7.23/2.72	YES
8.03/2.87	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
8.03/2.87	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
8.03/2.87	
8.03/2.87	
8.03/2.87	Termination w.r.t. Q of the given QTRS could be proven:
8.03/2.87	
8.03/2.87	(0) QTRS
8.03/2.87	(1) DependencyPairsProof [EQUIVALENT, 0 ms]
8.03/2.87	(2) QDP
8.03/2.87	(3) DependencyGraphProof [EQUIVALENT, 1 ms]
8.03/2.87	(4) QDP
8.03/2.87	(5) QDPSizeChangeProof [EQUIVALENT, 4 ms]
8.03/2.87	(6) YES
8.03/2.87	
8.03/2.87	
8.03/2.87	----------------------------------------
8.03/2.87	
8.03/2.87	(0)
8.03/2.87	Obligation:
8.03/2.87	Q restricted rewrite system:
8.03/2.87	The TRS R consists of the following rules:
8.03/2.87	
8.03/2.87	   a(x1) -> x1
8.03/2.87	   a(a(x1)) -> a(b(c(a(x1))))
8.03/2.87	   c(b(x1)) -> a(b(a(x1)))
8.03/2.87	
8.03/2.87	Q is empty.
8.03/2.87	
8.03/2.87	----------------------------------------
8.03/2.87	
8.03/2.87	(1) DependencyPairsProof (EQUIVALENT)
8.03/2.87	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
8.03/2.87	----------------------------------------
8.03/2.87	
8.03/2.87	(2)
8.03/2.87	Obligation:
8.03/2.87	Q DP problem:
8.03/2.87	The TRS P consists of the following rules:
8.03/2.87	
8.03/2.87	   A(a(x1)) -> A(b(c(a(x1))))
8.03/2.87	   A(a(x1)) -> C(a(x1))
8.03/2.87	   C(b(x1)) -> A(b(a(x1)))
8.03/2.87	   C(b(x1)) -> A(x1)
8.03/2.87	
8.03/2.87	The TRS R consists of the following rules:
8.03/2.87	
8.03/2.87	   a(x1) -> x1
8.03/2.87	   a(a(x1)) -> a(b(c(a(x1))))
8.03/2.87	   c(b(x1)) -> a(b(a(x1)))
8.03/2.87	
8.03/2.87	Q is empty.
8.03/2.87	We have to consider all minimal (P,Q,R)-chains.
8.03/2.87	----------------------------------------
8.03/2.87	
8.03/2.87	(3) DependencyGraphProof (EQUIVALENT)
8.03/2.87	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
8.03/2.87	----------------------------------------
8.03/2.87	
8.03/2.87	(4)
8.03/2.87	Obligation:
8.03/2.87	Q DP problem:
8.03/2.87	The TRS P consists of the following rules:
8.03/2.87	
8.03/2.87	   A(a(x1)) -> C(a(x1))
8.03/2.87	   C(b(x1)) -> A(x1)
8.03/2.87	
8.03/2.87	The TRS R consists of the following rules:
8.03/2.87	
8.03/2.87	   a(x1) -> x1
8.03/2.87	   a(a(x1)) -> a(b(c(a(x1))))
8.03/2.87	   c(b(x1)) -> a(b(a(x1)))
8.03/2.87	
8.03/2.87	Q is empty.
8.03/2.87	We have to consider all minimal (P,Q,R)-chains.
8.03/2.87	----------------------------------------
8.03/2.87	
8.03/2.87	(5) QDPSizeChangeProof (EQUIVALENT)
8.03/2.87	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. 
8.03/2.87	
8.03/2.87	From the DPs we obtained the following set of size-change graphs:
8.03/2.87	*C(b(x1)) -> A(x1)
8.03/2.87	The graph contains the following edges 1 > 1
8.03/2.87	
8.03/2.87	
8.03/2.87	*A(a(x1)) -> C(a(x1))
8.03/2.87	The graph contains the following edges 1 >= 1
8.03/2.87	
8.03/2.87	
8.03/2.87	----------------------------------------
8.03/2.87	
8.03/2.87	(6)
8.03/2.87	YES
8.20/2.97	EOF