33.60/9.49	YES
33.98/9.57	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
33.98/9.57	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
33.98/9.57	
33.98/9.57	
33.98/9.57	Termination w.r.t. Q of the given QTRS could be proven:
33.98/9.57	
33.98/9.57	(0) QTRS
33.98/9.57	(1) QTRS Reverse [EQUIVALENT, 0 ms]
33.98/9.57	(2) QTRS
33.98/9.57	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
33.98/9.57	(4) QDP
33.98/9.57	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
33.98/9.57	(6) QDP
33.98/9.57	(7) QDPOrderProof [EQUIVALENT, 143 ms]
33.98/9.57	(8) QDP
33.98/9.57	(9) QDPOrderProof [EQUIVALENT, 76 ms]
33.98/9.57	(10) QDP
33.98/9.57	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
33.98/9.57	(12) TRUE
33.98/9.57	
33.98/9.57	
33.98/9.57	----------------------------------------
33.98/9.57	
33.98/9.57	(0)
33.98/9.57	Obligation:
33.98/9.57	Q restricted rewrite system:
33.98/9.57	The TRS R consists of the following rules:
33.98/9.57	
33.98/9.57	   a(x1) -> x1
33.98/9.57	   a(a(b(x1))) -> c(c(a(c(x1))))
33.98/9.57	   c(c(x1)) -> b(a(x1))
33.98/9.57	
33.98/9.57	Q is empty.
33.98/9.57	
33.98/9.57	----------------------------------------
33.98/9.57	
33.98/9.57	(1) QTRS Reverse (EQUIVALENT)
33.98/9.57	We applied the QTRS Reverse Processor [REVERSE].
33.98/9.57	----------------------------------------
33.98/9.57	
33.98/9.57	(2)
33.98/9.57	Obligation:
33.98/9.57	Q restricted rewrite system:
33.98/9.57	The TRS R consists of the following rules:
33.98/9.57	
33.98/9.57	   a(x1) -> x1
33.98/9.57	   b(a(a(x1))) -> c(a(c(c(x1))))
33.98/9.58	   c(c(x1)) -> a(b(x1))
33.98/9.58	
33.98/9.58	Q is empty.
33.98/9.58	
33.98/9.58	----------------------------------------
33.98/9.58	
33.98/9.58	(3) DependencyPairsProof (EQUIVALENT)
33.98/9.58	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
33.98/9.58	----------------------------------------
33.98/9.58	
33.98/9.58	(4)
33.98/9.58	Obligation:
33.98/9.58	Q DP problem:
33.98/9.58	The TRS P consists of the following rules:
33.98/9.58	
33.98/9.58	   B(a(a(x1))) -> C(a(c(c(x1))))
33.98/9.58	   B(a(a(x1))) -> A(c(c(x1)))
33.98/9.58	   B(a(a(x1))) -> C(c(x1))
33.98/9.58	   B(a(a(x1))) -> C(x1)
33.98/9.58	   C(c(x1)) -> A(b(x1))
33.98/9.58	   C(c(x1)) -> B(x1)
33.98/9.58	
33.98/9.58	The TRS R consists of the following rules:
33.98/9.58	
33.98/9.58	   a(x1) -> x1
33.98/9.58	   b(a(a(x1))) -> c(a(c(c(x1))))
33.98/9.58	   c(c(x1)) -> a(b(x1))
33.98/9.58	
33.98/9.58	Q is empty.
33.98/9.58	We have to consider all minimal (P,Q,R)-chains.
33.98/9.58	----------------------------------------
33.98/9.58	
33.98/9.58	(5) DependencyGraphProof (EQUIVALENT)
33.98/9.58	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
33.98/9.58	----------------------------------------
33.98/9.58	
33.98/9.58	(6)
33.98/9.58	Obligation:
33.98/9.58	Q DP problem:
33.98/9.58	The TRS P consists of the following rules:
33.98/9.58	
33.98/9.58	   C(c(x1)) -> B(x1)
33.98/9.58	   B(a(a(x1))) -> C(a(c(c(x1))))
33.98/9.58	   B(a(a(x1))) -> C(c(x1))
33.98/9.58	   B(a(a(x1))) -> C(x1)
33.98/9.58	
33.98/9.58	The TRS R consists of the following rules:
33.98/9.58	
33.98/9.58	   a(x1) -> x1
33.98/9.58	   b(a(a(x1))) -> c(a(c(c(x1))))
33.98/9.58	   c(c(x1)) -> a(b(x1))
33.98/9.58	
33.98/9.58	Q is empty.
33.98/9.58	We have to consider all minimal (P,Q,R)-chains.
33.98/9.58	----------------------------------------
33.98/9.58	
33.98/9.58	(7) QDPOrderProof (EQUIVALENT)
33.98/9.58	We use the reduction pair processor [LPAR04,JAR06].
33.98/9.58	
33.98/9.58	
33.98/9.58	The following pairs can be oriented strictly and are deleted.
33.98/9.58	
33.98/9.58	   B(a(a(x1))) -> C(x1)
33.98/9.58	The remaining pairs can at least be oriented weakly.
33.98/9.58	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
33.98/9.58	
33.98/9.58	   <<<
33.98/9.58	 POL(C(x_1)) =  	[[-I]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
33.98/9.58	>>>
33.98/9.58	
33.98/9.58	   <<<
33.98/9.58	 POL(c(x_1)) =  	[[-I], [0A], [0A]] 	 +  	[[0A, 0A, -I], [1A, 0A, 0A], [1A, -I, -I]] 	* 	x_1
33.98/9.58	>>>
33.98/9.58	
33.98/9.58	   <<<
33.98/9.58	 POL(B(x_1)) =  	[[0A]] 	 +  	[[1A, -I, -I]] 	* 	x_1
33.98/9.58	>>>
33.98/9.58	
33.98/9.58	   <<<
33.98/9.58	 POL(a(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, -I, 0A], [1A, 1A, 0A], [0A, 0A, 0A]] 	* 	x_1
33.98/9.58	>>>
33.98/9.58	
33.98/9.58	   <<<
33.98/9.58	 POL(b(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, 0A, -I], [0A, 0A, -I], [0A, 0A, -I]] 	* 	x_1
33.98/9.58	>>>
33.98/9.58	
33.98/9.58	
33.98/9.58	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
33.98/9.58	
33.98/9.58	   c(c(x1)) -> a(b(x1))
33.98/9.58	   a(x1) -> x1
33.98/9.58	   b(a(a(x1))) -> c(a(c(c(x1))))
33.98/9.58	
33.98/9.58	
33.98/9.58	----------------------------------------
33.98/9.58	
33.98/9.58	(8)
33.98/9.58	Obligation:
33.98/9.58	Q DP problem:
33.98/9.58	The TRS P consists of the following rules:
33.98/9.58	
33.98/9.58	   C(c(x1)) -> B(x1)
33.98/9.58	   B(a(a(x1))) -> C(a(c(c(x1))))
33.98/9.58	   B(a(a(x1))) -> C(c(x1))
33.98/9.58	
33.98/9.58	The TRS R consists of the following rules:
33.98/9.58	
33.98/9.58	   a(x1) -> x1
33.98/9.58	   b(a(a(x1))) -> c(a(c(c(x1))))
33.98/9.58	   c(c(x1)) -> a(b(x1))
33.98/9.58	
33.98/9.58	Q is empty.
33.98/9.58	We have to consider all minimal (P,Q,R)-chains.
33.98/9.58	----------------------------------------
33.98/9.58	
33.98/9.58	(9) QDPOrderProof (EQUIVALENT)
33.98/9.58	We use the reduction pair processor [LPAR04,JAR06].
33.98/9.58	
33.98/9.58	
33.98/9.58	The following pairs can be oriented strictly and are deleted.
33.98/9.58	
33.98/9.58	   C(c(x1)) -> B(x1)
33.98/9.58	The remaining pairs can at least be oriented weakly.
33.98/9.58	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
33.98/9.58	
33.98/9.58	   <<<
33.98/9.58	 POL(C(x_1)) =  	[[-I]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
33.98/9.58	>>>
33.98/9.58	
33.98/9.58	   <<<
33.98/9.58	 POL(c(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[0A, 1A, 0A], [-I, 0A, -I], [0A, 0A, 0A]] 	* 	x_1
33.98/9.58	>>>
33.98/9.58	
33.98/9.58	   <<<
33.98/9.58	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
33.98/9.58	>>>
33.98/9.58	
33.98/9.58	   <<<
33.98/9.58	 POL(a(x_1)) =  	[[0A], [-I], [1A]] 	 +  	[[0A, 0A, 0A], [-I, 0A, 0A], [0A, 1A, 0A]] 	* 	x_1
33.98/9.58	>>>
33.98/9.58	
33.98/9.58	   <<<
33.98/9.58	 POL(b(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[0A, 1A, 0A], [-I, 0A, -I], [-I, 0A, -I]] 	* 	x_1
33.98/9.58	>>>
33.98/9.58	
33.98/9.58	
33.98/9.58	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
33.98/9.58	
33.98/9.58	   c(c(x1)) -> a(b(x1))
33.98/9.58	   a(x1) -> x1
33.98/9.58	   b(a(a(x1))) -> c(a(c(c(x1))))
33.98/9.58	
33.98/9.58	
33.98/9.58	----------------------------------------
33.98/9.58	
33.98/9.58	(10)
33.98/9.58	Obligation:
33.98/9.58	Q DP problem:
33.98/9.58	The TRS P consists of the following rules:
33.98/9.58	
33.98/9.58	   B(a(a(x1))) -> C(a(c(c(x1))))
33.98/9.58	   B(a(a(x1))) -> C(c(x1))
33.98/9.58	
33.98/9.58	The TRS R consists of the following rules:
33.98/9.58	
33.98/9.58	   a(x1) -> x1
33.98/9.58	   b(a(a(x1))) -> c(a(c(c(x1))))
33.98/9.58	   c(c(x1)) -> a(b(x1))
33.98/9.58	
33.98/9.58	Q is empty.
33.98/9.58	We have to consider all minimal (P,Q,R)-chains.
33.98/9.58	----------------------------------------
33.98/9.58	
33.98/9.58	(11) DependencyGraphProof (EQUIVALENT)
33.98/9.58	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
33.98/9.58	----------------------------------------
33.98/9.58	
33.98/9.58	(12)
33.98/9.58	TRUE
34.36/9.84	EOF