5.98/2.29	YES
6.68/2.43	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
6.68/2.43	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.68/2.43	
6.68/2.43	
6.68/2.43	Termination w.r.t. Q of the given QTRS could be proven:
6.68/2.43	
6.68/2.43	(0) QTRS
6.68/2.43	(1) DependencyPairsProof [EQUIVALENT, 0 ms]
6.68/2.43	(2) QDP
6.68/2.43	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
6.68/2.43	(4) QDP
6.68/2.43	(5) QDPSizeChangeProof [EQUIVALENT, 0 ms]
6.68/2.43	(6) YES
6.68/2.43	
6.68/2.43	
6.68/2.43	----------------------------------------
6.68/2.43	
6.68/2.43	(0)
6.68/2.43	Obligation:
6.68/2.43	Q restricted rewrite system:
6.68/2.43	The TRS R consists of the following rules:
6.68/2.43	
6.68/2.43	   a(x1) -> x1
6.68/2.43	   a(a(x1)) -> a(b(a(b(c(a(x1))))))
6.68/2.43	   c(b(x1)) -> a(x1)
6.68/2.43	
6.68/2.43	Q is empty.
6.68/2.43	
6.68/2.43	----------------------------------------
6.68/2.43	
6.68/2.43	(1) DependencyPairsProof (EQUIVALENT)
6.68/2.43	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
6.68/2.43	----------------------------------------
6.68/2.43	
6.68/2.43	(2)
6.68/2.43	Obligation:
6.68/2.43	Q DP problem:
6.68/2.43	The TRS P consists of the following rules:
6.68/2.43	
6.68/2.43	   A(a(x1)) -> A(b(a(b(c(a(x1))))))
6.68/2.43	   A(a(x1)) -> A(b(c(a(x1))))
6.68/2.43	   A(a(x1)) -> C(a(x1))
6.68/2.43	   C(b(x1)) -> A(x1)
6.68/2.43	
6.68/2.43	The TRS R consists of the following rules:
6.68/2.43	
6.68/2.43	   a(x1) -> x1
6.68/2.43	   a(a(x1)) -> a(b(a(b(c(a(x1))))))
6.68/2.43	   c(b(x1)) -> a(x1)
6.68/2.43	
6.68/2.43	Q is empty.
6.68/2.43	We have to consider all minimal (P,Q,R)-chains.
6.68/2.43	----------------------------------------
6.68/2.43	
6.68/2.43	(3) DependencyGraphProof (EQUIVALENT)
6.68/2.43	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
6.68/2.43	----------------------------------------
6.68/2.43	
6.68/2.43	(4)
6.68/2.43	Obligation:
6.68/2.43	Q DP problem:
6.68/2.43	The TRS P consists of the following rules:
6.68/2.43	
6.68/2.43	   A(a(x1)) -> C(a(x1))
6.68/2.43	   C(b(x1)) -> A(x1)
6.68/2.43	
6.68/2.43	The TRS R consists of the following rules:
6.68/2.43	
6.68/2.43	   a(x1) -> x1
6.68/2.43	   a(a(x1)) -> a(b(a(b(c(a(x1))))))
6.68/2.43	   c(b(x1)) -> a(x1)
6.68/2.43	
6.68/2.43	Q is empty.
6.68/2.43	We have to consider all minimal (P,Q,R)-chains.
6.68/2.43	----------------------------------------
6.68/2.43	
6.68/2.43	(5) QDPSizeChangeProof (EQUIVALENT)
6.68/2.43	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. 
6.68/2.43	
6.68/2.43	From the DPs we obtained the following set of size-change graphs:
6.68/2.43	*C(b(x1)) -> A(x1)
6.68/2.43	The graph contains the following edges 1 > 1
6.68/2.43	
6.68/2.43	
6.68/2.43	*A(a(x1)) -> C(a(x1))
6.68/2.43	The graph contains the following edges 1 >= 1
6.68/2.43	
6.68/2.43	
6.68/2.43	----------------------------------------
6.68/2.43	
6.68/2.43	(6)
6.68/2.43	YES
6.68/2.48	EOF