48.37/13.28	YES
48.37/13.30	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
48.37/13.30	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
48.37/13.30	
48.37/13.30	
48.37/13.30	Termination w.r.t. Q of the given QTRS could be proven:
48.37/13.30	
48.37/13.30	(0) QTRS
48.37/13.30	(1) QTRS Reverse [EQUIVALENT, 0 ms]
48.37/13.30	(2) QTRS
48.37/13.30	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
48.37/13.30	(4) QDP
48.37/13.30	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
48.37/13.30	(6) QDP
48.37/13.30	(7) QDPOrderProof [EQUIVALENT, 147 ms]
48.37/13.30	(8) QDP
48.37/13.30	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
48.37/13.30	(10) QDP
48.37/13.30	(11) QDPOrderProof [EQUIVALENT, 40 ms]
48.37/13.30	(12) QDP
48.37/13.30	(13) QDPOrderProof [EQUIVALENT, 27 ms]
48.37/13.30	(14) QDP
48.37/13.30	(15) PisEmptyProof [EQUIVALENT, 0 ms]
48.37/13.30	(16) YES
48.37/13.30	
48.37/13.30	
48.37/13.30	----------------------------------------
48.37/13.30	
48.37/13.30	(0)
48.37/13.30	Obligation:
48.37/13.30	Q restricted rewrite system:
48.37/13.30	The TRS R consists of the following rules:
48.37/13.30	
48.37/13.30	   a(x1) -> x1
48.37/13.30	   a(b(x1)) -> b(b(a(c(x1))))
48.37/13.30	   b(x1) -> x1
48.37/13.30	   c(c(x1)) -> b(a(x1))
48.37/13.30	
48.37/13.30	Q is empty.
48.37/13.30	
48.37/13.30	----------------------------------------
48.37/13.30	
48.37/13.30	(1) QTRS Reverse (EQUIVALENT)
48.37/13.30	We applied the QTRS Reverse Processor [REVERSE].
48.37/13.30	----------------------------------------
48.37/13.30	
48.37/13.30	(2)
48.37/13.30	Obligation:
48.37/13.30	Q restricted rewrite system:
48.37/13.30	The TRS R consists of the following rules:
48.37/13.30	
48.37/13.30	   a(x1) -> x1
48.37/13.30	   b(a(x1)) -> c(a(b(b(x1))))
48.37/13.30	   b(x1) -> x1
48.37/13.30	   c(c(x1)) -> a(b(x1))
48.37/13.30	
48.37/13.30	Q is empty.
48.37/13.30	
48.37/13.30	----------------------------------------
48.37/13.30	
48.37/13.30	(3) DependencyPairsProof (EQUIVALENT)
48.37/13.30	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
48.37/13.30	----------------------------------------
48.37/13.30	
48.37/13.30	(4)
48.37/13.30	Obligation:
48.37/13.30	Q DP problem:
48.37/13.30	The TRS P consists of the following rules:
48.37/13.30	
48.37/13.30	   B(a(x1)) -> C(a(b(b(x1))))
48.37/13.30	   B(a(x1)) -> A(b(b(x1)))
48.37/13.30	   B(a(x1)) -> B(b(x1))
48.37/13.30	   B(a(x1)) -> B(x1)
48.37/13.30	   C(c(x1)) -> A(b(x1))
48.37/13.30	   C(c(x1)) -> B(x1)
48.37/13.30	
48.37/13.30	The TRS R consists of the following rules:
48.37/13.30	
48.37/13.30	   a(x1) -> x1
48.37/13.30	   b(a(x1)) -> c(a(b(b(x1))))
48.37/13.30	   b(x1) -> x1
48.37/13.30	   c(c(x1)) -> a(b(x1))
48.37/13.30	
48.37/13.30	Q is empty.
48.37/13.30	We have to consider all minimal (P,Q,R)-chains.
48.37/13.30	----------------------------------------
48.37/13.30	
48.37/13.30	(5) DependencyGraphProof (EQUIVALENT)
48.37/13.30	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
48.37/13.30	----------------------------------------
48.37/13.30	
48.37/13.30	(6)
48.37/13.30	Obligation:
48.37/13.30	Q DP problem:
48.37/13.30	The TRS P consists of the following rules:
48.37/13.30	
48.37/13.30	   C(c(x1)) -> B(x1)
48.37/13.30	   B(a(x1)) -> C(a(b(b(x1))))
48.37/13.30	   B(a(x1)) -> B(b(x1))
48.37/13.30	   B(a(x1)) -> B(x1)
48.37/13.30	
48.37/13.30	The TRS R consists of the following rules:
48.37/13.30	
48.37/13.30	   a(x1) -> x1
48.37/13.30	   b(a(x1)) -> c(a(b(b(x1))))
48.37/13.30	   b(x1) -> x1
48.37/13.30	   c(c(x1)) -> a(b(x1))
48.37/13.30	
48.37/13.30	Q is empty.
48.37/13.30	We have to consider all minimal (P,Q,R)-chains.
48.37/13.30	----------------------------------------
48.37/13.30	
48.37/13.30	(7) QDPOrderProof (EQUIVALENT)
48.37/13.30	We use the reduction pair processor [LPAR04,JAR06].
48.37/13.30	
48.37/13.30	
48.37/13.30	The following pairs can be oriented strictly and are deleted.
48.37/13.30	
48.37/13.30	   C(c(x1)) -> B(x1)
48.37/13.30	The remaining pairs can at least be oriented weakly.
48.37/13.30	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
48.37/13.30	
48.37/13.30	   <<<
48.37/13.30	 POL(C(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
48.37/13.30	>>>
48.37/13.30	
48.37/13.30	   <<<
48.37/13.30	 POL(c(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[1A, 1A, 1A], [0A, 0A, 0A], [0A, -I, 0A]] 	* 	x_1
48.37/13.30	>>>
48.37/13.30	
48.37/13.30	   <<<
48.37/13.30	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
48.37/13.30	>>>
48.37/13.30	
48.37/13.30	   <<<
48.37/13.30	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, -I], [0A, 0A, 0A], [0A, 0A, 1A]] 	* 	x_1
48.37/13.30	>>>
48.37/13.30	
48.37/13.30	   <<<
48.37/13.30	 POL(b(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, 0A, 1A], [0A, 0A, 0A], [-I, -I, 0A]] 	* 	x_1
48.37/13.30	>>>
48.37/13.30	
48.37/13.30	
48.37/13.30	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
48.37/13.30	
48.37/13.30	   b(a(x1)) -> c(a(b(b(x1))))
48.37/13.30	   b(x1) -> x1
48.37/13.30	   a(x1) -> x1
48.37/13.30	   c(c(x1)) -> a(b(x1))
48.37/13.30	
48.37/13.30	
48.37/13.30	----------------------------------------
48.37/13.30	
48.37/13.30	(8)
48.37/13.30	Obligation:
48.37/13.30	Q DP problem:
48.37/13.30	The TRS P consists of the following rules:
48.37/13.30	
48.37/13.30	   B(a(x1)) -> C(a(b(b(x1))))
48.37/13.30	   B(a(x1)) -> B(b(x1))
48.37/13.30	   B(a(x1)) -> B(x1)
48.37/13.30	
48.37/13.30	The TRS R consists of the following rules:
48.37/13.30	
48.37/13.30	   a(x1) -> x1
48.37/13.30	   b(a(x1)) -> c(a(b(b(x1))))
48.37/13.30	   b(x1) -> x1
48.37/13.30	   c(c(x1)) -> a(b(x1))
48.37/13.30	
48.37/13.30	Q is empty.
48.37/13.30	We have to consider all minimal (P,Q,R)-chains.
48.37/13.30	----------------------------------------
48.37/13.30	
48.37/13.30	(9) DependencyGraphProof (EQUIVALENT)
48.37/13.30	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
48.37/13.30	----------------------------------------
48.37/13.30	
48.37/13.30	(10)
48.37/13.30	Obligation:
48.37/13.30	Q DP problem:
48.37/13.30	The TRS P consists of the following rules:
48.37/13.30	
48.37/13.30	   B(a(x1)) -> B(x1)
48.37/13.30	   B(a(x1)) -> B(b(x1))
48.37/13.30	
48.37/13.30	The TRS R consists of the following rules:
48.37/13.30	
48.37/13.30	   a(x1) -> x1
48.37/13.30	   b(a(x1)) -> c(a(b(b(x1))))
48.37/13.30	   b(x1) -> x1
48.37/13.30	   c(c(x1)) -> a(b(x1))
48.37/13.30	
48.37/13.30	Q is empty.
48.37/13.30	We have to consider all minimal (P,Q,R)-chains.
48.37/13.30	----------------------------------------
48.37/13.30	
48.37/13.30	(11) QDPOrderProof (EQUIVALENT)
48.37/13.30	We use the reduction pair processor [LPAR04,JAR06].
48.37/13.30	
48.37/13.30	
48.37/13.30	The following pairs can be oriented strictly and are deleted.
48.37/13.30	
48.37/13.30	   B(a(x1)) -> B(x1)
48.37/13.30	The remaining pairs can at least be oriented weakly.
48.37/13.30	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
48.37/13.30	
48.37/13.30	   <<<
48.37/13.30	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, -I, -I]] 	* 	x_1
48.37/13.30	>>>
48.37/13.30	
48.37/13.30	   <<<
48.37/13.30	 POL(a(x_1)) =  	[[1A], [0A], [-I]] 	 +  	[[1A, 1A, 0A], [0A, 0A, 0A], [0A, 0A, 1A]] 	* 	x_1
48.37/13.30	>>>
48.37/13.30	
48.37/13.30	   <<<
48.37/13.30	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
48.37/13.30	>>>
48.37/13.30	
48.37/13.30	   <<<
48.37/13.30	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 1A, 0A], [0A, -I, 0A], [0A, 1A, 0A]] 	* 	x_1
48.37/13.30	>>>
48.37/13.30	
48.37/13.30	
48.37/13.30	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
48.37/13.30	
48.37/13.30	   b(a(x1)) -> c(a(b(b(x1))))
48.37/13.30	   b(x1) -> x1
48.37/13.30	   c(c(x1)) -> a(b(x1))
48.37/13.30	   a(x1) -> x1
48.37/13.30	
48.37/13.30	
48.37/13.30	----------------------------------------
48.37/13.30	
48.37/13.30	(12)
48.37/13.30	Obligation:
48.37/13.30	Q DP problem:
48.37/13.30	The TRS P consists of the following rules:
48.37/13.30	
48.37/13.30	   B(a(x1)) -> B(b(x1))
48.37/13.30	
48.37/13.30	The TRS R consists of the following rules:
48.37/13.30	
48.37/13.30	   a(x1) -> x1
48.37/13.30	   b(a(x1)) -> c(a(b(b(x1))))
48.37/13.30	   b(x1) -> x1
48.37/13.30	   c(c(x1)) -> a(b(x1))
48.37/13.30	
48.37/13.30	Q is empty.
48.37/13.30	We have to consider all minimal (P,Q,R)-chains.
48.37/13.30	----------------------------------------
48.37/13.30	
48.37/13.30	(13) QDPOrderProof (EQUIVALENT)
48.37/13.30	We use the reduction pair processor [LPAR04,JAR06].
48.37/13.30	
48.37/13.30	
48.37/13.30	The following pairs can be oriented strictly and are deleted.
48.37/13.30	
48.37/13.30	   B(a(x1)) -> B(b(x1))
48.37/13.30	The remaining pairs can at least be oriented weakly.
48.37/13.30	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
48.37/13.30	
48.37/13.30	   <<<
48.37/13.30	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
48.37/13.30	>>>
48.37/13.30	
48.37/13.30	   <<<
48.37/13.30	 POL(a(x_1)) =  	[[0A], [1A], [-I]] 	 +  	[[0A, 0A, 0A], [1A, 1A, 1A], [-I, -I, 0A]] 	* 	x_1
48.37/13.30	>>>
48.37/13.30	
48.37/13.30	   <<<
48.37/13.30	 POL(b(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, 0A, -I], [0A, 0A, -I], [-I, -I, 0A]] 	* 	x_1
48.37/13.30	>>>
48.37/13.30	
48.37/13.30	   <<<
48.37/13.30	 POL(c(x_1)) =  	[[1A], [-I], [0A]] 	 +  	[[-I, 0A, 0A], [1A, 0A, -I], [-I, -I, 0A]] 	* 	x_1
48.37/13.30	>>>
48.37/13.30	
48.37/13.30	
48.37/13.30	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
48.37/13.30	
48.37/13.30	   b(a(x1)) -> c(a(b(b(x1))))
48.37/13.30	   b(x1) -> x1
48.37/13.30	   c(c(x1)) -> a(b(x1))
48.37/13.30	   a(x1) -> x1
48.37/13.30	
48.37/13.30	
48.37/13.30	----------------------------------------
48.37/13.30	
48.37/13.30	(14)
48.37/13.30	Obligation:
48.37/13.30	Q DP problem:
48.37/13.30	P is empty.
48.37/13.30	The TRS R consists of the following rules:
48.37/13.30	
48.37/13.30	   a(x1) -> x1
48.37/13.30	   b(a(x1)) -> c(a(b(b(x1))))
48.37/13.30	   b(x1) -> x1
48.37/13.30	   c(c(x1)) -> a(b(x1))
48.37/13.30	
48.37/13.30	Q is empty.
48.37/13.30	We have to consider all minimal (P,Q,R)-chains.
48.37/13.30	----------------------------------------
48.37/13.30	
48.37/13.30	(15) PisEmptyProof (EQUIVALENT)
48.37/13.30	The TRS P is empty. Hence, there is no (P,Q,R) chain.
48.37/13.30	----------------------------------------
48.37/13.30	
48.37/13.30	(16)
48.37/13.30	YES
48.74/13.40	EOF