8.53/2.98	YES
9.08/3.08	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
9.08/3.08	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.08/3.08	
9.08/3.08	
9.08/3.08	Termination w.r.t. Q of the given QTRS could be proven:
9.08/3.08	
9.08/3.08	(0) QTRS
9.08/3.08	(1) QTRS Reverse [EQUIVALENT, 0 ms]
9.08/3.08	(2) QTRS
9.08/3.08	(3) DependencyPairsProof [EQUIVALENT, 1 ms]
9.08/3.08	(4) QDP
9.08/3.08	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
9.08/3.08	(6) QDP
9.08/3.08	(7) QDPSizeChangeProof [EQUIVALENT, 2 ms]
9.08/3.08	(8) YES
9.08/3.08	
9.08/3.08	
9.08/3.08	----------------------------------------
9.08/3.08	
9.08/3.08	(0)
9.08/3.08	Obligation:
9.08/3.08	Q restricted rewrite system:
9.08/3.08	The TRS R consists of the following rules:
9.08/3.08	
9.08/3.08	   a(x1) -> x1
9.08/3.08	   a(a(x1)) -> a(b(c(x1)))
9.08/3.08	   c(b(x1)) -> a(c(a(x1)))
9.08/3.08	
9.08/3.08	Q is empty.
9.08/3.08	
9.08/3.08	----------------------------------------
9.08/3.08	
9.08/3.08	(1) QTRS Reverse (EQUIVALENT)
9.08/3.08	We applied the QTRS Reverse Processor [REVERSE].
9.08/3.08	----------------------------------------
9.08/3.08	
9.08/3.08	(2)
9.08/3.08	Obligation:
9.08/3.08	Q restricted rewrite system:
9.08/3.08	The TRS R consists of the following rules:
9.08/3.08	
9.08/3.08	   a(x1) -> x1
9.08/3.08	   a(a(x1)) -> c(b(a(x1)))
9.08/3.08	   b(c(x1)) -> a(c(a(x1)))
9.08/3.08	
9.08/3.08	Q is empty.
9.08/3.08	
9.08/3.08	----------------------------------------
9.08/3.08	
9.08/3.08	(3) DependencyPairsProof (EQUIVALENT)
9.08/3.08	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
9.08/3.08	----------------------------------------
9.08/3.08	
9.08/3.08	(4)
9.08/3.08	Obligation:
9.08/3.08	Q DP problem:
9.08/3.08	The TRS P consists of the following rules:
9.08/3.08	
9.08/3.08	   A(a(x1)) -> B(a(x1))
9.08/3.08	   B(c(x1)) -> A(c(a(x1)))
9.08/3.08	   B(c(x1)) -> A(x1)
9.08/3.08	
9.08/3.08	The TRS R consists of the following rules:
9.08/3.08	
9.08/3.08	   a(x1) -> x1
9.08/3.08	   a(a(x1)) -> c(b(a(x1)))
9.08/3.08	   b(c(x1)) -> a(c(a(x1)))
9.08/3.08	
9.08/3.08	Q is empty.
9.08/3.08	We have to consider all minimal (P,Q,R)-chains.
9.08/3.08	----------------------------------------
9.08/3.08	
9.08/3.08	(5) DependencyGraphProof (EQUIVALENT)
9.08/3.08	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
9.08/3.08	----------------------------------------
9.08/3.08	
9.08/3.08	(6)
9.08/3.08	Obligation:
9.08/3.08	Q DP problem:
9.08/3.08	The TRS P consists of the following rules:
9.08/3.08	
9.08/3.08	   B(c(x1)) -> A(x1)
9.08/3.08	   A(a(x1)) -> B(a(x1))
9.08/3.08	
9.08/3.08	The TRS R consists of the following rules:
9.08/3.08	
9.08/3.08	   a(x1) -> x1
9.08/3.08	   a(a(x1)) -> c(b(a(x1)))
9.08/3.08	   b(c(x1)) -> a(c(a(x1)))
9.08/3.08	
9.08/3.08	Q is empty.
9.08/3.08	We have to consider all minimal (P,Q,R)-chains.
9.08/3.08	----------------------------------------
9.08/3.08	
9.08/3.08	(7) QDPSizeChangeProof (EQUIVALENT)
9.08/3.08	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. 
9.08/3.08	
9.08/3.08	From the DPs we obtained the following set of size-change graphs:
9.08/3.08	*A(a(x1)) -> B(a(x1))
9.08/3.08	The graph contains the following edges 1 >= 1
9.08/3.08	
9.08/3.08	
9.08/3.08	*B(c(x1)) -> A(x1)
9.08/3.08	The graph contains the following edges 1 > 1
9.08/3.08	
9.08/3.08	
9.08/3.08	----------------------------------------
9.08/3.08	
9.08/3.08	(8)
9.08/3.08	YES
9.08/3.13	EOF