43.89/12.07	YES
44.29/12.11	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
44.29/12.11	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
44.29/12.11	
44.29/12.11	
44.29/12.11	Termination w.r.t. Q of the given QTRS could be proven:
44.29/12.11	
44.29/12.11	(0) QTRS
44.29/12.11	(1) QTRS Reverse [EQUIVALENT, 0 ms]
44.29/12.11	(2) QTRS
44.29/12.11	(3) DependencyPairsProof [EQUIVALENT, 8 ms]
44.29/12.11	(4) QDP
44.29/12.11	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
44.29/12.11	(6) QDP
44.29/12.11	(7) QDPOrderProof [EQUIVALENT, 109 ms]
44.29/12.11	(8) QDP
44.29/12.11	(9) QDPOrderProof [EQUIVALENT, 36 ms]
44.29/12.11	(10) QDP
44.29/12.11	(11) QDPOrderProof [EQUIVALENT, 26 ms]
44.29/12.11	(12) QDP
44.29/12.11	(13) DependencyGraphProof [EQUIVALENT, 0 ms]
44.29/12.11	(14) TRUE
44.29/12.11	
44.29/12.11	
44.29/12.11	----------------------------------------
44.29/12.11	
44.29/12.11	(0)
44.29/12.11	Obligation:
44.29/12.11	Q restricted rewrite system:
44.29/12.11	The TRS R consists of the following rules:
44.29/12.11	
44.29/12.11	   a(x1) -> x1
44.29/12.11	   a(b(x1)) -> b(c(x1))
44.29/12.11	   b(x1) -> x1
44.29/12.11	   c(c(x1)) -> a(c(b(a(x1))))
44.29/12.11	
44.29/12.11	Q is empty.
44.29/12.11	
44.29/12.11	----------------------------------------
44.29/12.11	
44.29/12.11	(1) QTRS Reverse (EQUIVALENT)
44.29/12.11	We applied the QTRS Reverse Processor [REVERSE].
44.29/12.11	----------------------------------------
44.29/12.11	
44.29/12.11	(2)
44.29/12.11	Obligation:
44.29/12.11	Q restricted rewrite system:
44.29/12.11	The TRS R consists of the following rules:
44.29/12.11	
44.29/12.11	   a(x1) -> x1
44.29/12.11	   b(a(x1)) -> c(b(x1))
44.29/12.11	   b(x1) -> x1
44.29/12.11	   c(c(x1)) -> a(b(c(a(x1))))
44.29/12.11	
44.29/12.11	Q is empty.
44.29/12.11	
44.29/12.11	----------------------------------------
44.29/12.11	
44.29/12.11	(3) DependencyPairsProof (EQUIVALENT)
44.29/12.11	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
44.29/12.11	----------------------------------------
44.29/12.11	
44.29/12.11	(4)
44.29/12.11	Obligation:
44.29/12.11	Q DP problem:
44.29/12.11	The TRS P consists of the following rules:
44.29/12.11	
44.29/12.11	   B(a(x1)) -> C(b(x1))
44.29/12.11	   B(a(x1)) -> B(x1)
44.29/12.11	   C(c(x1)) -> A(b(c(a(x1))))
44.29/12.11	   C(c(x1)) -> B(c(a(x1)))
44.29/12.11	   C(c(x1)) -> C(a(x1))
44.29/12.11	   C(c(x1)) -> A(x1)
44.29/12.11	
44.29/12.11	The TRS R consists of the following rules:
44.29/12.11	
44.29/12.11	   a(x1) -> x1
44.29/12.11	   b(a(x1)) -> c(b(x1))
44.29/12.11	   b(x1) -> x1
44.29/12.11	   c(c(x1)) -> a(b(c(a(x1))))
44.29/12.11	
44.29/12.11	Q is empty.
44.29/12.11	We have to consider all minimal (P,Q,R)-chains.
44.29/12.11	----------------------------------------
44.29/12.11	
44.29/12.11	(5) DependencyGraphProof (EQUIVALENT)
44.29/12.11	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
44.29/12.11	----------------------------------------
44.29/12.11	
44.29/12.11	(6)
44.29/12.11	Obligation:
44.29/12.11	Q DP problem:
44.29/12.11	The TRS P consists of the following rules:
44.29/12.11	
44.29/12.11	   C(c(x1)) -> B(c(a(x1)))
44.29/12.11	   B(a(x1)) -> C(b(x1))
44.29/12.11	   C(c(x1)) -> C(a(x1))
44.29/12.11	   B(a(x1)) -> B(x1)
44.29/12.11	
44.29/12.11	The TRS R consists of the following rules:
44.29/12.11	
44.29/12.11	   a(x1) -> x1
44.29/12.11	   b(a(x1)) -> c(b(x1))
44.29/12.11	   b(x1) -> x1
44.29/12.11	   c(c(x1)) -> a(b(c(a(x1))))
44.29/12.11	
44.29/12.11	Q is empty.
44.29/12.11	We have to consider all minimal (P,Q,R)-chains.
44.29/12.11	----------------------------------------
44.29/12.11	
44.29/12.11	(7) QDPOrderProof (EQUIVALENT)
44.29/12.11	We use the reduction pair processor [LPAR04,JAR06].
44.29/12.11	
44.29/12.11	
44.29/12.11	The following pairs can be oriented strictly and are deleted.
44.29/12.11	
44.29/12.11	   B(a(x1)) -> B(x1)
44.29/12.11	The remaining pairs can at least be oriented weakly.
44.29/12.11	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
44.29/12.11	
44.29/12.11	   <<<
44.29/12.11	 POL(C(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
44.29/12.11	>>>
44.29/12.11	
44.29/12.11	   <<<
44.29/12.11	 POL(c(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[1A, 1A, -I], [0A, 0A, -I], [0A, 0A, -I]] 	* 	x_1
44.29/12.11	>>>
44.29/12.11	
44.29/12.11	   <<<
44.29/12.11	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
44.29/12.11	>>>
44.29/12.11	
44.29/12.11	   <<<
44.29/12.11	 POL(a(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[0A, 0A, -I], [0A, 0A, -I], [0A, 0A, 1A]] 	* 	x_1
44.29/12.11	>>>
44.29/12.11	
44.29/12.11	   <<<
44.29/12.11	 POL(b(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[0A, 0A, 1A], [0A, 0A, 1A], [-I, 0A, 0A]] 	* 	x_1
44.29/12.11	>>>
44.29/12.11	
44.29/12.11	
44.29/12.11	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
44.29/12.11	
44.29/12.11	   a(x1) -> x1
44.29/12.11	   c(c(x1)) -> a(b(c(a(x1))))
44.29/12.11	   b(a(x1)) -> c(b(x1))
44.29/12.11	   b(x1) -> x1
44.29/12.11	
44.29/12.11	
44.29/12.11	----------------------------------------
44.29/12.11	
44.29/12.11	(8)
44.29/12.11	Obligation:
44.29/12.11	Q DP problem:
44.29/12.11	The TRS P consists of the following rules:
44.29/12.11	
44.29/12.11	   C(c(x1)) -> B(c(a(x1)))
44.29/12.11	   B(a(x1)) -> C(b(x1))
44.29/12.11	   C(c(x1)) -> C(a(x1))
44.29/12.11	
44.29/12.11	The TRS R consists of the following rules:
44.29/12.11	
44.29/12.11	   a(x1) -> x1
44.29/12.11	   b(a(x1)) -> c(b(x1))
44.29/12.11	   b(x1) -> x1
44.29/12.11	   c(c(x1)) -> a(b(c(a(x1))))
44.29/12.11	
44.29/12.11	Q is empty.
44.29/12.11	We have to consider all minimal (P,Q,R)-chains.
44.29/12.11	----------------------------------------
44.29/12.11	
44.29/12.11	(9) QDPOrderProof (EQUIVALENT)
44.29/12.11	We use the reduction pair processor [LPAR04,JAR06].
44.29/12.11	
44.29/12.11	
44.29/12.11	The following pairs can be oriented strictly and are deleted.
44.29/12.11	
44.29/12.11	   C(c(x1)) -> C(a(x1))
44.29/12.11	The remaining pairs can at least be oriented weakly.
44.29/12.11	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
44.29/12.11	
44.29/12.11	   <<<
44.29/12.11	 POL(C(x_1)) =  	[[-I]] 	 +  	[[0A, -I, -I]] 	* 	x_1
44.29/12.11	>>>
44.29/12.11	
44.29/12.11	   <<<
44.29/12.11	 POL(c(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[1A, -I, -I], [0A, -I, -I], [1A, -I, -I]] 	* 	x_1
44.29/12.11	>>>
44.29/12.11	
44.29/12.11	   <<<
44.29/12.11	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
44.29/12.11	>>>
44.29/12.11	
44.29/12.11	   <<<
44.29/12.11	 POL(a(x_1)) =  	[[-I], [0A], [0A]] 	 +  	[[0A, -I, -I], [0A, 0A, 0A], [1A, 1A, 1A]] 	* 	x_1
44.29/12.11	>>>
44.29/12.11	
44.29/12.11	   <<<
44.29/12.11	 POL(b(x_1)) =  	[[-I], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
44.29/12.11	>>>
44.29/12.11	
44.29/12.11	
44.29/12.11	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
44.29/12.11	
44.29/12.11	   a(x1) -> x1
44.29/12.11	   c(c(x1)) -> a(b(c(a(x1))))
44.29/12.11	   b(a(x1)) -> c(b(x1))
44.29/12.11	   b(x1) -> x1
44.29/12.11	
44.29/12.11	
44.29/12.11	----------------------------------------
44.29/12.11	
44.29/12.11	(10)
44.29/12.11	Obligation:
44.29/12.11	Q DP problem:
44.29/12.11	The TRS P consists of the following rules:
44.29/12.11	
44.29/12.11	   C(c(x1)) -> B(c(a(x1)))
44.29/12.11	   B(a(x1)) -> C(b(x1))
44.29/12.11	
44.29/12.11	The TRS R consists of the following rules:
44.29/12.11	
44.29/12.11	   a(x1) -> x1
44.29/12.11	   b(a(x1)) -> c(b(x1))
44.29/12.11	   b(x1) -> x1
44.29/12.11	   c(c(x1)) -> a(b(c(a(x1))))
44.29/12.11	
44.29/12.11	Q is empty.
44.29/12.11	We have to consider all minimal (P,Q,R)-chains.
44.29/12.11	----------------------------------------
44.29/12.11	
44.29/12.11	(11) QDPOrderProof (EQUIVALENT)
44.29/12.11	We use the reduction pair processor [LPAR04,JAR06].
44.29/12.11	
44.29/12.11	
44.29/12.11	The following pairs can be oriented strictly and are deleted.
44.29/12.11	
44.29/12.11	   B(a(x1)) -> C(b(x1))
44.29/12.11	The remaining pairs can at least be oriented weakly.
44.29/12.11	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
44.29/12.11	
44.29/12.11	   <<<
44.29/12.11	 POL(C(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
44.29/12.11	>>>
44.29/12.11	
44.29/12.11	   <<<
44.29/12.11	 POL(c(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, 0A, -I], [0A, 1A, -I], [0A, 1A, -I]] 	* 	x_1
44.29/12.11	>>>
44.29/12.11	
44.29/12.11	   <<<
44.29/12.11	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
44.29/12.11	>>>
44.29/12.11	
44.29/12.11	   <<<
44.29/12.11	 POL(a(x_1)) =  	[[-I], [-I], [1A]] 	 +  	[[0A, 0A, -I], [-I, 0A, -I], [1A, 1A, 1A]] 	* 	x_1
44.29/12.11	>>>
44.29/12.11	
44.29/12.11	   <<<
44.29/12.11	 POL(b(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [-I, -I, 0A]] 	* 	x_1
44.29/12.11	>>>
44.29/12.11	
44.29/12.11	
44.29/12.11	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
44.29/12.11	
44.29/12.11	   a(x1) -> x1
44.29/12.11	   c(c(x1)) -> a(b(c(a(x1))))
44.29/12.11	   b(a(x1)) -> c(b(x1))
44.29/12.11	   b(x1) -> x1
44.29/12.11	
44.29/12.11	
44.29/12.11	----------------------------------------
44.29/12.11	
44.29/12.11	(12)
44.29/12.11	Obligation:
44.29/12.11	Q DP problem:
44.29/12.11	The TRS P consists of the following rules:
44.29/12.11	
44.29/12.11	   C(c(x1)) -> B(c(a(x1)))
44.29/12.11	
44.29/12.11	The TRS R consists of the following rules:
44.29/12.11	
44.29/12.11	   a(x1) -> x1
44.29/12.11	   b(a(x1)) -> c(b(x1))
44.29/12.11	   b(x1) -> x1
44.29/12.11	   c(c(x1)) -> a(b(c(a(x1))))
44.29/12.11	
44.29/12.11	Q is empty.
44.29/12.11	We have to consider all minimal (P,Q,R)-chains.
44.29/12.11	----------------------------------------
44.29/12.11	
44.29/12.11	(13) DependencyGraphProof (EQUIVALENT)
44.29/12.11	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
44.29/12.11	----------------------------------------
44.29/12.11	
44.29/12.11	(14)
44.29/12.11	TRUE
44.29/12.15	EOF