ms]
4.78/2.09	(10) QDP
4.78/2.09	(11) QReductionProof [EQUIVALENT, 0 ms]
4.78/2.09	(12) QDP
4.78/2.09	(13) NonTerminationLoopProof [COMPLETE, 0 ms]
4.78/2.09	(14) NO
4.78/2.09	
4.78/2.09	
4.78/2.09	----------------------------------------
4.78/2.09	
4.78/2.09	(0)
4.78/2.09	Obligation:
4.78/2.09	Conditional term rewrite system:
4.78/2.09	The TRS R is empty.
4.78/2.09	The conditional TRS C consists of the following conditional rules:
4.78/2.09	
4.78/2.09	   a -> b <= a -> b
4.78/2.09	
4.78/2.09	
4.78/2.09	----------------------------------------
4.78/2.09	
4.78/2.09	(1) CTRSToQTRSProof (SOUND)
4.78/2.09	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
4.78/2.09	----------------------------------------
4.78/2.09	
4.78/2.09	(2)
4.78/2.09	Obligation:
4.78/2.09	Q restricted rewrite system:
4.78/2.09	The TRS R consists of the following rules:
4.78/2.09	
4.78/2.09	   a -> U1(a)
4.78/2.09	   U1(b) -> b
4.78/2.09	
4.78/2.09	Q is empty.
4.78/2.09	
4.78/2.09	----------------------------------------
4.78/2.09	
4.78/2.09	(3) QTRSRRRProof (EQUIVALENT)
4.78/2.09	Used ordering:
4.78/2.09	Polynomial interpretation [POLO]:
4.78/2.09	
4.78/2.09	   POL(U1(x_1)) = 2*x_1
4.78/2.09	   POL(a) = 0
4.78/2.09	   POL(b) = 1
4.78/2.09	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.78/2.09	
4.78/2.09	   U1(b) -> b
4.78/2.09	
4.78/2.09	
4.78/2.09	
4.78/2.09	
4.78/2.09	----------------------------------------
4.78/2.09	
4.78/2.09	(4)
4.78/2.09	Obligation:
4.78/2.09	Q restricted rewrite system:
4.78/2.09	The TRS R consists of the following rules:
4.78/2.09	
4.78/2.09	   a -> U1(a)
4.78/2.09	
4.78/2.09	Q is empty.
4.78/2.09	
4.78/2.09	----------------------------------------
4.78/2.09	
4.78/2.09	(5) Overlay + Local Confluence (EQUIVALENT)
4.78/2.09	The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
4.78/2.09	----------------------------------------
4.78/2.09	
4.78/2.09	(6)
4.78/2.09	Obligation:
4.78/2.09	Q restricted rewrite system:
4.78/2.09	The TRS R consists of the following rules:
4.78/2.09	
4.78/2.09	   a -> U1(a)
4.78/2.09	
4.78/2.09	The set Q consists of the following terms:
4.78/2.09	
4.78/2.09	   a
4.78/2.09	
4.78/2.09	
4.78/2.09	----------------------------------------
4.78/2.09	
4.78/2.09	(7) DependencyPairsProof (EQUIVALENT)
4.78/2.09	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
4.78/2.09	----------------------------------------
4.78/2.09	
4.78/2.09	(8)
4.78/2.09	Obligation:
4.78/2.09	Q DP problem:
4.78/2.09	The TRS P consists of the following rules:
4.78/2.09	
4.78/2.09	   A -> A
4.78/2.09	
4.78/2.09	The TRS R consists of the following rules:
4.78/2.09	
4.78/2.09	   a -> U1(a)
4.78/2.09	
4.78/2.09	The set Q consists of the following terms:
4.78/2.09	
4.78/2.09	   a
4.78/2.09	
4.78/2.09	We have to consider all minimal (P,Q,R)-chains.
4.78/2.09	----------------------------------------
4.78/2.09	
4.78/2.09	(9) UsableRulesProof (EQUIVALENT)
4.78/2.09	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
4.78/2.09	----------------------------------------
4.78/2.09	
4.78/2.09	(10)
4.78/2.09	Obligation:
4.78/2.09	Q DP problem:
4.78/2.09	The TRS P consists of the following rules:
4.78/2.09	
4.78/2.09	   A -> A
4.78/2.09	
4.78/2.09	R is empty.
4.78/2.09	The set Q consists of the following terms:
4.78/2.09	
4.78/2.09	   a
4.78/2.09	
4.78/2.09	We have to consider all minimal (P,Q,R)-chains.
4.78/2.09	----------------------------------------
4.78/2.09	
4.78/2.09	(11) QReductionProof (EQUIVALENT)
4.78/2.09	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
4.78/2.09	
4.78/2.09	   a
4.78/2.09	
4.78/2.09	
4.78/2.09	----------------------------------------
4.78/2.09	
4.78/2.09	(12)
4.78/2.09	Obligation:
4.78/2.09	Q DP problem:
4.78/2.09	The TRS P consists of the following rules:
4.78/2.09	
4.78/2.09	   A -> A
4.78/2.09	
4.78/2.09	R is empty.
4.78/2.09	Q is empty.
4.78/2.09	We have to consider all minimal (P,Q,R)-chains.
4.78/2.09	----------------------------------------
4.78/2.09	
4.78/2.09	(13) NonTerminationLoopProof (COMPLETE)
4.78/2.09	We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
4.78/2.09	Found a loop by semiunifying a rule from P directly.
4.78/2.09	
4.78/2.09	s = A evaluates to  t =A
4.78/2.09	
4.78/2.09	Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
4.78/2.09	* Matcher: [ ]
4.78/2.09	* Semiunifier: [ ]
4.78/2.09	
4.78/2.09	--------------------------------------------------------------------------------
4.78/2.09	Rewriting sequence
4.78/2.09	
4.78/2.09	The DP semiunifies directly so there is only one rewrite step from A to A.
4.78/2.09	
4.78/2.09	
4.78/2.09	
4.78/2.09	
4.78/2.09	----------------------------------------
4.78/2.09	
4.78/2.09	(14)
4.78/2.09	NO
4.96/2.14	EOF