43.91/12.09	YES
44.51/12.28	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
44.51/12.28	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
44.51/12.28	
44.51/12.28	
44.51/12.28	Termination w.r.t. Q of the given QTRS could be proven:
44.51/12.28	
44.51/12.28	(0) QTRS
44.51/12.28	(1) QTRS Reverse [EQUIVALENT, 0 ms]
44.51/12.28	(2) QTRS
44.51/12.28	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
44.51/12.28	(4) QDP
44.51/12.28	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
44.51/12.28	(6) QDP
44.51/12.28	(7) QDPOrderProof [EQUIVALENT, 173 ms]
44.51/12.28	(8) QDP
44.51/12.28	(9) QDPOrderProof [EQUIVALENT, 174 ms]
44.51/12.28	(10) QDP
44.51/12.28	(11) PisEmptyProof [EQUIVALENT, 0 ms]
44.51/12.28	(12) YES
44.51/12.28	
44.51/12.28	
44.51/12.28	----------------------------------------
44.51/12.28	
44.51/12.28	(0)
44.51/12.28	Obligation:
44.51/12.28	Q restricted rewrite system:
44.51/12.28	The TRS R consists of the following rules:
44.51/12.28	
44.51/12.28	   a(x1) -> x1
44.51/12.28	   a(b(x1)) -> b(c(b(a(x1))))
44.51/12.28	   b(x1) -> a(x1)
44.51/12.28	   c(c(c(x1))) -> x1
44.51/12.28	
44.51/12.28	Q is empty.
44.51/12.28	
44.51/12.28	----------------------------------------
44.51/12.28	
44.51/12.28	(1) QTRS Reverse (EQUIVALENT)
44.51/12.28	We applied the QTRS Reverse Processor [REVERSE].
44.51/12.28	----------------------------------------
44.51/12.28	
44.51/12.28	(2)
44.51/12.28	Obligation:
44.51/12.28	Q restricted rewrite system:
44.51/12.28	The TRS R consists of the following rules:
44.51/12.28	
44.51/12.28	   a(x1) -> x1
44.51/12.28	   b(a(x1)) -> a(b(c(b(x1))))
44.51/12.28	   b(x1) -> a(x1)
44.51/12.28	   c(c(c(x1))) -> x1
44.51/12.28	
44.51/12.28	Q is empty.
44.51/12.28	
44.51/12.28	----------------------------------------
44.51/12.28	
44.51/12.28	(3) DependencyPairsProof (EQUIVALENT)
44.51/12.28	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
44.51/12.28	----------------------------------------
44.51/12.28	
44.51/12.28	(4)
44.51/12.28	Obligation:
44.51/12.28	Q DP problem:
44.51/12.28	The TRS P consists of the following rules:
44.51/12.28	
44.51/12.28	   B(a(x1)) -> A(b(c(b(x1))))
44.51/12.28	   B(a(x1)) -> B(c(b(x1)))
44.51/12.28	   B(a(x1)) -> C(b(x1))
44.51/12.28	   B(a(x1)) -> B(x1)
44.51/12.28	   B(x1) -> A(x1)
44.51/12.28	
44.51/12.28	The TRS R consists of the following rules:
44.51/12.28	
44.51/12.28	   a(x1) -> x1
44.51/12.28	   b(a(x1)) -> a(b(c(b(x1))))
44.51/12.28	   b(x1) -> a(x1)
44.51/12.28	   c(c(c(x1))) -> x1
44.51/12.28	
44.51/12.28	Q is empty.
44.51/12.28	We have to consider all minimal (P,Q,R)-chains.
44.51/12.28	----------------------------------------
44.51/12.28	
44.51/12.28	(5) DependencyGraphProof (EQUIVALENT)
44.51/12.28	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
44.51/12.28	----------------------------------------
44.51/12.28	
44.51/12.28	(6)
44.51/12.28	Obligation:
44.51/12.28	Q DP problem:
44.51/12.28	The TRS P consists of the following rules:
44.51/12.28	
44.51/12.28	   B(a(x1)) -> B(x1)
44.51/12.28	   B(a(x1)) -> B(c(b(x1)))
44.51/12.28	
44.51/12.28	The TRS R consists of the following rules:
44.51/12.28	
44.51/12.28	   a(x1) -> x1
44.51/12.28	   b(a(x1)) -> a(b(c(b(x1))))
44.51/12.28	   b(x1) -> a(x1)
44.51/12.28	   c(c(c(x1))) -> x1
44.51/12.28	
44.51/12.28	Q is empty.
44.51/12.28	We have to consider all minimal (P,Q,R)-chains.
44.51/12.28	----------------------------------------
44.51/12.28	
44.51/12.28	(7) QDPOrderProof (EQUIVALENT)
44.51/12.28	We use the reduction pair processor [LPAR04,JAR06].
44.51/12.28	
44.51/12.28	
44.51/12.28	The following pairs can be oriented strictly and are deleted.
44.51/12.28	
44.51/12.28	   B(a(x1)) -> B(x1)
44.51/12.28	The remaining pairs can at least be oriented weakly.
44.51/12.28	Used ordering:  Matrix interpretation [MATRO] with arctic integers [ARCTIC,STERNAGEL_THIEMANN_RTA14]:
44.51/12.28	
44.51/12.28	   <<<
44.51/12.28	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-1A, 0A, -1A]] 	* 	x_1
44.51/12.28	>>>
44.51/12.28	
44.51/12.28	   <<<
44.51/12.28	 POL(a(x_1)) =  	[[-1A], [1A], [-I]] 	 +  	[[0A, -1A, -I], [0A, 1A, 0A], [-I, -I, 0A]] 	* 	x_1
44.51/12.28	>>>
44.51/12.28	
44.51/12.28	   <<<
44.51/12.28	 POL(c(x_1)) =  	[[2A], [-I], [-I]] 	 +  	[[-I, -I, -1A], [-I, -I, -1A], [0A, 0A, 1A]] 	* 	x_1
44.51/12.28	>>>
44.51/12.28	
44.51/12.28	   <<<
44.51/12.28	 POL(b(x_1)) =  	[[0A], [1A], [-I]] 	 +  	[[0A, 1A, -I], [0A, 2A, 0A], [-I, 1A, 0A]] 	* 	x_1
44.51/12.28	>>>
44.51/12.28	
44.51/12.28	
44.51/12.28	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
44.51/12.28	
44.51/12.28	   b(a(x1)) -> a(b(c(b(x1))))
44.51/12.28	   b(x1) -> a(x1)
44.51/12.28	   c(c(c(x1))) -> x1
44.51/12.28	   a(x1) -> x1
44.51/12.28	
44.51/12.28	
44.51/12.28	----------------------------------------
44.51/12.28	
44.51/12.28	(8)
44.51/12.28	Obligation:
44.51/12.28	Q DP problem:
44.51/12.28	The TRS P consists of the following rules:
44.51/12.28	
44.51/12.28	   B(a(x1)) -> B(c(b(x1)))
44.51/12.28	
44.51/12.28	The TRS R consists of the following rules:
44.51/12.28	
44.51/12.28	   a(x1) -> x1
44.51/12.28	   b(a(x1)) -> a(b(c(b(x1))))
44.51/12.28	   b(x1) -> a(x1)
44.51/12.28	   c(c(c(x1))) -> x1
44.51/12.28	
44.51/12.28	Q is empty.
44.51/12.28	We have to consider all minimal (P,Q,R)-chains.
44.51/12.28	----------------------------------------
44.51/12.28	
44.51/12.28	(9) QDPOrderProof (EQUIVALENT)
44.51/12.28	We use the reduction pair processor [LPAR04,JAR06].
44.51/12.28	
44.51/12.28	
44.51/12.28	The following pairs can be oriented strictly and are deleted.
44.51/12.28	
44.51/12.28	   B(a(x1)) -> B(c(b(x1)))
44.51/12.28	The remaining pairs can at least be oriented weakly.
44.51/12.28	Used ordering:  Matrix interpretation [MATRO] with arctic integers [ARCTIC,STERNAGEL_THIEMANN_RTA14]:
44.51/12.28	
44.51/12.28	   <<<
44.51/12.28	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, 2A, -I]] 	* 	x_1
44.51/12.28	>>>
44.51/12.28	
44.51/12.28	   <<<
44.51/12.28	 POL(a(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -1A, -I], [1A, 1A, -I], [-I, -I, 0A]] 	* 	x_1
44.51/12.28	>>>
44.51/12.28	
44.51/12.28	   <<<
44.51/12.28	 POL(c(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[1A, -1A, -I], [0A, -I, -I], [-1A, -1A, 0A]] 	* 	x_1
44.51/12.28	>>>
44.51/12.28	
44.51/12.28	   <<<
44.51/12.28	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, -I], [1A, 2A, -I], [-I, 1A, 0A]] 	* 	x_1
44.51/12.28	>>>
44.51/12.28	
44.51/12.28	
44.51/12.28	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
44.51/12.28	
44.51/12.28	   b(a(x1)) -> a(b(c(b(x1))))
44.51/12.28	   b(x1) -> a(x1)
44.51/12.28	   c(c(c(x1))) -> x1
44.51/12.28	   a(x1) -> x1
44.51/12.28	
44.51/12.28	
44.51/12.28	----------------------------------------
44.51/12.28	
44.51/12.28	(10)
44.51/12.28	Obligation:
44.51/12.28	Q DP problem:
44.51/12.28	P is empty.
44.51/12.28	The TRS R consists of the following rules:
44.51/12.28	
44.51/12.28	   a(x1) -> x1
44.51/12.28	   b(a(x1)) -> a(b(c(b(x1))))
44.51/12.28	   b(x1) -> a(x1)
44.51/12.28	   c(c(c(x1))) -> x1
44.51/12.28	
44.51/12.28	Q is empty.
44.51/12.28	We have to consider all minimal (P,Q,R)-chains.
44.51/12.28	----------------------------------------
44.51/12.28	
44.51/12.28	(11) PisEmptyProof (EQUIVALENT)
44.51/12.28	The TRS P is empty. Hence, there is no (P,Q,R) chain.
44.51/12.28	----------------------------------------
44.51/12.28	
44.51/12.28	(12)
44.51/12.28	YES
44.79/12.36	EOF