29.71/8.62	YES
34.44/10.43	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
34.44/10.43	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
34.44/10.43	
34.44/10.43	
34.44/10.43	Termination w.r.t. Q of the given QTRS could be proven:
34.44/10.43	
34.44/10.43	(0) QTRS
34.44/10.43	(1) QTRS Reverse [EQUIVALENT, 0 ms]
34.44/10.43	(2) QTRS
34.44/10.43	(3) DependencyPairsProof [EQUIVALENT, 33 ms]
34.44/10.43	(4) QDP
34.44/10.43	(5) QDPOrderProof [EQUIVALENT, 40 ms]
34.44/10.43	(6) QDP
34.44/10.43	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
34.44/10.43	(8) TRUE
34.44/10.43	
34.44/10.43	
34.44/10.43	----------------------------------------
34.44/10.43	
34.44/10.43	(0)
34.44/10.43	Obligation:
34.44/10.43	Q restricted rewrite system:
34.44/10.43	The TRS R consists of the following rules:
34.44/10.43	
34.44/10.43	   b(a(a(x1))) -> a(b(c(x1)))
34.44/10.43	   c(a(x1)) -> a(c(x1))
34.44/10.43	   b(c(a(x1))) -> a(b(c(x1)))
34.44/10.43	   c(b(x1)) -> b(a(x1))
34.44/10.43	   a(c(b(x1))) -> c(b(a(x1)))
34.44/10.43	
34.44/10.43	Q is empty.
34.44/10.43	
34.44/10.43	----------------------------------------
34.44/10.43	
34.44/10.43	(1) QTRS Reverse (EQUIVALENT)
34.44/10.43	We applied the QTRS Reverse Processor [REVERSE].
34.44/10.43	----------------------------------------
34.44/10.43	
34.44/10.43	(2)
34.44/10.43	Obligation:
34.44/10.43	Q restricted rewrite system:
34.44/10.43	The TRS R consists of the following rules:
34.44/10.43	
34.44/10.43	   a(a(b(x1))) -> c(b(a(x1)))
34.44/10.43	   a(c(x1)) -> c(a(x1))
34.44/10.43	   a(c(b(x1))) -> c(b(a(x1)))
34.44/10.43	   b(c(x1)) -> a(b(x1))
34.44/10.43	   b(c(a(x1))) -> a(b(c(x1)))
34.44/10.43	
34.44/10.43	Q is empty.
34.44/10.43	
34.44/10.43	----------------------------------------
34.44/10.43	
34.44/10.43	(3) DependencyPairsProof (EQUIVALENT)
34.44/10.43	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
34.44/10.43	----------------------------------------
34.44/10.43	
34.44/10.43	(4)
34.44/10.43	Obligation:
34.44/10.43	Q DP problem:
34.44/10.43	The TRS P consists of the following rules:
34.44/10.43	
34.44/10.43	   A(a(b(x1))) -> B(a(x1))
34.44/10.43	   A(a(b(x1))) -> A(x1)
34.44/10.43	   A(c(x1)) -> A(x1)
34.44/10.43	   A(c(b(x1))) -> B(a(x1))
34.44/10.43	   A(c(b(x1))) -> A(x1)
34.44/10.43	   B(c(x1)) -> A(b(x1))
34.44/10.43	   B(c(x1)) -> B(x1)
34.44/10.43	   B(c(a(x1))) -> A(b(c(x1)))
34.44/10.43	   B(c(a(x1))) -> B(c(x1))
34.44/10.43	
34.44/10.43	The TRS R consists of the following rules:
34.44/10.43	
34.44/10.43	   a(a(b(x1))) -> c(b(a(x1)))
34.44/10.43	   a(c(x1)) -> c(a(x1))
34.44/10.43	   a(c(b(x1))) -> c(b(a(x1)))
34.44/10.43	   b(c(x1)) -> a(b(x1))
34.44/10.43	   b(c(a(x1))) -> a(b(c(x1)))
34.44/10.43	
34.44/10.43	Q is empty.
34.44/10.43	We have to consider all minimal (P,Q,R)-chains.
34.44/10.43	----------------------------------------
34.44/10.43	
34.44/10.43	(5) QDPOrderProof (EQUIVALENT)
34.44/10.43	We use the reduction pair processor [LPAR04,JAR06].
34.44/10.43	
34.44/10.43	
34.44/10.43	The following pairs can be oriented strictly and are deleted.
34.44/10.43	
34.44/10.43	   A(a(b(x1))) -> B(a(x1))
34.44/10.43	   A(a(b(x1))) -> A(x1)
34.44/10.43	   A(c(x1)) -> A(x1)
34.44/10.43	   A(c(b(x1))) -> B(a(x1))
34.44/10.43	   A(c(b(x1))) -> A(x1)
34.44/10.43	   B(c(x1)) -> B(x1)
34.44/10.43	   B(c(a(x1))) -> B(c(x1))
34.44/10.43	The remaining pairs can at least be oriented weakly.
34.44/10.43	Used ordering:  Polynomial interpretation [POLO]:
34.44/10.43	
34.44/10.43	   POL(A(x_1)) = 1 + x_1
34.44/10.43	   POL(B(x_1)) = 1 + x_1
34.44/10.43	   POL(a(x_1)) = 1 + x_1
34.44/10.43	   POL(b(x_1)) = 1 + x_1
34.44/10.43	   POL(c(x_1)) = 1 + x_1
34.44/10.43	
34.44/10.43	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
34.44/10.43	
34.44/10.43	   a(a(b(x1))) -> c(b(a(x1)))
34.44/10.43	   a(c(x1)) -> c(a(x1))
34.44/10.43	   a(c(b(x1))) -> c(b(a(x1)))
34.44/10.43	   b(c(x1)) -> a(b(x1))
34.44/10.43	   b(c(a(x1))) -> a(b(c(x1)))
34.44/10.43	
34.44/10.43	
34.44/10.43	----------------------------------------
34.44/10.43	
34.44/10.43	(6)
34.44/10.43	Obligation:
34.44/10.43	Q DP problem:
34.44/10.43	The TRS P consists of the following rules:
34.44/10.43	
34.44/10.43	   B(c(x1)) -> A(b(x1))
34.44/10.43	   B(c(a(x1))) -> A(b(c(x1)))
34.44/10.43	
34.44/10.43	The TRS R consists of the following rules:
34.44/10.43	
34.44/10.43	   a(a(b(x1))) -> c(b(a(x1)))
34.44/10.43	   a(c(x1)) -> c(a(x1))
34.44/10.43	   a(c(b(x1))) -> c(b(a(x1)))
34.44/10.43	   b(c(x1)) -> a(b(x1))
34.44/10.43	   b(c(a(x1))) -> a(b(c(x1)))
34.44/10.43	
34.44/10.43	Q is empty.
34.44/10.43	We have to consider all minimal (P,Q,R)-chains.
34.44/10.43	----------------------------------------
34.44/10.43	
34.44/10.43	(7) DependencyGraphProof (EQUIVALENT)
34.44/10.43	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
34.44/10.43	----------------------------------------
34.44/10.43	
34.44/10.43	(8)
34.44/10.43	TRUE
34.67/10.52	EOF