4.79/2.19	MAYBE
4.79/2.20	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
4.79/2.20	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.79/2.20	
4.79/2.20	
4.79/2.20	Quasi decreasingness of the given CTRS could not be shown:
4.79/2.20	
4.79/2.20	(0) CTRS
4.79/2.20	(1) CTRSToQTRSProof [SOUND, 0 ms]
4.79/2.20	(2) QTRS
4.79/2.20	(3) QTRSRRRProof [EQUIVALENT, 29 ms]
4.79/2.20	(4) QTRS
4.79/2.20	(5) AAECC Innermost [EQUIVALENT, 0 ms]
4.79/2.20	(6) QTRS
4.79/2.20	(7) DependencyPairsProof [EQUIVALENT, 0 ms]
4.79/2.20	(8) QDP
4.79/2.20	(9) UsableRulesProof [EQUIVALENT, 0 ms]
4.79/2.20	(10) QDP
4.79/2.20	(11) QReductionProof [EQUIVALENT, 0 ms]
4.79/2.20	(12) QDP
4.79/2.20	(13) NonTerminationLoopProof [COMPLETE, 0 ms]
4.79/2.20	(14) NO
4.79/2.20	
4.79/2.20	
4.79/2.20	----------------------------------------
4.79/2.20	
4.79/2.20	(0)
4.79/2.20	Obligation:
4.79/2.20	Conditional term rewrite system:
4.79/2.20	The TRS R consists of the following rules:
4.79/2.20	
4.79/2.20	   g(s(x)) -> x
4.79/2.20	   h(s(x)) -> x
4.79/2.20	   a -> d
4.79/2.20	   b -> d
4.79/2.20	   e -> e
4.79/2.20	
4.79/2.20	The conditional TRS C consists of the following conditional rules:
4.79/2.20	
4.79/2.20	   f(x, y) -> g(x) <= a -> d
4.79/2.20	   f(x, y) -> h(x) <= b -> d
4.79/2.20	
4.79/2.20	
4.79/2.20	----------------------------------------
4.79/2.20	
4.79/2.20	(1) CTRSToQTRSProof (SOUND)
4.79/2.20	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
4.79/2.20	----------------------------------------
4.79/2.20	
4.79/2.20	(2)
4.79/2.20	Obligation:
4.79/2.20	Q restricted rewrite system:
4.79/2.20	The TRS R consists of the following rules:
4.79/2.20	
4.79/2.20	   f(x, y) -> U1(a, x)
4.79/2.20	   U1(d, x) -> g(x)
4.79/2.20	   f(x, y) -> U2(b, x)
4.79/2.20	   U2(d, x) -> h(x)
4.79/2.20	   g(s(x)) -> x
4.79/2.20	   h(s(x)) -> x
4.79/2.20	   a -> d
4.79/2.20	   b -> d
4.79/2.20	   e -> e
4.79/2.20	
4.79/2.20	Q is empty.
4.79/2.20	
4.79/2.20	----------------------------------------
4.79/2.20	
4.79/2.20	(3) QTRSRRRProof (EQUIVALENT)
4.79/2.20	Used ordering:
4.79/2.20	Knuth-Bendix order [KBO] with precedence:b > f_2 > U2_2 > g_1 > d > a > U1_2 > e > s_1 > h_1
4.79/2.20	
4.79/2.20	and weight map:
4.79/2.20	
4.79/2.20	   a=2
4.79/2.20	   d=1
4.79/2.20	   b=2
4.79/2.20	   e=1
4.79/2.20	   g_1=1
4.79/2.20	   h_1=1
4.79/2.20	   s_1=1
4.79/2.20	   f_2=2
4.79/2.20	   U1_2=1
4.79/2.20	   U2_2=1
4.79/2.20	
4.79/2.20	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.79/2.20	
4.79/2.20	   f(x, y) -> U1(a, x)
4.79/2.20	   U1(d, x) -> g(x)
4.79/2.20	   f(x, y) -> U2(b, x)
4.79/2.20	   U2(d, x) -> h(x)
4.79/2.20	   g(s(x)) -> x
4.79/2.20	   h(s(x)) -> x
4.79/2.20	   a -> d
4.79/2.20	   b -> d
4.79/2.20	
4.79/2.20	
4.79/2.20	
4.79/2.20	
4.79/2.20	----------------------------------------
4.79/2.20	
4.79/2.20	(4)
4.79/2.20	Obligation:
4.79/2.20	Q restricted rewrite system:
4.79/2.20	The TRS R consists of the following rules:
4.79/2.20	
4.79/2.20	   e -> e
4.79/2.20	
4.79/2.20	Q is empty.
4.79/2.20	
4.79/2.20	----------------------------------------
4.79/2.20	
4.79/2.20	(5) AAECC Innermost (EQUIVALENT)
4.79/2.20	We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is none
4.79/2.20	
4.79/2.20	The TRS R 2 is 
4.79/2.20	   e -> e
4.79/2.20	
4.79/2.20	The signature Sigma is {e}
4.79/2.20	----------------------------------------
4.79/2.20	
4.79/2.20	(6)
4.79/2.20	Obligation:
4.79/2.20	Q restricted rewrite system:
4.79/2.20	The TRS R consists of the following rules:
4.79/2.20	
4.79/2.20	   e -> e
4.79/2.20	
4.79/2.20	The set Q consists of the following terms:
4.79/2.20	
4.79/2.20	   e
4.79/2.20	
4.79/2.20	
4.79/2.20	----------------------------------------
4.79/2.20	
4.79/2.20	(7) DependencyPairsProof (EQUIVALENT)
4.79/2.20	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
4.79/2.20	----------------------------------------
4.79/2.20	
4.79/2.20	(8)
4.79/2.20	Obligation:
4.79/2.20	Q DP problem:
4.79/2.20	The TRS P consists of the following rules:
4.79/2.20	
4.79/2.20	   E -> E
4.79/2.20	
4.79/2.20	The TRS R consists of the following rules:
4.79/2.20	
4.79/2.20	   e -> e
4.79/2.20	
4.79/2.20	The set Q consists of the following terms:
4.79/2.20	
4.79/2.20	   e
4.79/2.20	
4.79/2.20	We have to consider all minimal (P,Q,R)-chains.
4.79/2.20	----------------------------------------
4.79/2.20	
4.79/2.20	(9) UsableRulesProof (EQUIVALENT)
4.79/2.20	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.
4.79/2.20	----------------------------------------
4.79/2.20	
4.79/2.20	(10)
4.79/2.20	Obligation:
4.79/2.20	Q DP problem:
4.79/2.20	The TRS P consists of the following rules:
4.79/2.20	
4.79/2.20	   E -> E
4.79/2.20	
4.79/2.20	R is empty.
4.79/2.20	The set Q consists of the following terms:
4.79/2.20	
4.79/2.20	   e
4.79/2.20	
4.79/2.20	We have to consider all minimal (P,Q,R)-chains.
4.79/2.20	----------------------------------------
4.79/2.20	
4.79/2.20	(11) QReductionProof (EQUIVALENT)
4.79/2.20	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
4.79/2.20	
4.79/2.20	   e
4.79/2.20	
4.79/2.20	
4.79/2.20	----------------------------------------
4.79/2.20	
4.79/2.20	(12)
4.79/2.20	Obligation:
4.79/2.20	Q DP problem:
4.79/2.20	The TRS P consists of the following rules:
4.79/2.20	
4.79/2.20	   E -> E
4.79/2.20	
4.79/2.20	R is empty.
4.79/2.20	Q is empty.
4.79/2.20	We have to consider all minimal (P,Q,R)-chains.
4.79/2.20	----------------------------------------
4.79/2.20	
4.79/2.20	(13) NonTerminationLoopProof (COMPLETE)
4.79/2.20	We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
4.79/2.20	Found a loop by semiunifying a rule from P directly.
4.79/2.20	
4.79/2.20	s = E evaluates to  t =E
4.79/2.20	
4.79/2.20	Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
4.79/2.20	* Matcher: [ ]
4.79/2.20	* Semiunifier: [ ]
4.79/2.20	
4.79/2.20	--------------------------------------------------------------------------------
4.79/2.20	Rewriting sequence
4.79/2.20	
4.79/2.20	The DP semiunifies directly so there is only one rewrite step from E to E.
4.79/2.20	
4.79/2.20	
4.79/2.20	
4.79/2.20	
4.79/2.20	----------------------------------------
4.79/2.20	
4.79/2.20	(14)
4.79/2.20	NO
4.79/2.25	EOF