11.08/3.73	YES
11.08/3.78	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
11.08/3.78	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.08/3.78	
11.08/3.78	
11.08/3.78	Termination w.r.t. Q of the given QTRS could be proven:
11.08/3.78	
11.08/3.78	(0) QTRS
11.08/3.78	(1) QTRSRRRProof [EQUIVALENT, 53 ms]
11.08/3.78	(2) QTRS
11.08/3.78	(3) Overlay + Local Confluence [EQUIVALENT, 5 ms]
11.08/3.78	(4) QTRS
11.08/3.78	(5) DependencyPairsProof [EQUIVALENT, 2 ms]
11.08/3.78	(6) QDP
11.08/3.78	(7) UsableRulesProof [EQUIVALENT, 1 ms]
11.08/3.78	(8) QDP
11.08/3.78	(9) QReductionProof [EQUIVALENT, 0 ms]
11.08/3.78	(10) QDP
11.08/3.78	(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.08/3.78	(12) YES
11.08/3.78	
11.08/3.78	
11.08/3.78	----------------------------------------
11.08/3.78	
11.08/3.78	(0)
11.08/3.78	Obligation:
11.08/3.78	Q restricted rewrite system:
11.08/3.78	The TRS R consists of the following rules:
11.08/3.78	
11.08/3.78	   b(a(x1)) -> a(b(x1))
11.08/3.78	   a(a(a(x1))) -> b(a(a(b(x1))))
11.08/3.78	   b(b(b(b(x1)))) -> a(x1)
11.08/3.78	
11.08/3.78	Q is empty.
11.08/3.78	
11.08/3.78	----------------------------------------
11.08/3.78	
11.08/3.78	(1) QTRSRRRProof (EQUIVALENT)
11.08/3.78	Used ordering:
11.08/3.78	Polynomial interpretation [POLO]:
11.08/3.78	
11.08/3.78	   POL(a(x_1)) = 3 + x_1
11.08/3.78	   POL(b(x_1)) = 1 + x_1
11.08/3.78	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
11.08/3.78	
11.08/3.78	   a(a(a(x1))) -> b(a(a(b(x1))))
11.08/3.78	   b(b(b(b(x1)))) -> a(x1)
11.08/3.78	
11.08/3.78	
11.08/3.78	
11.08/3.78	
11.08/3.78	----------------------------------------
11.08/3.78	
11.08/3.78	(2)
11.08/3.78	Obligation:
11.08/3.78	Q restricted rewrite system:
11.08/3.78	The TRS R consists of the following rules:
11.08/3.78	
11.08/3.78	   b(a(x1)) -> a(b(x1))
11.08/3.78	
11.08/3.78	Q is empty.
11.08/3.78	
11.08/3.78	----------------------------------------
11.08/3.78	
11.08/3.78	(3) Overlay + Local Confluence (EQUIVALENT)
11.08/3.78	The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
11.08/3.78	----------------------------------------
11.08/3.78	
11.08/3.78	(4)
11.08/3.78	Obligation:
11.08/3.78	Q restricted rewrite system:
11.08/3.78	The TRS R consists of the following rules:
11.08/3.78	
11.08/3.78	   b(a(x1)) -> a(b(x1))
11.08/3.78	
11.08/3.78	The set Q consists of the following terms:
11.08/3.78	
11.08/3.78	   b(a(x0))
11.08/3.78	
11.08/3.78	
11.08/3.78	----------------------------------------
11.08/3.78	
11.08/3.78	(5) DependencyPairsProof (EQUIVALENT)
11.08/3.78	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
11.08/3.78	----------------------------------------
11.08/3.78	
11.08/3.78	(6)
11.08/3.78	Obligation:
11.08/3.78	Q DP problem:
11.08/3.78	The TRS P consists of the following rules:
11.08/3.78	
11.08/3.78	   B(a(x1)) -> B(x1)
11.08/3.78	
11.08/3.78	The TRS R consists of the following rules:
11.08/3.78	
11.08/3.78	   b(a(x1)) -> a(b(x1))
11.08/3.78	
11.08/3.78	The set Q consists of the following terms:
11.08/3.78	
11.08/3.78	   b(a(x0))
11.08/3.78	
11.08/3.78	We have to consider all minimal (P,Q,R)-chains.
11.08/3.78	----------------------------------------
11.08/3.78	
11.08/3.78	(7) UsableRulesProof (EQUIVALENT)
11.08/3.78	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.
11.08/3.78	----------------------------------------
11.08/3.78	
11.08/3.78	(8)
11.08/3.78	Obligation:
11.08/3.78	Q DP problem:
11.08/3.78	The TRS P consists of the following rules:
11.08/3.78	
11.08/3.78	   B(a(x1)) -> B(x1)
11.08/3.78	
11.08/3.78	R is empty.
11.08/3.78	The set Q consists of the following terms:
11.08/3.78	
11.08/3.78	   b(a(x0))
11.08/3.78	
11.08/3.78	We have to consider all minimal (P,Q,R)-chains.
11.08/3.78	----------------------------------------
11.08/3.78	
11.08/3.78	(9) QReductionProof (EQUIVALENT)
11.08/3.78	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
11.08/3.78	
11.08/3.78	   b(a(x0))
11.08/3.78	
11.08/3.78	
11.08/3.78	----------------------------------------
11.08/3.78	
11.08/3.78	(10)
11.08/3.78	Obligation:
11.08/3.78	Q DP problem:
11.08/3.78	The TRS P consists of the following rules:
11.08/3.78	
11.08/3.78	   B(a(x1)) -> B(x1)
11.08/3.78	
11.08/3.78	R is empty.
11.08/3.78	Q is empty.
11.08/3.78	We have to consider all minimal (P,Q,R)-chains.
11.08/3.78	----------------------------------------
11.08/3.78	
11.08/3.78	(11) QDPSizeChangeProof (EQUIVALENT)
11.08/3.78	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
11.08/3.78	
11.08/3.78	From the DPs we obtained the following set of size-change graphs:
11.08/3.78	*B(a(x1)) -> B(x1)
11.08/3.78	The graph contains the following edges 1 > 1
11.08/3.78	
11.08/3.78	
11.08/3.78	----------------------------------------
11.08/3.78	
11.08/3.78	(12)
11.08/3.78	YES
11.53/3.88	EOF