20.83/6.20	YES
20.83/6.21	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
20.83/6.21	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
20.83/6.21	
20.83/6.21	
20.83/6.21	Termination w.r.t. Q of the given QTRS could be proven:
20.83/6.21	
20.83/6.21	(0) QTRS
20.83/6.21	(1) QTRS Reverse [EQUIVALENT, 0 ms]
20.83/6.21	(2) QTRS
20.83/6.21	(3) Overlay + Local Confluence [EQUIVALENT, 0 ms]
20.83/6.21	(4) QTRS
20.83/6.21	(5) DependencyPairsProof [EQUIVALENT, 13 ms]
20.83/6.21	(6) QDP
20.83/6.21	(7) QDPOrderProof [EQUIVALENT, 244 ms]
20.83/6.21	(8) QDP
20.83/6.21	(9) QDPOrderProof [EQUIVALENT, 80 ms]
20.83/6.21	(10) QDP
20.83/6.21	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
20.83/6.21	(12) TRUE
20.83/6.21	
20.83/6.21	
20.83/6.21	----------------------------------------
20.83/6.21	
20.83/6.21	(0)
20.83/6.21	Obligation:
20.83/6.21	Q restricted rewrite system:
20.83/6.21	The TRS R consists of the following rules:
20.83/6.21	
20.83/6.21	   a(a(b(x1))) -> c(a(c(a(a(x1)))))
20.83/6.21	   a(c(x1)) -> b(a(x1))
20.83/6.21	
20.83/6.21	Q is empty.
20.83/6.21	
20.83/6.21	----------------------------------------
20.83/6.21	
20.83/6.21	(1) QTRS Reverse (EQUIVALENT)
20.83/6.21	We applied the QTRS Reverse Processor [REVERSE].
20.83/6.21	----------------------------------------
20.83/6.21	
20.83/6.21	(2)
20.83/6.21	Obligation:
20.83/6.21	Q restricted rewrite system:
20.83/6.21	The TRS R consists of the following rules:
20.83/6.21	
20.83/6.21	   b(a(a(x1))) -> a(a(c(a(c(x1)))))
20.83/6.21	   c(a(x1)) -> a(b(x1))
20.83/6.21	
20.83/6.21	Q is empty.
20.83/6.21	
20.83/6.21	----------------------------------------
20.83/6.21	
20.83/6.21	(3) Overlay + Local Confluence (EQUIVALENT)
20.83/6.21	The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
20.83/6.21	----------------------------------------
20.83/6.21	
20.83/6.21	(4)
20.83/6.21	Obligation:
20.83/6.21	Q restricted rewrite system:
20.83/6.21	The TRS R consists of the following rules:
20.83/6.21	
20.83/6.21	   b(a(a(x1))) -> a(a(c(a(c(x1)))))
20.83/6.21	   c(a(x1)) -> a(b(x1))
20.83/6.21	
20.83/6.21	The set Q consists of the following terms:
20.83/6.21	
20.83/6.21	   b(a(a(x0)))
20.83/6.21	   c(a(x0))
20.83/6.21	
20.83/6.21	
20.83/6.21	----------------------------------------
20.83/6.21	
20.83/6.21	(5) DependencyPairsProof (EQUIVALENT)
20.83/6.21	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
20.83/6.21	----------------------------------------
20.83/6.21	
20.83/6.21	(6)
20.83/6.21	Obligation:
20.83/6.21	Q DP problem:
20.83/6.21	The TRS P consists of the following rules:
20.83/6.21	
20.83/6.21	   B(a(a(x1))) -> C(a(c(x1)))
20.83/6.21	   B(a(a(x1))) -> C(x1)
20.83/6.21	   C(a(x1)) -> B(x1)
20.83/6.21	
20.83/6.21	The TRS R consists of the following rules:
20.83/6.21	
20.83/6.21	   b(a(a(x1))) -> a(a(c(a(c(x1)))))
20.83/6.21	   c(a(x1)) -> a(b(x1))
20.83/6.21	
20.83/6.21	The set Q consists of the following terms:
20.83/6.21	
20.83/6.21	   b(a(a(x0)))
20.83/6.21	   c(a(x0))
20.83/6.21	
20.83/6.21	We have to consider all minimal (P,Q,R)-chains.
20.83/6.21	----------------------------------------
20.83/6.21	
20.83/6.21	(7) QDPOrderProof (EQUIVALENT)
20.83/6.21	We use the reduction pair processor [LPAR04,JAR06].
20.83/6.21	
20.83/6.21	
20.83/6.21	The following pairs can be oriented strictly and are deleted.
20.83/6.21	
20.83/6.21	   B(a(a(x1))) -> C(x1)
20.83/6.21	The remaining pairs can at least be oriented weakly.
20.83/6.21	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
20.83/6.21	
20.83/6.21	   <<<
20.83/6.21	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
20.83/6.21	>>>
20.83/6.21	
20.83/6.21	   <<<
20.83/6.21	 POL(a(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[0A, 0A, -I], [0A, 0A, 0A], [1A, 0A, 0A]] 	* 	x_1
20.83/6.21	>>>
20.83/6.21	
20.83/6.21	   <<<
20.83/6.21	 POL(C(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
20.83/6.21	>>>
20.83/6.21	
20.83/6.21	   <<<
20.83/6.21	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, -I, -I], [0A, -I, -I], [1A, 0A, 0A]] 	* 	x_1
20.83/6.21	>>>
20.83/6.21	
20.83/6.21	   <<<
20.83/6.21	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, -I], [0A, 0A, -I], [0A, 0A, -I]] 	* 	x_1
20.83/6.21	>>>
20.83/6.21	
20.83/6.21	
20.83/6.21	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
20.83/6.21	
20.83/6.21	   c(a(x1)) -> a(b(x1))
20.83/6.21	   b(a(a(x1))) -> a(a(c(a(c(x1)))))
20.83/6.21	
20.83/6.21	
20.83/6.21	----------------------------------------
20.83/6.21	
20.83/6.21	(8)
20.83/6.21	Obligation:
20.83/6.21	Q DP problem:
20.83/6.21	The TRS P consists of the following rules:
20.83/6.21	
20.83/6.21	   B(a(a(x1))) -> C(a(c(x1)))
20.83/6.21	   C(a(x1)) -> B(x1)
20.83/6.21	
20.83/6.21	The TRS R consists of the following rules:
20.83/6.21	
20.83/6.21	   b(a(a(x1))) -> a(a(c(a(c(x1)))))
20.83/6.21	   c(a(x1)) -> a(b(x1))
20.83/6.21	
20.83/6.21	The set Q consists of the following terms:
20.83/6.21	
20.83/6.21	   b(a(a(x0)))
20.83/6.21	   c(a(x0))
20.83/6.21	
20.83/6.21	We have to consider all minimal (P,Q,R)-chains.
20.83/6.21	----------------------------------------
20.83/6.21	
20.83/6.21	(9) QDPOrderProof (EQUIVALENT)
20.83/6.21	We use the reduction pair processor [LPAR04,JAR06].
20.83/6.21	
20.83/6.21	
20.83/6.21	The following pairs can be oriented strictly and are deleted.
20.83/6.21	
20.83/6.21	   B(a(a(x1))) -> C(a(c(x1)))
20.83/6.21	The remaining pairs can at least be oriented weakly.
20.83/6.21	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
20.83/6.21	
20.83/6.21	   <<<
20.83/6.21	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
20.83/6.21	>>>
20.83/6.21	
20.83/6.21	   <<<
20.83/6.21	 POL(a(x_1)) =  	[[-I], [1A], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 1A], [0A, -I, 0A]] 	* 	x_1
20.83/6.21	>>>
20.83/6.21	
20.83/6.21	   <<<
20.83/6.21	 POL(C(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
20.83/6.21	>>>
20.83/6.21	
20.83/6.21	   <<<
20.83/6.21	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, 0A], [0A, 0A, 1A], [-I, -I, 0A]] 	* 	x_1
20.83/6.21	>>>
20.83/6.21	
20.83/6.21	   <<<
20.83/6.21	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, -I, 0A], [0A, -I, 0A], [0A, -I, 0A]] 	* 	x_1
20.83/6.21	>>>
20.83/6.21	
20.83/6.21	
20.83/6.21	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
20.83/6.21	
20.83/6.21	   c(a(x1)) -> a(b(x1))
20.83/6.21	   b(a(a(x1))) -> a(a(c(a(c(x1)))))
20.83/6.21	
20.83/6.21	
20.83/6.21	----------------------------------------
20.83/6.21	
20.83/6.21	(10)
20.83/6.21	Obligation:
20.83/6.21	Q DP problem:
20.83/6.21	The TRS P consists of the following rules:
20.83/6.21	
20.83/6.21	   C(a(x1)) -> B(x1)
20.83/6.21	
20.83/6.21	The TRS R consists of the following rules:
20.83/6.21	
20.83/6.21	   b(a(a(x1))) -> a(a(c(a(c(x1)))))
20.83/6.21	   c(a(x1)) -> a(b(x1))
20.83/6.21	
20.83/6.21	The set Q consists of the following terms:
20.83/6.21	
20.83/6.21	   b(a(a(x0)))
20.83/6.21	   c(a(x0))
20.83/6.21	
20.83/6.21	We have to consider all minimal (P,Q,R)-chains.
20.83/6.21	----------------------------------------
20.83/6.21	
20.83/6.21	(11) DependencyGraphProof (EQUIVALENT)
20.83/6.21	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
20.83/6.21	----------------------------------------
20.83/6.21	
20.83/6.21	(12)
20.83/6.21	TRUE
20.99/6.28	EOF