191.41/49.31	YES
191.54/49.32	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
191.54/49.32	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
191.54/49.32	
191.54/49.32	
191.54/49.32	Termination w.r.t. Q of the given QTRS could be proven:
191.54/49.32	
191.54/49.32	(0) QTRS
191.54/49.32	(1) QTRS Reverse [EQUIVALENT, 0 ms]
191.54/49.32	(2) QTRS
191.54/49.32	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
191.54/49.32	(4) QDP
191.54/49.32	(5) MRRProof [EQUIVALENT, 81 ms]
191.54/49.32	(6) QDP
191.54/49.32	(7) QDPOrderProof [EQUIVALENT, 4177 ms]
191.54/49.32	(8) QDP
191.54/49.32	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
191.54/49.32	(10) TRUE
191.54/49.32	
191.54/49.32	
191.54/49.32	----------------------------------------
191.54/49.32	
191.54/49.32	(0)
191.54/49.32	Obligation:
191.54/49.32	Q restricted rewrite system:
191.54/49.32	The TRS R consists of the following rules:
191.54/49.32	
191.54/49.32	   a(a(b(x1))) -> b(a(a(a(x1))))
191.54/49.32	   b(a(b(a(x1)))) -> a(b(b(x1)))
191.54/49.32	
191.54/49.32	Q is empty.
191.54/49.32	
191.54/49.32	----------------------------------------
191.54/49.32	
191.54/49.32	(1) QTRS Reverse (EQUIVALENT)
191.54/49.32	We applied the QTRS Reverse Processor [REVERSE].
191.54/49.32	----------------------------------------
191.54/49.32	
191.54/49.32	(2)
191.54/49.32	Obligation:
191.54/49.32	Q restricted rewrite system:
191.54/49.32	The TRS R consists of the following rules:
191.54/49.32	
191.54/49.32	   b(a(a(x1))) -> a(a(a(b(x1))))
191.54/49.32	   a(b(a(b(x1)))) -> b(b(a(x1)))
191.54/49.32	
191.54/49.32	Q is empty.
191.54/49.32	
191.54/49.32	----------------------------------------
191.54/49.32	
191.54/49.32	(3) DependencyPairsProof (EQUIVALENT)
191.54/49.32	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
191.54/49.32	----------------------------------------
191.54/49.32	
191.54/49.32	(4)
191.54/49.32	Obligation:
191.54/49.32	Q DP problem:
191.54/49.32	The TRS P consists of the following rules:
191.54/49.32	
191.54/49.32	   B(a(a(x1))) -> A(a(a(b(x1))))
191.54/49.32	   B(a(a(x1))) -> A(a(b(x1)))
191.54/49.32	   B(a(a(x1))) -> A(b(x1))
191.54/49.32	   B(a(a(x1))) -> B(x1)
191.54/49.32	   A(b(a(b(x1)))) -> B(b(a(x1)))
191.54/49.32	   A(b(a(b(x1)))) -> B(a(x1))
191.54/49.32	   A(b(a(b(x1)))) -> A(x1)
191.54/49.32	
191.54/49.32	The TRS R consists of the following rules:
191.54/49.32	
191.54/49.32	   b(a(a(x1))) -> a(a(a(b(x1))))
191.54/49.32	   a(b(a(b(x1)))) -> b(b(a(x1)))
191.54/49.32	
191.54/49.32	Q is empty.
191.54/49.32	We have to consider all minimal (P,Q,R)-chains.
191.54/49.32	----------------------------------------
191.54/49.32	
191.54/49.32	(5) MRRProof (EQUIVALENT)
191.54/49.32	By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
191.54/49.32	
191.54/49.32	Strictly oriented dependency pairs:
191.54/49.32	
191.54/49.32	   A(b(a(b(x1)))) -> B(a(x1))
191.54/49.32	   A(b(a(b(x1)))) -> A(x1)
191.54/49.32	
191.54/49.32	
191.54/49.32	Used ordering: Polynomial interpretation [POLO]:
191.54/49.32	
191.54/49.32	   POL(A(x_1)) = x_1
191.54/49.32	   POL(B(x_1)) = 1 + x_1
191.54/49.32	   POL(a(x_1)) = x_1
191.54/49.32	   POL(b(x_1)) = 1 + x_1
191.54/49.32	
191.54/49.32	
191.54/49.32	----------------------------------------
191.54/49.32	
191.54/49.32	(6)
191.54/49.32	Obligation:
191.54/49.32	Q DP problem:
191.54/49.32	The TRS P consists of the following rules:
191.54/49.32	
191.54/49.32	   B(a(a(x1))) -> A(a(a(b(x1))))
191.54/49.32	   B(a(a(x1))) -> A(a(b(x1)))
191.54/49.32	   B(a(a(x1))) -> A(b(x1))
191.54/49.32	   B(a(a(x1))) -> B(x1)
191.54/49.32	   A(b(a(b(x1)))) -> B(b(a(x1)))
191.54/49.32	
191.54/49.32	The TRS R consists of the following rules:
191.54/49.32	
191.54/49.32	   b(a(a(x1))) -> a(a(a(b(x1))))
191.54/49.32	   a(b(a(b(x1)))) -> b(b(a(x1)))
191.54/49.32	
191.54/49.32	Q is empty.
191.54/49.32	We have to consider all minimal (P,Q,R)-chains.
191.54/49.32	----------------------------------------
191.54/49.32	
191.54/49.32	(7) QDPOrderProof (EQUIVALENT)
191.54/49.32	We use the reduction pair processor [LPAR04,JAR06].
191.54/49.32	
191.54/49.32	
191.54/49.32	The following pairs can be oriented strictly and are deleted.
191.54/49.32	
191.54/49.32	   B(a(a(x1))) -> A(a(b(x1)))
191.54/49.32	   B(a(a(x1))) -> A(b(x1))
191.54/49.32	   B(a(a(x1))) -> B(x1)
191.54/49.32	   A(b(a(b(x1)))) -> B(b(a(x1)))
191.54/49.32	The remaining pairs can at least be oriented weakly.
191.54/49.32	Used ordering:  Polynomial interpretation [POLO,RATPOLO]:
191.54/49.32	
191.54/49.32	   POL(A(x_1)) = [1/2] + [1/2]x_1
191.54/49.32	   POL(B(x_1)) = [1/2] + [3/4]x_1
191.54/49.32	   POL(a(x_1)) = [1/4] + x_1
191.54/49.33	   POL(b(x_1)) = [1/4] + [3/2]x_1
191.54/49.33	The value of delta used in the strict ordering is 1/32.
191.54/49.33	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
191.54/49.33	
191.54/49.33	   a(b(a(b(x1)))) -> b(b(a(x1)))
191.54/49.33	   b(a(a(x1))) -> a(a(a(b(x1))))
191.54/49.33	
191.54/49.33	
191.54/49.33	----------------------------------------
191.54/49.33	
191.54/49.33	(8)
191.54/49.33	Obligation:
191.54/49.33	Q DP problem:
191.54/49.33	The TRS P consists of the following rules:
191.54/49.33	
191.54/49.33	   B(a(a(x1))) -> A(a(a(b(x1))))
191.54/49.33	
191.54/49.33	The TRS R consists of the following rules:
191.54/49.33	
191.54/49.33	   b(a(a(x1))) -> a(a(a(b(x1))))
191.54/49.33	   a(b(a(b(x1)))) -> b(b(a(x1)))
191.54/49.33	
191.54/49.33	Q is empty.
191.54/49.33	We have to consider all minimal (P,Q,R)-chains.
191.54/49.33	----------------------------------------
191.54/49.33	
191.54/49.33	(9) DependencyGraphProof (EQUIVALENT)
191.54/49.33	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
191.54/49.33	----------------------------------------
191.54/49.33	
191.54/49.33	(10)
191.54/49.33	TRUE
191.54/49.41	EOF