5.76/2.35	YES
6.34/2.49	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
6.34/2.49	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.34/2.49	
6.34/2.49	
6.34/2.49	Termination w.r.t. Q of the given QTRS could be proven:
6.34/2.49	
6.34/2.49	(0) QTRS
6.34/2.49	(1) QTRSRRRProof [EQUIVALENT, 41 ms]
6.34/2.49	(2) QTRS
6.34/2.49	(3) Overlay + Local Confluence [EQUIVALENT, 5 ms]
6.34/2.49	(4) QTRS
6.34/2.49	(5) DependencyPairsProof [EQUIVALENT, 23 ms]
6.34/2.49	(6) QDP
6.34/2.49	(7) UsableRulesProof [EQUIVALENT, 1 ms]
6.34/2.49	(8) QDP
6.34/2.49	(9) QReductionProof [EQUIVALENT, 0 ms]
6.34/2.49	(10) QDP
6.34/2.49	(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
6.34/2.49	(12) YES
6.34/2.49	
6.34/2.49	
6.34/2.49	----------------------------------------
6.34/2.49	
6.34/2.49	(0)
6.34/2.49	Obligation:
6.34/2.49	Q restricted rewrite system:
6.34/2.49	The TRS R consists of the following rules:
6.34/2.49	
6.34/2.49	   b(c(x1)) -> c(b(x1))
6.34/2.49	   c(b(x1)) -> a(a(a(x1)))
6.34/2.49	   a(a(a(a(x1)))) -> b(c(x1))
6.34/2.49	
6.34/2.49	Q is empty.
6.34/2.49	
6.34/2.49	----------------------------------------
6.34/2.49	
6.34/2.49	(1) QTRSRRRProof (EQUIVALENT)
6.34/2.49	Used ordering:
6.34/2.49	Polynomial interpretation [POLO]:
6.34/2.49	
6.34/2.49	   POL(a(x_1)) = 2 + x_1
6.34/2.49	   POL(b(x_1)) = x_1
6.34/2.49	   POL(c(x_1)) = 7 + x_1
6.34/2.49	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
6.34/2.49	
6.34/2.49	   c(b(x1)) -> a(a(a(x1)))
6.34/2.49	   a(a(a(a(x1)))) -> b(c(x1))
6.34/2.49	
6.34/2.49	
6.34/2.49	
6.34/2.49	
6.34/2.49	----------------------------------------
6.34/2.49	
6.34/2.49	(2)
6.34/2.49	Obligation:
6.34/2.49	Q restricted rewrite system:
6.34/2.49	The TRS R consists of the following rules:
6.34/2.49	
6.34/2.49	   b(c(x1)) -> c(b(x1))
6.34/2.49	
6.34/2.49	Q is empty.
6.34/2.49	
6.34/2.49	----------------------------------------
6.34/2.49	
6.34/2.49	(3) Overlay + Local Confluence (EQUIVALENT)
6.34/2.49	The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
6.34/2.49	----------------------------------------
6.34/2.49	
6.34/2.49	(4)
6.34/2.49	Obligation:
6.34/2.49	Q restricted rewrite system:
6.34/2.49	The TRS R consists of the following rules:
6.34/2.49	
6.34/2.49	   b(c(x1)) -> c(b(x1))
6.34/2.49	
6.34/2.49	The set Q consists of the following terms:
6.34/2.49	
6.34/2.49	   b(c(x0))
6.34/2.49	
6.34/2.49	
6.34/2.49	----------------------------------------
6.34/2.49	
6.34/2.49	(5) DependencyPairsProof (EQUIVALENT)
6.34/2.49	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
6.34/2.49	----------------------------------------
6.34/2.49	
6.34/2.49	(6)
6.34/2.49	Obligation:
6.34/2.49	Q DP problem:
6.34/2.49	The TRS P consists of the following rules:
6.34/2.49	
6.34/2.49	   B(c(x1)) -> B(x1)
6.34/2.49	
6.34/2.49	The TRS R consists of the following rules:
6.34/2.49	
6.34/2.49	   b(c(x1)) -> c(b(x1))
6.34/2.49	
6.34/2.49	The set Q consists of the following terms:
6.34/2.49	
6.34/2.49	   b(c(x0))
6.34/2.49	
6.34/2.49	We have to consider all minimal (P,Q,R)-chains.
6.34/2.49	----------------------------------------
6.34/2.49	
6.34/2.49	(7) UsableRulesProof (EQUIVALENT)
6.34/2.49	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.
6.34/2.49	----------------------------------------
6.34/2.49	
6.34/2.49	(8)
6.34/2.49	Obligation:
6.34/2.49	Q DP problem:
6.34/2.49	The TRS P consists of the following rules:
6.34/2.49	
6.34/2.49	   B(c(x1)) -> B(x1)
6.34/2.49	
6.34/2.49	R is empty.
6.34/2.49	The set Q consists of the following terms:
6.34/2.49	
6.34/2.49	   b(c(x0))
6.34/2.49	
6.34/2.49	We have to consider all minimal (P,Q,R)-chains.
6.34/2.49	----------------------------------------
6.34/2.49	
6.34/2.49	(9) QReductionProof (EQUIVALENT)
6.34/2.49	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
6.34/2.49	
6.34/2.49	   b(c(x0))
6.34/2.49	
6.34/2.49	
6.34/2.49	----------------------------------------
6.34/2.49	
6.34/2.49	(10)
6.34/2.49	Obligation:
6.34/2.49	Q DP problem:
6.34/2.49	The TRS P consists of the following rules:
6.34/2.49	
6.34/2.49	   B(c(x1)) -> B(x1)
6.34/2.49	
6.34/2.49	R is empty.
6.34/2.49	Q is empty.
6.34/2.49	We have to consider all minimal (P,Q,R)-chains.
6.34/2.49	----------------------------------------
6.34/2.49	
6.34/2.49	(11) QDPSizeChangeProof (EQUIVALENT)
6.34/2.49	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. 
6.34/2.49	
6.34/2.49	From the DPs we obtained the following set of size-change graphs:
6.34/2.49	*B(c(x1)) -> B(x1)
6.34/2.49	The graph contains the following edges 1 > 1
6.34/2.49	
6.34/2.49	
6.34/2.49	----------------------------------------
6.34/2.49	
6.34/2.49	(12)
6.34/2.49	YES
6.34/2.54	EOF