36.85/10.42	YES
39.43/11.52	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
39.43/11.52	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
39.43/11.52	
39.43/11.52	
39.43/11.52	Termination w.r.t. Q of the given QTRS could be proven:
39.43/11.52	
39.43/11.52	(0) QTRS
39.43/11.52	(1) QTRS Reverse [EQUIVALENT, 0 ms]
39.43/11.52	(2) QTRS
39.43/11.52	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
39.43/11.52	(4) QDP
39.43/11.52	(5) QDPOrderProof [EQUIVALENT, 20 ms]
39.43/11.52	(6) QDP
39.43/11.52	(7) QDPOrderProof [EQUIVALENT, 19 ms]
39.43/11.52	(8) QDP
39.43/11.52	(9) MRRProof [EQUIVALENT, 33 ms]
39.43/11.52	(10) QDP
39.43/11.52	(11) QDPOrderProof [EQUIVALENT, 109 ms]
39.43/11.52	(12) QDP
39.43/11.52	(13) DependencyGraphProof [EQUIVALENT, 0 ms]
39.43/11.52	(14) TRUE
39.43/11.52	
39.43/11.52	
39.43/11.52	----------------------------------------
39.43/11.52	
39.43/11.52	(0)
39.43/11.52	Obligation:
39.43/11.52	Q restricted rewrite system:
39.43/11.52	The TRS R consists of the following rules:
39.43/11.52	
39.43/11.52	   c(b(a(a(x1)))) -> a(a(b(c(x1))))
39.43/11.52	   b(a(a(a(x1)))) -> a(a(a(b(x1))))
39.43/11.52	   a(b(c(x1))) -> c(b(a(x1)))
39.43/11.52	   c(c(b(b(x1)))) -> b(b(c(c(x1))))
39.43/11.52	
39.43/11.52	Q is empty.
39.43/11.52	
39.43/11.52	----------------------------------------
39.43/11.52	
39.43/11.52	(1) QTRS Reverse (EQUIVALENT)
39.43/11.52	We applied the QTRS Reverse Processor [REVERSE].
39.43/11.52	----------------------------------------
39.43/11.52	
39.43/11.52	(2)
39.43/11.52	Obligation:
39.43/11.52	Q restricted rewrite system:
39.43/11.52	The TRS R consists of the following rules:
39.43/11.52	
39.43/11.52	   a(a(b(c(x1)))) -> c(b(a(a(x1))))
39.43/11.52	   a(a(a(b(x1)))) -> b(a(a(a(x1))))
39.43/11.52	   c(b(a(x1))) -> a(b(c(x1)))
39.43/11.52	   b(b(c(c(x1)))) -> c(c(b(b(x1))))
39.43/11.52	
39.43/11.52	Q is empty.
39.43/11.52	
39.43/11.52	----------------------------------------
39.43/11.52	
39.43/11.52	(3) DependencyPairsProof (EQUIVALENT)
39.43/11.52	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
39.43/11.52	----------------------------------------
39.43/11.52	
39.43/11.52	(4)
39.43/11.52	Obligation:
39.43/11.52	Q DP problem:
39.43/11.52	The TRS P consists of the following rules:
39.43/11.52	
39.43/11.52	   A(a(b(c(x1)))) -> C(b(a(a(x1))))
39.43/11.52	   A(a(b(c(x1)))) -> B(a(a(x1)))
39.43/11.52	   A(a(b(c(x1)))) -> A(a(x1))
39.43/11.52	   A(a(b(c(x1)))) -> A(x1)
39.43/11.52	   A(a(a(b(x1)))) -> B(a(a(a(x1))))
39.43/11.52	   A(a(a(b(x1)))) -> A(a(a(x1)))
39.43/11.52	   A(a(a(b(x1)))) -> A(a(x1))
39.43/11.52	   A(a(a(b(x1)))) -> A(x1)
39.43/11.52	   C(b(a(x1))) -> A(b(c(x1)))
39.43/11.52	   C(b(a(x1))) -> B(c(x1))
39.43/11.52	   C(b(a(x1))) -> C(x1)
39.43/11.52	   B(b(c(c(x1)))) -> C(c(b(b(x1))))
39.43/11.52	   B(b(c(c(x1)))) -> C(b(b(x1)))
39.43/11.52	   B(b(c(c(x1)))) -> B(b(x1))
39.43/11.52	   B(b(c(c(x1)))) -> B(x1)
39.43/11.52	
39.43/11.52	The TRS R consists of the following rules:
39.43/11.52	
39.43/11.52	   a(a(b(c(x1)))) -> c(b(a(a(x1))))
39.43/11.52	   a(a(a(b(x1)))) -> b(a(a(a(x1))))
39.43/11.52	   c(b(a(x1))) -> a(b(c(x1)))
39.43/11.52	   b(b(c(c(x1)))) -> c(c(b(b(x1))))
39.43/11.52	
39.43/11.52	Q is empty.
39.43/11.52	We have to consider all minimal (P,Q,R)-chains.
39.43/11.52	----------------------------------------
39.43/11.52	
39.43/11.52	(5) QDPOrderProof (EQUIVALENT)
39.43/11.52	We use the reduction pair processor [LPAR04,JAR06].
39.43/11.52	
39.43/11.52	
39.43/11.52	The following pairs can be oriented strictly and are deleted.
39.43/11.52	
39.43/11.52	   A(a(b(c(x1)))) -> A(x1)
39.43/11.52	   A(a(a(b(x1)))) -> A(a(x1))
39.43/11.52	   A(a(a(b(x1)))) -> A(x1)
39.43/11.52	   C(b(a(x1))) -> B(c(x1))
39.43/11.52	   C(b(a(x1))) -> C(x1)
39.43/11.52	The remaining pairs can at least be oriented weakly.
39.43/11.52	Used ordering:  Polynomial interpretation [POLO]:
39.43/11.52	
39.43/11.52	   POL(A(x_1)) = 1 + x_1
39.43/11.52	   POL(B(x_1)) = x_1
39.43/11.52	   POL(C(x_1)) = x_1
39.43/11.52	   POL(a(x_1)) = 1 + x_1
39.43/11.52	   POL(b(x_1)) = x_1
39.43/11.52	   POL(c(x_1)) = x_1
39.43/11.52	
39.43/11.52	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
39.43/11.52	
39.43/11.52	   c(b(a(x1))) -> a(b(c(x1)))
39.43/11.52	   a(a(b(c(x1)))) -> c(b(a(a(x1))))
39.43/11.52	   a(a(a(b(x1)))) -> b(a(a(a(x1))))
39.43/11.52	   b(b(c(c(x1)))) -> c(c(b(b(x1))))
39.43/11.52	
39.43/11.52	
39.43/11.52	----------------------------------------
39.43/11.52	
39.43/11.52	(6)
39.43/11.52	Obligation:
39.43/11.52	Q DP problem:
39.43/11.52	The TRS P consists of the following rules:
39.43/11.52	
39.43/11.52	   A(a(b(c(x1)))) -> C(b(a(a(x1))))
39.43/11.52	   A(a(b(c(x1)))) -> B(a(a(x1)))
39.43/11.52	   A(a(b(c(x1)))) -> A(a(x1))
39.43/11.52	   A(a(a(b(x1)))) -> B(a(a(a(x1))))
39.43/11.52	   A(a(a(b(x1)))) -> A(a(a(x1)))
39.43/11.52	   C(b(a(x1))) -> A(b(c(x1)))
39.43/11.52	   B(b(c(c(x1)))) -> C(c(b(b(x1))))
39.43/11.52	   B(b(c(c(x1)))) -> C(b(b(x1)))
39.43/11.52	   B(b(c(c(x1)))) -> B(b(x1))
39.43/11.52	   B(b(c(c(x1)))) -> B(x1)
39.43/11.52	
39.43/11.52	The TRS R consists of the following rules:
39.43/11.52	
39.43/11.52	   a(a(b(c(x1)))) -> c(b(a(a(x1))))
39.43/11.52	   a(a(a(b(x1)))) -> b(a(a(a(x1))))
39.43/11.52	   c(b(a(x1))) -> a(b(c(x1)))
39.43/11.52	   b(b(c(c(x1)))) -> c(c(b(b(x1))))
39.43/11.52	
39.43/11.52	Q is empty.
39.43/11.52	We have to consider all minimal (P,Q,R)-chains.
39.43/11.52	----------------------------------------
39.43/11.52	
39.43/11.52	(7) QDPOrderProof (EQUIVALENT)
39.43/11.52	We use the reduction pair processor [LPAR04,JAR06].
39.43/11.52	
39.43/11.52	
39.43/11.52	The following pairs can be oriented strictly and are deleted.
39.43/11.52	
39.43/11.52	   A(a(b(c(x1)))) -> B(a(a(x1)))
39.43/11.52	   A(a(b(c(x1)))) -> A(a(x1))
39.43/11.52	   B(b(c(c(x1)))) -> C(b(b(x1)))
39.43/11.52	   B(b(c(c(x1)))) -> B(b(x1))
39.43/11.52	   B(b(c(c(x1)))) -> B(x1)
39.43/11.52	The remaining pairs can at least be oriented weakly.
39.43/11.52	Used ordering:  Polynomial interpretation [POLO]:
39.43/11.52	
39.43/11.52	   POL(A(x_1)) = x_1
39.43/11.52	   POL(B(x_1)) = x_1
39.43/11.52	   POL(C(x_1)) = 1 + x_1
39.43/11.52	   POL(a(x_1)) = x_1
39.43/11.52	   POL(b(x_1)) = x_1
39.43/11.52	   POL(c(x_1)) = 1 + x_1
39.43/11.52	
39.43/11.52	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
39.43/11.52	
39.43/11.52	   c(b(a(x1))) -> a(b(c(x1)))
39.43/11.52	   a(a(b(c(x1)))) -> c(b(a(a(x1))))
39.43/11.52	   a(a(a(b(x1)))) -> b(a(a(a(x1))))
39.43/11.52	   b(b(c(c(x1)))) -> c(c(b(b(x1))))
39.43/11.52	
39.43/11.52	
39.43/11.52	----------------------------------------
39.43/11.52	
39.43/11.52	(8)
39.43/11.52	Obligation:
39.43/11.52	Q DP problem:
39.43/11.52	The TRS P consists of the following rules:
39.43/11.52	
39.43/11.52	   A(a(b(c(x1)))) -> C(b(a(a(x1))))
39.43/11.52	   A(a(a(b(x1)))) -> B(a(a(a(x1))))
39.43/11.52	   A(a(a(b(x1)))) -> A(a(a(x1)))
39.43/11.52	   C(b(a(x1))) -> A(b(c(x1)))
39.43/11.52	   B(b(c(c(x1)))) -> C(c(b(b(x1))))
39.43/11.52	
39.43/11.52	The TRS R consists of the following rules:
39.43/11.52	
39.43/11.52	   a(a(b(c(x1)))) -> c(b(a(a(x1))))
39.43/11.52	   a(a(a(b(x1)))) -> b(a(a(a(x1))))
39.43/11.52	   c(b(a(x1))) -> a(b(c(x1)))
39.43/11.52	   b(b(c(c(x1)))) -> c(c(b(b(x1))))
39.43/11.52	
39.43/11.52	Q is empty.
39.43/11.52	We have to consider all minimal (P,Q,R)-chains.
39.43/11.52	----------------------------------------
39.43/11.52	
39.43/11.52	(9) MRRProof (EQUIVALENT)
39.43/11.52	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.
39.43/11.52	
39.43/11.52	Strictly oriented dependency pairs:
39.43/11.52	
39.43/11.52	   A(a(a(b(x1)))) -> A(a(a(x1)))
39.43/11.52	
39.43/11.52	
39.43/11.52	Used ordering: Polynomial interpretation [POLO]:
39.43/11.52	
39.43/11.52	   POL(A(x_1)) = x_1
39.43/11.52	   POL(B(x_1)) = 2 + 2*x_1
39.43/11.52	   POL(C(x_1)) = 2 + 2*x_1
39.43/11.52	   POL(a(x_1)) = x_1
39.43/11.52	   POL(b(x_1)) = 2 + 2*x_1
39.43/11.52	   POL(c(x_1)) = 2 + 2*x_1
39.43/11.52	
39.43/11.52	
39.43/11.52	----------------------------------------
39.43/11.52	
39.43/11.52	(10)
39.43/11.52	Obligation:
39.43/11.52	Q DP problem:
39.43/11.52	The TRS P consists of the following rules:
39.43/11.52	
39.43/11.52	   A(a(b(c(x1)))) -> C(b(a(a(x1))))
39.43/11.52	   A(a(a(b(x1)))) -> B(a(a(a(x1))))
39.43/11.52	   C(b(a(x1))) -> A(b(c(x1)))
39.43/11.52	   B(b(c(c(x1)))) -> C(c(b(b(x1))))
39.43/11.52	
39.43/11.52	The TRS R consists of the following rules:
39.43/11.52	
39.43/11.52	   a(a(b(c(x1)))) -> c(b(a(a(x1))))
39.43/11.52	   a(a(a(b(x1)))) -> b(a(a(a(x1))))
39.43/11.52	   c(b(a(x1))) -> a(b(c(x1)))
39.43/11.52	   b(b(c(c(x1)))) -> c(c(b(b(x1))))
39.43/11.52	
39.43/11.52	Q is empty.
39.43/11.52	We have to consider all minimal (P,Q,R)-chains.
39.43/11.52	----------------------------------------
39.43/11.52	
39.43/11.52	(11) QDPOrderProof (EQUIVALENT)
39.43/11.52	We use the reduction pair processor [LPAR04,JAR06].
39.43/11.52	
39.43/11.52	
39.43/11.52	The following pairs can be oriented strictly and are deleted.
39.43/11.52	
39.43/11.52	   C(b(a(x1))) -> A(b(c(x1)))
39.43/11.52	The remaining pairs can at least be oriented weakly.
39.43/11.52	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
39.43/11.52	
39.43/11.52	   <<<
39.43/11.52	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
39.43/11.52	>>>
39.43/11.52	
39.43/11.52	   <<<
39.43/11.52	 POL(a(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, -I, -I], [-I, -I, 0A], [0A, 0A, 0A]] 	* 	x_1
39.43/11.52	>>>
39.43/11.52	
39.43/11.52	   <<<
39.43/11.52	 POL(b(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, -I, -I], [0A, -I, 0A], [0A, 0A, -I]] 	* 	x_1
39.43/11.52	>>>
39.43/11.52	
39.43/11.52	   <<<
39.43/11.52	 POL(c(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, -I, -I], [0A, -I, -I], [1A, 0A, 0A]] 	* 	x_1
39.43/11.52	>>>
39.43/11.52	
39.43/11.52	   <<<
39.43/11.52	 POL(C(x_1)) =  	[[-I]] 	 +  	[[1A, -I, -I]] 	* 	x_1
39.43/11.52	>>>
39.43/11.52	
39.43/11.52	   <<<
39.43/11.52	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
39.43/11.52	>>>
39.43/11.52	
39.43/11.52	
39.43/11.52	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
39.43/11.52	
39.43/11.52	   c(b(a(x1))) -> a(b(c(x1)))
39.43/11.52	   a(a(b(c(x1)))) -> c(b(a(a(x1))))
39.43/11.52	   a(a(a(b(x1)))) -> b(a(a(a(x1))))
39.43/11.52	   b(b(c(c(x1)))) -> c(c(b(b(x1))))
39.43/11.52	
39.43/11.52	
39.43/11.52	----------------------------------------
39.43/11.52	
39.43/11.52	(12)
39.43/11.52	Obligation:
39.43/11.52	Q DP problem:
39.43/11.52	The TRS P consists of the following rules:
39.43/11.52	
39.43/11.52	   A(a(b(c(x1)))) -> C(b(a(a(x1))))
39.43/11.52	   A(a(a(b(x1)))) -> B(a(a(a(x1))))
39.43/11.52	   B(b(c(c(x1)))) -> C(c(b(b(x1))))
39.43/11.52	
39.43/11.52	The TRS R consists of the following rules:
39.43/11.52	
39.43/11.52	   a(a(b(c(x1)))) -> c(b(a(a(x1))))
39.43/11.52	   a(a(a(b(x1)))) -> b(a(a(a(x1))))
39.43/11.52	   c(b(a(x1))) -> a(b(c(x1)))
39.43/11.52	   b(b(c(c(x1)))) -> c(c(b(b(x1))))
39.43/11.52	
39.43/11.52	Q is empty.
39.43/11.52	We have to consider all minimal (P,Q,R)-chains.
39.43/11.52	----------------------------------------
39.43/11.52	
39.43/11.52	(13) DependencyGraphProof (EQUIVALENT)
39.43/11.52	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.
39.43/11.52	----------------------------------------
39.43/11.52	
39.43/11.52	(14)
39.43/11.52	TRUE
39.66/11.74	EOF