28.43/8.19	YES
28.93/8.37	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
28.93/8.37	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
28.93/8.37	
28.93/8.37	
28.93/8.37	Termination w.r.t. Q of the given QTRS could be proven:
28.93/8.37	
28.93/8.37	(0) QTRS
28.93/8.37	(1) QTRS Reverse [EQUIVALENT, 0 ms]
28.93/8.37	(2) QTRS
28.93/8.37	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
28.93/8.37	(4) QDP
28.93/8.37	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
28.93/8.37	(6) QDP
28.93/8.37	(7) QDPOrderProof [EQUIVALENT, 107 ms]
28.93/8.37	(8) QDP
28.93/8.37	(9) QDPOrderProof [EQUIVALENT, 47 ms]
28.93/8.37	(10) QDP
28.93/8.37	(11) PisEmptyProof [EQUIVALENT, 0 ms]
28.93/8.37	(12) YES
28.93/8.37	
28.93/8.37	
28.93/8.37	----------------------------------------
28.93/8.37	
28.93/8.37	(0)
28.93/8.37	Obligation:
28.93/8.37	Q restricted rewrite system:
28.93/8.37	The TRS R consists of the following rules:
28.93/8.37	
28.93/8.37	   a(x1) -> x1
28.93/8.37	   a(b(x1)) -> x1
28.93/8.37	   b(a(x1)) -> a(a(b(c(x1))))
28.93/8.37	   c(c(x1)) -> b(x1)
28.93/8.37	
28.93/8.37	Q is empty.
28.93/8.37	
28.93/8.37	----------------------------------------
28.93/8.37	
28.93/8.37	(1) QTRS Reverse (EQUIVALENT)
28.93/8.37	We applied the QTRS Reverse Processor [REVERSE].
28.93/8.37	----------------------------------------
28.93/8.37	
28.93/8.37	(2)
28.93/8.37	Obligation:
28.93/8.37	Q restricted rewrite system:
28.93/8.37	The TRS R consists of the following rules:
28.93/8.37	
28.93/8.37	   a(x1) -> x1
28.93/8.37	   b(a(x1)) -> x1
28.93/8.37	   a(b(x1)) -> c(b(a(a(x1))))
28.93/8.37	   c(c(x1)) -> b(x1)
28.93/8.37	
28.93/8.37	Q is empty.
28.93/8.37	
28.93/8.37	----------------------------------------
28.93/8.37	
28.93/8.37	(3) DependencyPairsProof (EQUIVALENT)
28.93/8.37	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
28.93/8.37	----------------------------------------
28.93/8.37	
28.93/8.37	(4)
28.93/8.37	Obligation:
28.93/8.37	Q DP problem:
28.93/8.37	The TRS P consists of the following rules:
28.93/8.37	
28.93/8.37	   A(b(x1)) -> C(b(a(a(x1))))
28.93/8.37	   A(b(x1)) -> B(a(a(x1)))
28.93/8.37	   A(b(x1)) -> A(a(x1))
28.93/8.37	   A(b(x1)) -> A(x1)
28.93/8.37	   C(c(x1)) -> B(x1)
28.93/8.37	
28.93/8.37	The TRS R consists of the following rules:
28.93/8.37	
28.93/8.37	   a(x1) -> x1
28.93/8.37	   b(a(x1)) -> x1
28.93/8.37	   a(b(x1)) -> c(b(a(a(x1))))
28.93/8.37	   c(c(x1)) -> b(x1)
28.93/8.37	
28.93/8.37	Q is empty.
28.93/8.37	We have to consider all minimal (P,Q,R)-chains.
28.93/8.37	----------------------------------------
28.93/8.37	
28.93/8.37	(5) DependencyGraphProof (EQUIVALENT)
28.93/8.37	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
28.93/8.37	----------------------------------------
28.93/8.37	
28.93/8.37	(6)
28.93/8.37	Obligation:
28.93/8.37	Q DP problem:
28.93/8.37	The TRS P consists of the following rules:
28.93/8.37	
28.93/8.37	   A(b(x1)) -> A(x1)
28.93/8.37	   A(b(x1)) -> A(a(x1))
28.93/8.37	
28.93/8.37	The TRS R consists of the following rules:
29.21/8.37	
29.21/8.37	   a(x1) -> x1
29.21/8.37	   b(a(x1)) -> x1
29.21/8.37	   a(b(x1)) -> c(b(a(a(x1))))
29.21/8.37	   c(c(x1)) -> b(x1)
29.21/8.37	
29.21/8.37	Q is empty.
29.21/8.37	We have to consider all minimal (P,Q,R)-chains.
29.21/8.37	----------------------------------------
29.21/8.37	
29.21/8.37	(7) QDPOrderProof (EQUIVALENT)
29.21/8.37	We use the reduction pair processor [LPAR04,JAR06].
29.21/8.37	
29.21/8.37	
29.21/8.37	The following pairs can be oriented strictly and are deleted.
29.21/8.37	
29.21/8.37	   A(b(x1)) -> A(x1)
29.21/8.37	The remaining pairs can at least be oriented weakly.
29.21/8.37	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
29.21/8.37	
29.21/8.37	   <<<
29.21/8.37	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
29.21/8.37	>>>
29.21/8.37	
29.21/8.37	   <<<
29.21/8.37	 POL(b(x_1)) =  	[[0A], [1A], [-I]] 	 +  	[[0A, 0A, -I], [0A, 1A, 0A], [1A, 0A, 1A]] 	* 	x_1
29.21/8.37	>>>
29.21/8.37	
29.21/8.37	   <<<
29.21/8.37	 POL(a(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
29.21/8.37	>>>
29.21/8.37	
29.21/8.37	   <<<
29.21/8.37	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [1A, 0A, 0A], [1A, 0A, 0A]] 	* 	x_1
29.21/8.37	>>>
29.21/8.37	
29.21/8.37	
29.21/8.37	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
29.21/8.37	
29.21/8.37	   a(x1) -> x1
29.21/8.37	   a(b(x1)) -> c(b(a(a(x1))))
29.21/8.37	   b(a(x1)) -> x1
29.21/8.37	   c(c(x1)) -> b(x1)
29.21/8.37	
29.21/8.37	
29.21/8.37	----------------------------------------
29.21/8.37	
29.21/8.37	(8)
29.21/8.37	Obligation:
29.21/8.37	Q DP problem:
29.21/8.37	The TRS P consists of the following rules:
29.21/8.37	
29.21/8.37	   A(b(x1)) -> A(a(x1))
29.21/8.37	
29.21/8.37	The TRS R consists of the following rules:
29.21/8.37	
29.21/8.37	   a(x1) -> x1
29.21/8.37	   b(a(x1)) -> x1
29.21/8.37	   a(b(x1)) -> c(b(a(a(x1))))
29.21/8.37	   c(c(x1)) -> b(x1)
29.21/8.37	
29.21/8.37	Q is empty.
29.21/8.37	We have to consider all minimal (P,Q,R)-chains.
29.21/8.37	----------------------------------------
29.21/8.37	
29.21/8.37	(9) QDPOrderProof (EQUIVALENT)
29.21/8.37	We use the reduction pair processor [LPAR04,JAR06].
29.21/8.37	
29.21/8.37	
29.21/8.37	The following pairs can be oriented strictly and are deleted.
29.21/8.37	
29.21/8.37	   A(b(x1)) -> A(a(x1))
29.21/8.37	The remaining pairs can at least be oriented weakly.
29.21/8.37	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
29.21/8.37	
29.21/8.37	   <<<
29.21/8.37	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
29.21/8.37	>>>
29.21/8.37	
29.21/8.37	   <<<
29.21/8.37	 POL(b(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[1A, 0A, 1A], [0A, 0A, 0A], [0A, -I, 0A]] 	* 	x_1
29.21/8.37	>>>
29.21/8.37	
29.21/8.37	   <<<
29.21/8.37	 POL(a(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, 0A], [0A, 0A, 0A], [0A, -I, 0A]] 	* 	x_1
29.21/8.37	>>>
29.21/8.37	
29.21/8.37	   <<<
29.21/8.37	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 1A], [-I, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
29.21/8.37	>>>
29.21/8.37	
29.21/8.37	
29.21/8.37	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
29.21/8.37	
29.21/8.37	   a(x1) -> x1
29.21/8.37	   a(b(x1)) -> c(b(a(a(x1))))
29.21/8.37	   b(a(x1)) -> x1
29.21/8.37	   c(c(x1)) -> b(x1)
29.21/8.37	
29.21/8.37	
29.21/8.37	----------------------------------------
29.21/8.37	
29.21/8.37	(10)
29.21/8.37	Obligation:
29.21/8.37	Q DP problem:
29.21/8.37	P is empty.
29.21/8.37	The TRS R consists of the following rules:
29.21/8.37	
29.21/8.37	   a(x1) -> x1
29.21/8.37	   b(a(x1)) -> x1
29.21/8.37	   a(b(x1)) -> c(b(a(a(x1))))
29.21/8.37	   c(c(x1)) -> b(x1)
29.21/8.37	
29.21/8.37	Q is empty.
29.21/8.37	We have to consider all minimal (P,Q,R)-chains.
29.21/8.37	----------------------------------------
29.21/8.37	
29.21/8.37	(11) PisEmptyProof (EQUIVALENT)
29.21/8.37	The TRS P is empty. Hence, there is no (P,Q,R) chain.
29.21/8.37	----------------------------------------
29.21/8.37	
29.21/8.37	(12)
29.21/8.37	YES
29.29/8.43	EOF