38.37/10.78	YES
38.87/10.86	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
38.87/10.86	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
38.87/10.86	
38.87/10.86	
38.87/10.86	Termination w.r.t. Q of the given QTRS could be proven:
38.87/10.86	
38.87/10.86	(0) QTRS
38.87/10.86	(1) QTRS Reverse [EQUIVALENT, 0 ms]
38.87/10.86	(2) QTRS
38.87/10.86	(3) DependencyPairsProof [EQUIVALENT, 19 ms]
38.87/10.86	(4) QDP
38.87/10.86	(5) MRRProof [EQUIVALENT, 62 ms]
38.87/10.86	(6) QDP
38.87/10.86	(7) QDPOrderProof [EQUIVALENT, 8 ms]
38.87/10.86	(8) QDP
38.87/10.86	(9) QDPOrderProof [EQUIVALENT, 0 ms]
38.87/10.86	(10) QDP
38.87/10.86	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
38.87/10.86	(12) TRUE
38.87/10.86	
38.87/10.86	
38.87/10.86	----------------------------------------
38.87/10.86	
38.87/10.86	(0)
38.87/10.86	Obligation:
38.87/10.86	Q restricted rewrite system:
38.87/10.86	The TRS R consists of the following rules:
38.87/10.86	
38.87/10.86	   b(a(b(x1))) -> a(b(a(x1)))
38.87/10.86	   b(b(a(x1))) -> b(b(b(x1)))
38.87/10.86	   c(a(x1)) -> a(b(c(x1)))
38.87/10.86	   c(b(x1)) -> b(a(c(x1)))
38.87/10.86	
38.87/10.86	Q is empty.
38.87/10.86	
38.87/10.86	----------------------------------------
38.87/10.86	
38.87/10.86	(1) QTRS Reverse (EQUIVALENT)
38.87/10.86	We applied the QTRS Reverse Processor [REVERSE].
38.87/10.86	----------------------------------------
38.87/10.86	
38.87/10.86	(2)
38.87/10.86	Obligation:
38.87/10.86	Q restricted rewrite system:
38.87/10.86	The TRS R consists of the following rules:
38.87/10.86	
38.87/10.86	   b(a(b(x1))) -> a(b(a(x1)))
38.87/10.86	   a(b(b(x1))) -> b(b(b(x1)))
38.87/10.86	   a(c(x1)) -> c(b(a(x1)))
38.87/10.86	   b(c(x1)) -> c(a(b(x1)))
38.87/10.86	
38.87/10.86	Q is empty.
38.87/10.86	
38.87/10.86	----------------------------------------
38.87/10.86	
38.87/10.86	(3) DependencyPairsProof (EQUIVALENT)
38.87/10.86	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
38.87/10.86	----------------------------------------
38.87/10.86	
38.87/10.86	(4)
38.87/10.86	Obligation:
38.87/10.86	Q DP problem:
38.87/10.86	The TRS P consists of the following rules:
38.87/10.86	
38.87/10.86	   B(a(b(x1))) -> A(b(a(x1)))
38.87/10.86	   B(a(b(x1))) -> B(a(x1))
38.87/10.86	   B(a(b(x1))) -> A(x1)
38.87/10.86	   A(b(b(x1))) -> B(b(b(x1)))
38.87/10.86	   A(c(x1)) -> B(a(x1))
38.87/10.86	   A(c(x1)) -> A(x1)
38.87/10.86	   B(c(x1)) -> A(b(x1))
38.87/10.86	   B(c(x1)) -> B(x1)
38.87/10.86	
38.87/10.86	The TRS R consists of the following rules:
38.87/10.86	
38.87/10.86	   b(a(b(x1))) -> a(b(a(x1)))
38.87/10.86	   a(b(b(x1))) -> b(b(b(x1)))
38.87/10.86	   a(c(x1)) -> c(b(a(x1)))
38.87/10.86	   b(c(x1)) -> c(a(b(x1)))
38.87/10.86	
38.87/10.86	Q is empty.
38.87/10.86	We have to consider all minimal (P,Q,R)-chains.
38.87/10.86	----------------------------------------
38.87/10.86	
38.87/10.86	(5) MRRProof (EQUIVALENT)
38.87/10.86	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.
38.87/10.86	
38.87/10.86	Strictly oriented dependency pairs:
38.87/10.86	
38.87/10.86	   A(c(x1)) -> B(a(x1))
38.87/10.86	   A(c(x1)) -> A(x1)
38.87/10.86	   B(c(x1)) -> A(b(x1))
38.87/10.86	   B(c(x1)) -> B(x1)
38.87/10.86	
38.87/10.86	
38.87/10.86	Used ordering: Polynomial interpretation [POLO]:
38.87/10.86	
38.87/10.86	   POL(A(x_1)) = x_1
38.87/10.86	   POL(B(x_1)) = x_1
38.87/10.86	   POL(a(x_1)) = x_1
38.87/10.86	   POL(b(x_1)) = x_1
38.87/10.86	   POL(c(x_1)) = 2 + x_1
38.87/10.86	
38.87/10.86	
38.87/10.86	----------------------------------------
38.87/10.86	
38.87/10.86	(6)
38.87/10.86	Obligation:
38.87/10.86	Q DP problem:
38.87/10.86	The TRS P consists of the following rules:
38.87/10.86	
38.87/10.86	   B(a(b(x1))) -> A(b(a(x1)))
38.87/10.86	   B(a(b(x1))) -> B(a(x1))
38.87/10.86	   B(a(b(x1))) -> A(x1)
38.87/10.86	   A(b(b(x1))) -> B(b(b(x1)))
38.87/10.86	
38.87/10.86	The TRS R consists of the following rules:
38.87/10.86	
38.87/10.86	   b(a(b(x1))) -> a(b(a(x1)))
38.87/10.86	   a(b(b(x1))) -> b(b(b(x1)))
38.87/10.86	   a(c(x1)) -> c(b(a(x1)))
38.87/10.86	   b(c(x1)) -> c(a(b(x1)))
38.87/10.86	
38.87/10.86	Q is empty.
38.87/10.86	We have to consider all minimal (P,Q,R)-chains.
38.87/10.86	----------------------------------------
38.87/10.86	
38.87/10.86	(7) QDPOrderProof (EQUIVALENT)
38.87/10.86	We use the reduction pair processor [LPAR04,JAR06].
38.87/10.86	
38.87/10.86	
38.87/10.86	The following pairs can be oriented strictly and are deleted.
38.87/10.86	
38.87/10.86	   B(a(b(x1))) -> B(a(x1))
38.87/10.86	   B(a(b(x1))) -> A(x1)
38.87/10.86	The remaining pairs can at least be oriented weakly.
38.87/10.86	Used ordering:  Polynomial interpretation [POLO]:
38.87/10.86	
38.87/10.86	   POL(A(x_1)) = x_1
38.87/10.86	   POL(B(x_1)) = x_1
38.87/10.86	   POL(a(x_1)) = 1 + x_1
38.87/10.86	   POL(b(x_1)) = 1 + x_1
38.87/10.86	   POL(c(x_1)) = 0
38.87/10.86	
38.87/10.86	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
38.87/10.86	
38.87/10.86	   a(b(b(x1))) -> b(b(b(x1)))
38.87/10.86	   b(a(b(x1))) -> a(b(a(x1)))
38.87/10.86	   a(c(x1)) -> c(b(a(x1)))
38.87/10.86	   b(c(x1)) -> c(a(b(x1)))
38.87/10.86	
38.87/10.86	
38.87/10.86	----------------------------------------
38.87/10.86	
38.87/10.86	(8)
38.87/10.86	Obligation:
38.87/10.86	Q DP problem:
38.87/10.86	The TRS P consists of the following rules:
38.87/10.86	
38.87/10.86	   B(a(b(x1))) -> A(b(a(x1)))
38.87/10.86	   A(b(b(x1))) -> B(b(b(x1)))
38.87/10.86	
38.87/10.86	The TRS R consists of the following rules:
38.87/10.86	
38.87/10.86	   b(a(b(x1))) -> a(b(a(x1)))
38.87/10.86	   a(b(b(x1))) -> b(b(b(x1)))
38.87/10.86	   a(c(x1)) -> c(b(a(x1)))
38.87/10.86	   b(c(x1)) -> c(a(b(x1)))
38.87/10.86	
38.87/10.86	Q is empty.
38.87/10.86	We have to consider all minimal (P,Q,R)-chains.
38.87/10.86	----------------------------------------
38.87/10.86	
38.87/10.86	(9) QDPOrderProof (EQUIVALENT)
38.87/10.86	We use the reduction pair processor [LPAR04,JAR06].
38.87/10.86	
38.87/10.86	
38.87/10.86	The following pairs can be oriented strictly and are deleted.
38.87/10.86	
38.87/10.86	   A(b(b(x1))) -> B(b(b(x1)))
38.87/10.86	The remaining pairs can at least be oriented weakly.
38.87/10.86	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
38.87/10.86	
38.87/10.86	   <<<
38.87/10.86	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
38.87/10.86	>>>
38.87/10.86	
38.87/10.86	   <<<
38.87/10.86	 POL(a(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, 0A, -I], [-I, 0A, -I], [0A, 1A, 0A]] 	* 	x_1
38.87/10.86	>>>
38.87/10.86	
38.87/10.86	   <<<
38.87/10.86	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 0A, -I], [0A, 0A, 0A]] 	* 	x_1
38.87/10.86	>>>
38.87/10.86	
38.87/10.86	   <<<
38.87/10.86	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, 1A, 0A]] 	* 	x_1
38.87/10.86	>>>
38.87/10.86	
38.87/10.86	   <<<
38.87/10.86	 POL(c(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[-I, -I, -I], [-I, -I, -I], [-I, -I, -I]] 	* 	x_1
38.87/10.86	>>>
38.87/10.86	
38.87/10.86	
38.87/10.86	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
38.87/10.86	
38.87/10.86	   a(b(b(x1))) -> b(b(b(x1)))
38.87/10.86	   b(a(b(x1))) -> a(b(a(x1)))
38.87/10.86	   a(c(x1)) -> c(b(a(x1)))
38.87/10.86	   b(c(x1)) -> c(a(b(x1)))
38.87/10.86	
38.87/10.86	
38.87/10.86	----------------------------------------
38.87/10.86	
38.87/10.86	(10)
38.87/10.86	Obligation:
38.87/10.86	Q DP problem:
38.87/10.86	The TRS P consists of the following rules:
38.87/10.86	
38.87/10.86	   B(a(b(x1))) -> A(b(a(x1)))
38.87/10.86	
38.87/10.86	The TRS R consists of the following rules:
38.87/10.86	
38.87/10.86	   b(a(b(x1))) -> a(b(a(x1)))
38.87/10.86	   a(b(b(x1))) -> b(b(b(x1)))
38.87/10.86	   a(c(x1)) -> c(b(a(x1)))
38.87/10.86	   b(c(x1)) -> c(a(b(x1)))
38.87/10.86	
38.87/10.86	Q is empty.
38.87/10.86	We have to consider all minimal (P,Q,R)-chains.
38.87/10.86	----------------------------------------
38.87/10.86	
38.87/10.86	(11) DependencyGraphProof (EQUIVALENT)
38.87/10.86	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
38.87/10.86	----------------------------------------
38.87/10.86	
38.87/10.86	(12)
38.87/10.86	TRUE
39.09/10.90	EOF