11.70/3.88	YES
12.31/4.00	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
12.31/4.00	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.31/4.00	
12.31/4.00	
12.31/4.00	Termination w.r.t. Q of the given QTRS could be proven:
12.31/4.00	
12.31/4.00	(0) QTRS
12.31/4.00	(1) QTRS Reverse [EQUIVALENT, 0 ms]
12.31/4.00	(2) QTRS
12.31/4.00	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
12.31/4.00	(4) QDP
12.31/4.00	(5) MRRProof [EQUIVALENT, 24 ms]
12.31/4.00	(6) QDP
12.31/4.00	(7) QDPOrderProof [EQUIVALENT, 53 ms]
12.31/4.00	(8) QDP
12.31/4.00	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
12.31/4.00	(10) TRUE
12.31/4.00	
12.31/4.00	
12.31/4.00	----------------------------------------
12.31/4.00	
12.31/4.00	(0)
12.31/4.00	Obligation:
12.31/4.00	Q restricted rewrite system:
12.31/4.00	The TRS R consists of the following rules:
12.31/4.00	
12.31/4.00	   a(a(x1)) -> a(b(x1))
12.31/4.00	   a(c(x1)) -> b(x1)
12.31/4.00	   b(b(x1)) -> c(b(a(x1)))
12.31/4.00	
12.31/4.00	Q is empty.
12.31/4.00	
12.31/4.00	----------------------------------------
12.31/4.00	
12.31/4.00	(1) QTRS Reverse (EQUIVALENT)
12.31/4.00	We applied the QTRS Reverse Processor [REVERSE].
12.31/4.00	----------------------------------------
12.31/4.00	
12.31/4.00	(2)
12.31/4.00	Obligation:
12.31/4.00	Q restricted rewrite system:
12.31/4.00	The TRS R consists of the following rules:
12.31/4.00	
12.31/4.00	   a(a(x1)) -> b(a(x1))
12.31/4.00	   c(a(x1)) -> b(x1)
12.31/4.00	   b(b(x1)) -> a(b(c(x1)))
12.31/4.00	
12.31/4.00	Q is empty.
12.31/4.00	
12.31/4.00	----------------------------------------
12.31/4.00	
12.31/4.00	(3) DependencyPairsProof (EQUIVALENT)
12.31/4.00	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
12.31/4.00	----------------------------------------
12.31/4.00	
12.31/4.00	(4)
12.31/4.00	Obligation:
12.31/4.00	Q DP problem:
12.31/4.00	The TRS P consists of the following rules:
12.31/4.00	
12.31/4.00	   A(a(x1)) -> B(a(x1))
12.31/4.00	   C(a(x1)) -> B(x1)
12.31/4.00	   B(b(x1)) -> A(b(c(x1)))
12.31/4.00	   B(b(x1)) -> B(c(x1))
12.31/4.00	   B(b(x1)) -> C(x1)
12.31/4.00	
12.31/4.00	The TRS R consists of the following rules:
12.31/4.00	
12.31/4.00	   a(a(x1)) -> b(a(x1))
12.31/4.00	   c(a(x1)) -> b(x1)
12.31/4.00	   b(b(x1)) -> a(b(c(x1)))
12.31/4.00	
12.31/4.00	Q is empty.
12.31/4.00	We have to consider all minimal (P,Q,R)-chains.
12.31/4.00	----------------------------------------
12.31/4.00	
12.31/4.00	(5) MRRProof (EQUIVALENT)
12.31/4.00	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.
12.31/4.00	
12.31/4.00	Strictly oriented dependency pairs:
12.31/4.00	
12.31/4.00	   C(a(x1)) -> B(x1)
12.31/4.00	   B(b(x1)) -> B(c(x1))
12.31/4.00	   B(b(x1)) -> C(x1)
12.31/4.00	
12.31/4.00	
12.31/4.00	Used ordering: Polynomial interpretation [POLO]:
12.31/4.00	
12.31/4.00	   POL(A(x_1)) = 3 + 3*x_1
12.31/4.00	   POL(B(x_1)) = 3 + 3*x_1
12.31/4.00	   POL(C(x_1)) = 3 + 3*x_1
12.31/4.00	   POL(a(x_1)) = 3 + 2*x_1
12.31/4.00	   POL(b(x_1)) = 3 + 2*x_1
12.31/4.00	   POL(c(x_1)) = x_1
12.31/4.00	
12.31/4.00	
12.31/4.00	----------------------------------------
12.31/4.00	
12.31/4.00	(6)
12.31/4.00	Obligation:
12.31/4.00	Q DP problem:
12.31/4.00	The TRS P consists of the following rules:
12.31/4.00	
12.31/4.00	   A(a(x1)) -> B(a(x1))
12.31/4.00	   B(b(x1)) -> A(b(c(x1)))
12.31/4.00	
12.31/4.00	The TRS R consists of the following rules:
12.31/4.00	
12.31/4.00	   a(a(x1)) -> b(a(x1))
12.31/4.00	   c(a(x1)) -> b(x1)
12.31/4.00	   b(b(x1)) -> a(b(c(x1)))
12.31/4.00	
12.31/4.00	Q is empty.
12.31/4.00	We have to consider all minimal (P,Q,R)-chains.
12.31/4.00	----------------------------------------
12.31/4.00	
12.31/4.00	(7) QDPOrderProof (EQUIVALENT)
12.31/4.00	We use the reduction pair processor [LPAR04,JAR06].
12.31/4.00	
12.31/4.00	
12.31/4.00	The following pairs can be oriented strictly and are deleted.
12.31/4.00	
12.31/4.00	   A(a(x1)) -> B(a(x1))
12.31/4.00	The remaining pairs can at least be oriented weakly.
12.31/4.00	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
12.31/4.00	
12.31/4.00	   <<<
12.31/4.00	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 1A]] 	* 	x_1
12.31/4.00	>>>
12.31/4.00	
12.31/4.00	   <<<
12.31/4.00	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, -I, 1A], [0A, 0A, 1A]] 	* 	x_1
12.31/4.00	>>>
12.31/4.00	
12.31/4.00	   <<<
12.31/4.00	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
12.31/4.00	>>>
12.31/4.00	
12.31/4.00	   <<<
12.31/4.00	 POL(b(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 1A], [-I, 0A, -I]] 	* 	x_1
12.31/4.00	>>>
12.31/4.00	
12.31/4.00	   <<<
12.31/4.00	 POL(c(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, -I], [-I, -I, 0A], [-I, -I, 0A]] 	* 	x_1
12.31/4.00	>>>
12.31/4.00	
12.31/4.00	
12.31/4.00	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
12.31/4.00	
12.31/4.00	   b(b(x1)) -> a(b(c(x1)))
12.31/4.00	   a(a(x1)) -> b(a(x1))
12.31/4.00	   c(a(x1)) -> b(x1)
12.31/4.00	
12.31/4.00	
12.31/4.00	----------------------------------------
12.31/4.00	
12.31/4.00	(8)
12.31/4.00	Obligation:
12.31/4.00	Q DP problem:
12.31/4.00	The TRS P consists of the following rules:
12.31/4.00	
12.31/4.00	   B(b(x1)) -> A(b(c(x1)))
12.31/4.00	
12.31/4.00	The TRS R consists of the following rules:
12.31/4.00	
12.31/4.00	   a(a(x1)) -> b(a(x1))
12.31/4.00	   c(a(x1)) -> b(x1)
12.31/4.00	   b(b(x1)) -> a(b(c(x1)))
12.31/4.00	
12.31/4.00	Q is empty.
12.31/4.00	We have to consider all minimal (P,Q,R)-chains.
12.31/4.00	----------------------------------------
12.31/4.00	
12.31/4.00	(9) DependencyGraphProof (EQUIVALENT)
12.31/4.00	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
12.31/4.00	----------------------------------------
12.31/4.00	
12.31/4.00	(10)
12.31/4.00	TRUE
12.31/4.06	EOF