24.29/7.13	YES
24.29/7.14	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
24.29/7.14	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
24.29/7.14	
24.29/7.14	
24.29/7.14	Termination w.r.t. Q of the given QTRS could be proven:
24.29/7.14	
24.29/7.14	(0) QTRS
24.29/7.14	(1) QTRS Reverse [EQUIVALENT, 0 ms]
24.29/7.14	(2) QTRS
24.29/7.14	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
24.29/7.14	(4) QDP
24.29/7.14	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
24.29/7.14	(6) QDP
24.29/7.14	(7) QDPOrderProof [EQUIVALENT, 136 ms]
24.29/7.14	(8) QDP
24.29/7.14	(9) QDPOrderProof [EQUIVALENT, 53 ms]
24.29/7.14	(10) QDP
24.29/7.14	(11) PisEmptyProof [EQUIVALENT, 0 ms]
24.29/7.14	(12) YES
24.29/7.14	
24.29/7.14	
24.29/7.14	----------------------------------------
24.29/7.14	
24.29/7.14	(0)
24.29/7.14	Obligation:
24.29/7.14	Q restricted rewrite system:
24.29/7.14	The TRS R consists of the following rules:
24.29/7.14	
24.29/7.14	   a(x1) -> x1
24.29/7.14	   a(a(b(x1))) -> b(b(a(a(x1))))
24.29/7.14	   b(x1) -> a(c(a(x1)))
24.29/7.14	
24.29/7.14	Q is empty.
24.29/7.14	
24.29/7.14	----------------------------------------
24.29/7.14	
24.29/7.14	(1) QTRS Reverse (EQUIVALENT)
24.29/7.14	We applied the QTRS Reverse Processor [REVERSE].
24.29/7.14	----------------------------------------
24.29/7.14	
24.29/7.14	(2)
24.29/7.14	Obligation:
24.29/7.14	Q restricted rewrite system:
24.29/7.14	The TRS R consists of the following rules:
24.29/7.14	
24.29/7.14	   a(x1) -> x1
24.29/7.14	   b(a(a(x1))) -> a(a(b(b(x1))))
24.29/7.14	   b(x1) -> a(c(a(x1)))
24.29/7.14	
24.29/7.14	Q is empty.
24.29/7.14	
24.29/7.14	----------------------------------------
24.29/7.14	
24.29/7.14	(3) DependencyPairsProof (EQUIVALENT)
24.29/7.14	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
24.29/7.14	----------------------------------------
24.29/7.14	
24.29/7.14	(4)
24.29/7.14	Obligation:
24.29/7.14	Q DP problem:
24.29/7.14	The TRS P consists of the following rules:
24.29/7.14	
24.29/7.14	   B(a(a(x1))) -> A(a(b(b(x1))))
24.29/7.14	   B(a(a(x1))) -> A(b(b(x1)))
24.29/7.14	   B(a(a(x1))) -> B(b(x1))
24.29/7.14	   B(a(a(x1))) -> B(x1)
24.29/7.14	   B(x1) -> A(c(a(x1)))
24.29/7.14	   B(x1) -> A(x1)
24.29/7.14	
24.29/7.14	The TRS R consists of the following rules:
24.29/7.14	
24.29/7.14	   a(x1) -> x1
24.29/7.14	   b(a(a(x1))) -> a(a(b(b(x1))))
24.29/7.14	   b(x1) -> a(c(a(x1)))
24.29/7.14	
24.29/7.14	Q is empty.
24.29/7.14	We have to consider all minimal (P,Q,R)-chains.
24.29/7.14	----------------------------------------
24.29/7.14	
24.29/7.14	(5) DependencyGraphProof (EQUIVALENT)
24.29/7.14	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.
24.29/7.14	----------------------------------------
24.29/7.14	
24.29/7.14	(6)
24.29/7.14	Obligation:
24.29/7.14	Q DP problem:
24.29/7.14	The TRS P consists of the following rules:
24.29/7.14	
24.29/7.14	   B(a(a(x1))) -> B(x1)
24.29/7.14	   B(a(a(x1))) -> B(b(x1))
24.29/7.14	
24.29/7.14	The TRS R consists of the following rules:
24.29/7.14	
24.29/7.14	   a(x1) -> x1
24.29/7.14	   b(a(a(x1))) -> a(a(b(b(x1))))
24.29/7.14	   b(x1) -> a(c(a(x1)))
24.29/7.14	
24.29/7.14	Q is empty.
24.29/7.14	We have to consider all minimal (P,Q,R)-chains.
24.29/7.14	----------------------------------------
24.29/7.14	
24.29/7.14	(7) QDPOrderProof (EQUIVALENT)
24.29/7.14	We use the reduction pair processor [LPAR04,JAR06].
24.29/7.14	
24.29/7.14	
24.29/7.14	The following pairs can be oriented strictly and are deleted.
24.29/7.14	
24.29/7.14	   B(a(a(x1))) -> B(x1)
24.29/7.14	The remaining pairs can at least be oriented weakly.
24.29/7.14	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
24.29/7.14	
24.29/7.14	   <<<
24.29/7.14	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
24.29/7.14	>>>
24.29/7.14	
24.29/7.14	   <<<
24.29/7.14	 POL(a(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [1A, 0A, 1A], [0A, 0A, 0A]] 	* 	x_1
24.29/7.14	>>>
24.29/7.14	
24.29/7.14	   <<<
24.29/7.14	 POL(b(x_1)) =  	[[1A], [1A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
24.29/7.14	>>>
24.29/7.14	
24.29/7.14	   <<<
24.29/7.14	 POL(c(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[-I, -I, -I], [-I, -I, 0A], [-I, -I, -I]] 	* 	x_1
24.29/7.14	>>>
24.29/7.14	
24.29/7.14	
24.29/7.14	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
24.29/7.14	
24.29/7.14	   b(a(a(x1))) -> a(a(b(b(x1))))
24.29/7.14	   b(x1) -> a(c(a(x1)))
24.29/7.14	   a(x1) -> x1
24.29/7.14	
24.29/7.14	
24.29/7.14	----------------------------------------
24.29/7.14	
24.29/7.14	(8)
24.29/7.14	Obligation:
24.29/7.14	Q DP problem:
24.29/7.14	The TRS P consists of the following rules:
24.29/7.14	
24.29/7.14	   B(a(a(x1))) -> B(b(x1))
24.29/7.14	
24.29/7.14	The TRS R consists of the following rules:
24.29/7.14	
24.29/7.14	   a(x1) -> x1
24.29/7.14	   b(a(a(x1))) -> a(a(b(b(x1))))
24.29/7.14	   b(x1) -> a(c(a(x1)))
24.29/7.14	
24.29/7.14	Q is empty.
24.29/7.14	We have to consider all minimal (P,Q,R)-chains.
24.29/7.14	----------------------------------------
24.29/7.14	
24.29/7.14	(9) QDPOrderProof (EQUIVALENT)
24.29/7.14	We use the reduction pair processor [LPAR04,JAR06].
24.29/7.14	
24.29/7.14	
24.29/7.14	The following pairs can be oriented strictly and are deleted.
24.29/7.14	
24.29/7.14	   B(a(a(x1))) -> B(b(x1))
24.29/7.14	The remaining pairs can at least be oriented weakly.
24.29/7.14	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
24.29/7.14	
24.29/7.14	   <<<
24.29/7.14	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, 1A, 0A]] 	* 	x_1
24.29/7.14	>>>
24.29/7.14	
24.29/7.14	   <<<
24.29/7.14	 POL(a(x_1)) =  	[[1A], [-I], [-I]] 	 +  	[[0A, 1A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
24.29/7.14	>>>
24.29/7.14	
24.29/7.14	   <<<
24.29/7.14	 POL(b(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[0A, 1A, 0A], [-I, 0A, -I], [-I, -I, 0A]] 	* 	x_1
24.29/7.14	>>>
24.29/7.14	
24.29/7.14	   <<<
24.29/7.14	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, -I], [-I, -I, -I], [-I, -I, -I]] 	* 	x_1
24.29/7.14	>>>
24.29/7.14	
24.29/7.14	
24.29/7.14	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
24.29/7.14	
24.29/7.14	   b(a(a(x1))) -> a(a(b(b(x1))))
24.29/7.14	   b(x1) -> a(c(a(x1)))
24.29/7.14	   a(x1) -> x1
24.29/7.14	
24.29/7.14	
24.29/7.14	----------------------------------------
24.29/7.14	
24.29/7.14	(10)
24.29/7.14	Obligation:
24.29/7.14	Q DP problem:
24.29/7.14	P is empty.
24.29/7.14	The TRS R consists of the following rules:
24.29/7.14	
24.29/7.14	   a(x1) -> x1
24.29/7.14	   b(a(a(x1))) -> a(a(b(b(x1))))
24.29/7.14	   b(x1) -> a(c(a(x1)))
24.29/7.14	
24.29/7.14	Q is empty.
24.29/7.14	We have to consider all minimal (P,Q,R)-chains.
24.29/7.14	----------------------------------------
24.29/7.14	
24.29/7.14	(11) PisEmptyProof (EQUIVALENT)
24.29/7.14	The TRS P is empty. Hence, there is no (P,Q,R) chain.
24.29/7.14	----------------------------------------
24.29/7.14	
24.29/7.14	(12)
24.29/7.14	YES
24.64/7.20	EOF