4.94/2.28	MAYBE
4.94/2.29	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
4.94/2.29	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.94/2.29	
4.94/2.29	
4.94/2.29	Quasi decreasingness of the given CTRS could not be shown:
4.94/2.29	
4.94/2.29	(0) CTRS
4.94/2.29	(1) CTRSToQTRSProof [SOUND, 0 ms]
4.94/2.29	(2) QTRS
4.94/2.29	(3) QTRSRRRProof [EQUIVALENT, 25 ms]
4.94/2.29	(4) QTRS
4.94/2.29	(5) AAECC Innermost [EQUIVALENT, 0 ms]
4.94/2.29	(6) QTRS
4.94/2.29	(7) DependencyPairsProof [EQUIVALENT, 0 ms]
4.94/2.29	(8) QDP
4.94/2.29	(9) UsableRulesProof [EQUIVALENT, 0 ms]
4.94/2.29	(10) QDP
4.94/2.29	(11) QReductionProof [EQUIVALENT, 0 ms]
4.94/2.29	(12) QDP
4.94/2.29	(13) NonTerminationLoopProof [COMPLETE, 0 ms]
4.94/2.29	(14) NO
4.94/2.29	
4.94/2.29	
4.94/2.29	----------------------------------------
4.94/2.29	
4.94/2.29	(0)
4.94/2.29	Obligation:
4.94/2.29	Conditional term rewrite system:
4.94/2.29	The TRS R consists of the following rules:
4.94/2.29	
4.94/2.29	   p(q(x)) -> 0
4.94/2.29	   c -> c
4.94/2.29	
4.94/2.29	The conditional TRS C consists of the following conditional rules:
4.94/2.29	
4.94/2.29	   a(x) -> b <= p(x) -> 0
4.94/2.29	
4.94/2.29	
4.94/2.29	----------------------------------------
4.94/2.29	
4.94/2.29	(1) CTRSToQTRSProof (SOUND)
4.94/2.29	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
4.94/2.29	----------------------------------------
4.94/2.29	
4.94/2.29	(2)
4.94/2.29	Obligation:
4.94/2.29	Q restricted rewrite system:
4.94/2.29	The TRS R consists of the following rules:
4.94/2.29	
4.94/2.29	   a(x) -> U1(p(x))
4.94/2.29	   U1(0) -> b
4.94/2.29	   p(q(x)) -> 0
4.94/2.29	   c -> c
4.94/2.29	
4.94/2.29	Q is empty.
4.94/2.29	
4.94/2.29	----------------------------------------
4.94/2.29	
4.94/2.29	(3) QTRSRRRProof (EQUIVALENT)
4.94/2.29	Used ordering:
4.94/2.29	Knuth-Bendix order [KBO] with precedence:c > q_1 > a_1 > U1_1 > b > p_1 > 0
4.94/2.29	
4.94/2.29	and weight map:
4.94/2.29	
4.94/2.29	   0=3
4.94/2.29	   b=4
4.94/2.29	   c=1
4.94/2.29	   a_1=2
4.94/2.29	   U1_1=1
4.94/2.29	   p_1=1
4.94/2.29	   q_1=1
4.94/2.29	
4.94/2.29	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.94/2.29	
4.94/2.29	   a(x) -> U1(p(x))
4.94/2.29	   U1(0) -> b
4.94/2.29	   p(q(x)) -> 0
4.94/2.29	
4.94/2.29	
4.94/2.29	
4.94/2.29	
4.94/2.29	----------------------------------------
4.94/2.29	
4.94/2.29	(4)
4.94/2.29	Obligation:
4.94/2.29	Q restricted rewrite system:
4.94/2.29	The TRS R consists of the following rules:
4.94/2.29	
4.94/2.29	   c -> c
4.94/2.29	
4.94/2.29	Q is empty.
4.94/2.29	
4.94/2.29	----------------------------------------
4.94/2.29	
4.94/2.29	(5) AAECC Innermost (EQUIVALENT)
4.94/2.29	We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is none
4.94/2.29	
4.94/2.29	The TRS R 2 is 
4.94/2.29	   c -> c
4.94/2.29	
4.94/2.29	The signature Sigma is {c}
4.94/2.29	----------------------------------------
4.94/2.29	
4.94/2.29	(6)
4.94/2.29	Obligation:
4.94/2.29	Q restricted rewrite system:
4.94/2.29	The TRS R consists of the following rules:
4.94/2.29	
4.94/2.29	   c -> c
4.94/2.29	
4.94/2.29	The set Q consists of the following terms:
4.94/2.29	
4.94/2.29	   c
4.94/2.29	
4.94/2.29	
4.94/2.29	----------------------------------------
4.94/2.29	
4.94/2.29	(7) DependencyPairsProof (EQUIVALENT)
4.94/2.29	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
4.94/2.29	----------------------------------------
4.94/2.29	
4.94/2.29	(8)
4.94/2.29	Obligation:
4.94/2.29	Q DP problem:
4.94/2.29	The TRS P consists of the following rules:
4.94/2.29	
4.94/2.29	   C -> C
4.94/2.29	
4.94/2.29	The TRS R consists of the following rules:
4.94/2.29	
4.94/2.29	   c -> c
4.94/2.29	
4.94/2.29	The set Q consists of the following terms:
4.94/2.29	
4.94/2.29	   c
4.94/2.29	
4.94/2.29	We have to consider all minimal (P,Q,R)-chains.
4.94/2.29	----------------------------------------
4.94/2.29	
4.94/2.29	(9) UsableRulesProof (EQUIVALENT)
4.94/2.29	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.
4.94/2.29	----------------------------------------
4.94/2.29	
4.94/2.29	(10)
4.94/2.29	Obligation:
4.94/2.29	Q DP problem:
4.94/2.29	The TRS P consists of the following rules:
4.94/2.29	
4.94/2.29	   C -> C
4.94/2.29	
4.94/2.29	R is empty.
4.94/2.29	The set Q consists of the following terms:
4.94/2.29	
4.94/2.29	   c
4.94/2.29	
4.94/2.29	We have to consider all minimal (P,Q,R)-chains.
4.94/2.29	----------------------------------------
4.94/2.29	
4.94/2.29	(11) QReductionProof (EQUIVALENT)
4.94/2.29	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
4.94/2.29	
4.94/2.29	   c
4.94/2.29	
4.94/2.29	
4.94/2.29	----------------------------------------
4.94/2.29	
4.94/2.29	(12)
4.94/2.29	Obligation:
4.94/2.29	Q DP problem:
4.94/2.29	The TRS P consists of the following rules:
4.94/2.29	
4.94/2.29	   C -> C
4.94/2.29	
4.94/2.29	R is empty.
4.94/2.29	Q is empty.
4.94/2.29	We have to consider all minimal (P,Q,R)-chains.
4.94/2.29	----------------------------------------
4.94/2.29	
4.94/2.29	(13) NonTerminationLoopProof (COMPLETE)
4.94/2.29	We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
4.94/2.29	Found a loop by semiunifying a rule from P directly.
4.94/2.29	
4.94/2.29	s = C evaluates to  t =C
4.94/2.29	
4.94/2.29	Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
4.94/2.29	* Matcher: [ ]
4.94/2.29	* Semiunifier: [ ]
4.94/2.29	
4.94/2.29	--------------------------------------------------------------------------------
4.94/2.29	Rewriting sequence
4.94/2.29	
4.94/2.29	The DP semiunifies directly so there is only one rewrite step from C to C.
4.94/2.29	
4.94/2.29	
4.94/2.29	
4.94/2.29	
4.94/2.29	----------------------------------------
4.94/2.29	
4.94/2.29	(14)
4.94/2.29	NO
5.15/2.32	EOF