20.83/6.14	YES
20.83/6.16	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
20.83/6.16	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
20.83/6.16	
20.83/6.16	
20.83/6.16	Termination w.r.t. Q of the given QTRS could be proven:
20.83/6.16	
20.83/6.16	(0) QTRS
20.83/6.16	(1) QTRS Reverse [EQUIVALENT, 0 ms]
20.83/6.16	(2) QTRS
20.83/6.16	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
20.83/6.16	(4) QDP
20.83/6.16	(5) DependencyGraphProof [EQUIVALENT, 1 ms]
20.83/6.16	(6) QDP
20.83/6.16	(7) QDPOrderProof [EQUIVALENT, 115 ms]
20.83/6.16	(8) QDP
20.83/6.16	(9) QDPOrderProof [EQUIVALENT, 77 ms]
20.83/6.16	(10) QDP
20.83/6.16	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
20.83/6.16	(12) TRUE
20.83/6.16	
20.83/6.16	
20.83/6.16	----------------------------------------
20.83/6.16	
20.83/6.16	(0)
20.83/6.16	Obligation:
20.83/6.16	Q restricted rewrite system:
20.83/6.16	The TRS R consists of the following rules:
20.83/6.16	
20.83/6.16	   a(a(x1)) -> a(b(b(c(x1))))
20.83/6.16	   b(c(x1)) -> x1
20.83/6.16	   c(b(x1)) -> a(c(x1))
20.83/6.16	
20.83/6.16	Q is empty.
20.83/6.16	
20.83/6.16	----------------------------------------
20.83/6.16	
20.83/6.16	(1) QTRS Reverse (EQUIVALENT)
20.83/6.16	We applied the QTRS Reverse Processor [REVERSE].
20.83/6.16	----------------------------------------
20.83/6.16	
20.83/6.16	(2)
20.83/6.16	Obligation:
20.83/6.16	Q restricted rewrite system:
20.83/6.16	The TRS R consists of the following rules:
20.83/6.16	
20.83/6.16	   a(a(x1)) -> c(b(b(a(x1))))
20.83/6.16	   c(b(x1)) -> x1
20.83/6.16	   b(c(x1)) -> c(a(x1))
20.83/6.16	
20.83/6.16	Q is empty.
20.83/6.16	
20.83/6.16	----------------------------------------
20.83/6.16	
20.83/6.16	(3) DependencyPairsProof (EQUIVALENT)
20.83/6.16	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
20.83/6.16	----------------------------------------
20.83/6.16	
20.83/6.16	(4)
20.83/6.16	Obligation:
20.83/6.16	Q DP problem:
20.83/6.16	The TRS P consists of the following rules:
20.83/6.16	
20.83/6.16	   A(a(x1)) -> C(b(b(a(x1))))
20.83/6.16	   A(a(x1)) -> B(b(a(x1)))
20.83/6.16	   A(a(x1)) -> B(a(x1))
20.83/6.16	   B(c(x1)) -> C(a(x1))
20.83/6.16	   B(c(x1)) -> A(x1)
20.83/6.16	
20.83/6.16	The TRS R consists of the following rules:
20.83/6.16	
20.83/6.16	   a(a(x1)) -> c(b(b(a(x1))))
20.83/6.16	   c(b(x1)) -> x1
20.83/6.16	   b(c(x1)) -> c(a(x1))
20.83/6.16	
20.83/6.16	Q is empty.
20.83/6.16	We have to consider all minimal (P,Q,R)-chains.
20.83/6.16	----------------------------------------
20.83/6.16	
20.83/6.16	(5) DependencyGraphProof (EQUIVALENT)
20.83/6.16	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
20.83/6.16	----------------------------------------
20.83/6.16	
20.83/6.16	(6)
20.83/6.16	Obligation:
20.83/6.16	Q DP problem:
20.83/6.16	The TRS P consists of the following rules:
20.83/6.16	
20.83/6.16	   A(a(x1)) -> B(b(a(x1)))
20.83/6.16	   B(c(x1)) -> A(x1)
20.83/6.16	   A(a(x1)) -> B(a(x1))
20.83/6.16	
20.83/6.16	The TRS R consists of the following rules:
20.83/6.16	
20.83/6.16	   a(a(x1)) -> c(b(b(a(x1))))
20.83/6.16	   c(b(x1)) -> x1
20.83/6.16	   b(c(x1)) -> c(a(x1))
20.83/6.16	
20.83/6.16	Q is empty.
20.83/6.16	We have to consider all minimal (P,Q,R)-chains.
20.83/6.16	----------------------------------------
20.83/6.16	
20.83/6.16	(7) QDPOrderProof (EQUIVALENT)
20.83/6.16	We use the reduction pair processor [LPAR04,JAR06].
20.83/6.16	
20.83/6.16	
20.83/6.16	The following pairs can be oriented strictly and are deleted.
20.83/6.16	
20.83/6.16	   A(a(x1)) -> B(a(x1))
20.83/6.16	The remaining pairs can at least be oriented weakly.
20.83/6.16	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
20.83/6.16	
20.83/6.16	   <<<
20.83/6.16	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
20.83/6.16	>>>
20.83/6.16	
20.83/6.16	   <<<
20.83/6.16	 POL(a(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 1A, 0A], [-I, 0A, -I]] 	* 	x_1
20.83/6.16	>>>
20.83/6.16	
20.83/6.16	   <<<
20.83/6.16	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
20.83/6.16	>>>
20.83/6.16	
20.83/6.16	   <<<
20.83/6.16	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 1A]] 	* 	x_1
20.83/6.16	>>>
20.83/6.16	
20.83/6.16	   <<<
20.83/6.16	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, -I, -I], [0A, -I, 0A], [0A, 0A, -I]] 	* 	x_1
20.83/6.16	>>>
20.83/6.16	
20.83/6.16	
20.83/6.16	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
20.83/6.16	
20.83/6.16	   a(a(x1)) -> c(b(b(a(x1))))
20.83/6.16	   b(c(x1)) -> c(a(x1))
20.83/6.16	   c(b(x1)) -> x1
20.83/6.16	
20.83/6.16	
20.83/6.16	----------------------------------------
20.83/6.16	
20.83/6.16	(8)
20.83/6.16	Obligation:
20.83/6.16	Q DP problem:
20.83/6.16	The TRS P consists of the following rules:
20.83/6.16	
20.83/6.16	   A(a(x1)) -> B(b(a(x1)))
20.83/6.16	   B(c(x1)) -> A(x1)
20.83/6.16	
20.83/6.16	The TRS R consists of the following rules:
20.83/6.16	
20.83/6.16	   a(a(x1)) -> c(b(b(a(x1))))
20.83/6.16	   c(b(x1)) -> x1
20.83/6.16	   b(c(x1)) -> c(a(x1))
20.83/6.16	
20.83/6.16	Q is empty.
20.83/6.16	We have to consider all minimal (P,Q,R)-chains.
20.83/6.16	----------------------------------------
20.83/6.16	
20.83/6.16	(9) QDPOrderProof (EQUIVALENT)
20.83/6.16	We use the reduction pair processor [LPAR04,JAR06].
20.83/6.16	
20.83/6.16	
20.83/6.16	The following pairs can be oriented strictly and are deleted.
20.83/6.16	
20.83/6.16	   B(c(x1)) -> A(x1)
20.83/6.16	The remaining pairs can at least be oriented weakly.
20.83/6.16	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
20.83/6.16	
20.83/6.16	   <<<
20.83/6.16	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
20.83/6.16	>>>
20.83/6.16	
20.83/6.16	   <<<
20.83/6.16	 POL(a(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[-I, 0A, -I], [-I, 1A, 0A], [-I, 0A, 0A]] 	* 	x_1
20.83/6.16	>>>
20.83/6.16	
20.83/6.16	   <<<
20.83/6.16	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
20.83/6.16	>>>
20.83/6.16	
20.83/6.16	   <<<
20.83/6.16	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[1A, 0A, 0A], [-I, -I, 0A], [0A, -I, -I]] 	* 	x_1
20.83/6.16	>>>
20.83/6.16	
20.83/6.16	   <<<
20.83/6.16	 POL(c(x_1)) =  	[[1A], [-I], [0A]] 	 +  	[[-I, 1A, 0A], [0A, -I, 0A], [-I, 0A, 0A]] 	* 	x_1
20.83/6.16	>>>
20.83/6.16	
20.83/6.16	
20.83/6.16	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
20.83/6.16	
20.83/6.16	   a(a(x1)) -> c(b(b(a(x1))))
20.83/6.16	   b(c(x1)) -> c(a(x1))
20.83/6.16	   c(b(x1)) -> x1
20.83/6.16	
20.83/6.16	
20.83/6.16	----------------------------------------
20.83/6.16	
20.83/6.16	(10)
20.83/6.16	Obligation:
20.83/6.16	Q DP problem:
20.83/6.16	The TRS P consists of the following rules:
20.83/6.16	
20.83/6.16	   A(a(x1)) -> B(b(a(x1)))
20.83/6.16	
20.83/6.16	The TRS R consists of the following rules:
20.83/6.16	
20.83/6.16	   a(a(x1)) -> c(b(b(a(x1))))
20.83/6.16	   c(b(x1)) -> x1
20.83/6.16	   b(c(x1)) -> c(a(x1))
20.83/6.16	
20.83/6.16	Q is empty.
20.83/6.16	We have to consider all minimal (P,Q,R)-chains.
20.83/6.16	----------------------------------------
20.83/6.16	
20.83/6.16	(11) DependencyGraphProof (EQUIVALENT)
20.83/6.16	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
20.83/6.16	----------------------------------------
20.83/6.16	
20.83/6.16	(12)
20.83/6.16	TRUE
21.47/6.33	EOF