28.51/8.30	YES
29.42/11.54	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
29.42/11.54	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
29.42/11.54	
29.42/11.54	
29.42/11.54	Termination w.r.t. Q of the given QTRS could be proven:
29.42/11.54	
29.42/11.54	(0) QTRS
29.42/11.54	(1) QTRS Reverse [EQUIVALENT, 0 ms]
29.42/11.54	(2) QTRS
29.42/11.54	(3) QTRSRRRProof [EQUIVALENT, 27 ms]
29.42/11.54	(4) QTRS
29.42/11.54	(5) Overlay + Local Confluence [EQUIVALENT, 1 ms]
29.42/11.54	(6) QTRS
29.42/11.54	(7) DependencyPairsProof [EQUIVALENT, 2 ms]
29.42/11.54	(8) QDP
29.42/11.54	(9) UsableRulesProof [EQUIVALENT, 1 ms]
29.42/11.54	(10) QDP
29.42/11.54	(11) QReductionProof [EQUIVALENT, 0 ms]
29.42/11.54	(12) QDP
29.42/11.54	(13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
29.42/11.54	(14) YES
29.42/11.54	
29.42/11.54	
29.42/11.54	----------------------------------------
29.42/11.54	
29.42/11.54	(0)
29.42/11.54	Obligation:
29.42/11.54	Q restricted rewrite system:
29.42/11.54	The TRS R consists of the following rules:
29.42/11.54	
29.42/11.54	   b(c(x1)) -> a(x1)
29.42/11.54	   b(b(x1)) -> a(c(x1))
29.42/11.54	   a(x1) -> c(b(x1))
29.42/11.54	   c(c(c(x1))) -> b(x1)
29.42/11.54	
29.42/11.54	Q is empty.
29.42/11.54	
29.42/11.54	----------------------------------------
29.42/11.54	
29.42/11.54	(1) QTRS Reverse (EQUIVALENT)
29.42/11.54	We applied the QTRS Reverse Processor [REVERSE].
29.42/11.54	----------------------------------------
29.42/11.54	
29.42/11.54	(2)
29.42/11.54	Obligation:
29.42/11.54	Q restricted rewrite system:
29.42/11.54	The TRS R consists of the following rules:
29.42/11.54	
29.42/11.54	   c(b(x1)) -> a(x1)
29.42/11.54	   b(b(x1)) -> c(a(x1))
29.42/11.54	   a(x1) -> b(c(x1))
29.42/11.54	   c(c(c(x1))) -> b(x1)
29.42/11.54	
29.42/11.54	Q is empty.
29.42/11.54	
29.42/11.54	----------------------------------------
29.42/11.54	
29.42/11.54	(3) QTRSRRRProof (EQUIVALENT)
29.42/11.54	Used ordering:
29.42/11.54	Polynomial interpretation [POLO]:
29.42/11.54	
29.42/11.54	   POL(a(x_1)) = 7 + x_1
29.42/11.54	   POL(b(x_1)) = 5 + x_1
29.42/11.54	   POL(c(x_1)) = 2 + x_1
29.42/11.54	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
29.42/11.54	
29.42/11.54	   b(b(x1)) -> c(a(x1))
29.42/11.54	   c(c(c(x1))) -> b(x1)
29.42/11.54	
29.42/11.54	
29.42/11.54	
29.42/11.54	
29.42/11.54	----------------------------------------
29.42/11.54	
29.42/11.54	(4)
29.42/11.54	Obligation:
29.42/11.54	Q restricted rewrite system:
29.42/11.54	The TRS R consists of the following rules:
29.42/11.54	
29.42/11.54	   c(b(x1)) -> a(x1)
29.42/11.54	   a(x1) -> b(c(x1))
29.42/11.54	
29.42/11.54	Q is empty.
29.42/11.54	
29.42/11.54	----------------------------------------
29.42/11.54	
29.42/11.54	(5) Overlay + Local Confluence (EQUIVALENT)
29.42/11.54	The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
29.42/11.54	----------------------------------------
29.42/11.54	
29.42/11.54	(6)
29.42/11.54	Obligation:
29.42/11.54	Q restricted rewrite system:
29.42/11.54	The TRS R consists of the following rules:
29.42/11.54	
29.42/11.54	   c(b(x1)) -> a(x1)
29.42/11.54	   a(x1) -> b(c(x1))
29.42/11.54	
29.42/11.54	The set Q consists of the following terms:
29.42/11.54	
29.42/11.54	   c(b(x0))
29.42/11.54	   a(x0)
29.42/11.54	
29.42/11.54	
29.42/11.54	----------------------------------------
29.42/11.54	
29.42/11.54	(7) DependencyPairsProof (EQUIVALENT)
29.42/11.54	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
29.42/11.54	----------------------------------------
29.42/11.54	
29.42/11.54	(8)
29.42/11.54	Obligation:
29.42/11.54	Q DP problem:
29.42/11.54	The TRS P consists of the following rules:
29.42/11.54	
29.42/11.54	   C(b(x1)) -> A(x1)
29.42/11.54	   A(x1) -> C(x1)
29.42/11.54	
29.42/11.54	The TRS R consists of the following rules:
29.42/11.54	
29.42/11.54	   c(b(x1)) -> a(x1)
29.42/11.54	   a(x1) -> b(c(x1))
29.42/11.54	
29.42/11.54	The set Q consists of the following terms:
29.42/11.54	
29.42/11.54	   c(b(x0))
29.42/11.54	   a(x0)
29.42/11.54	
29.42/11.54	We have to consider all minimal (P,Q,R)-chains.
29.42/11.54	----------------------------------------
29.42/11.54	
29.42/11.54	(9) UsableRulesProof (EQUIVALENT)
29.42/11.54	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.
29.42/11.54	----------------------------------------
29.42/11.54	
29.42/11.54	(10)
29.42/11.54	Obligation:
29.42/11.54	Q DP problem:
29.42/11.54	The TRS P consists of the following rules:
29.42/11.54	
29.42/11.54	   C(b(x1)) -> A(x1)
29.42/11.54	   A(x1) -> C(x1)
29.42/11.54	
29.42/11.54	R is empty.
29.42/11.54	The set Q consists of the following terms:
29.42/11.54	
29.42/11.54	   c(b(x0))
29.42/11.54	   a(x0)
29.42/11.54	
29.42/11.54	We have to consider all minimal (P,Q,R)-chains.
29.42/11.54	----------------------------------------
29.42/11.54	
29.42/11.54	(11) QReductionProof (EQUIVALENT)
29.42/11.54	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
29.42/11.54	
29.42/11.54	   c(b(x0))
29.42/11.54	   a(x0)
29.42/11.54	
29.42/11.54	
29.42/11.54	----------------------------------------
29.42/11.54	
29.42/11.54	(12)
29.42/11.54	Obligation:
29.42/11.54	Q DP problem:
29.42/11.54	The TRS P consists of the following rules:
29.42/11.54	
29.42/11.54	   C(b(x1)) -> A(x1)
29.42/11.54	   A(x1) -> C(x1)
29.42/11.54	
29.42/11.54	R is empty.
29.42/11.54	Q is empty.
29.42/11.54	We have to consider all minimal (P,Q,R)-chains.
29.42/11.54	----------------------------------------
29.42/11.54	
29.42/11.54	(13) QDPSizeChangeProof (EQUIVALENT)
29.42/11.54	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. 
29.42/11.54	
29.42/11.54	From the DPs we obtained the following set of size-change graphs:
29.42/11.54	*A(x1) -> C(x1)
29.42/11.54	The graph contains the following edges 1 >= 1
29.42/11.54	
29.42/11.54	
29.42/11.54	*C(b(x1)) -> A(x1)
29.42/11.54	The graph contains the following edges 1 > 1
29.42/11.54	
29.42/11.54	
29.42/11.54	----------------------------------------
29.42/11.54	
29.42/11.54	(14)
29.42/11.54	YES
29.57/11.58	EOF