27.19/7.82	YES
29.34/8.98	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
29.34/8.98	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
29.34/8.98	
29.34/8.98	
29.34/8.98	Termination w.r.t. Q of the given QTRS could be proven:
29.34/8.98	
29.34/8.98	(0) QTRS
29.34/8.98	(1) DependencyPairsProof [EQUIVALENT, 26 ms]
29.34/8.98	(2) QDP
29.34/8.98	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
29.34/8.98	(4) QDP
29.34/8.98	(5) QDPOrderProof [EQUIVALENT, 143 ms]
29.34/8.98	(6) QDP
29.34/8.98	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
29.34/8.98	(8) QDP
29.34/8.98	(9) QDPOrderProof [EQUIVALENT, 23 ms]
29.34/8.98	(10) QDP
29.34/8.98	(11) PisEmptyProof [EQUIVALENT, 0 ms]
29.34/8.98	(12) YES
29.34/8.98	
29.34/8.98	
29.34/8.98	----------------------------------------
29.34/8.98	
29.34/8.98	(0)
29.34/8.98	Obligation:
29.34/8.98	Q restricted rewrite system:
29.34/8.98	The TRS R consists of the following rules:
29.34/8.98	
29.34/8.98	   a(a(x1)) -> x1
29.34/8.98	   b(b(x1)) -> a(b(c(x1)))
29.34/8.98	   c(c(x1)) -> b(b(b(x1)))
29.34/8.98	
29.34/8.98	Q is empty.
29.34/8.98	
29.34/8.98	----------------------------------------
29.34/8.98	
29.34/8.98	(1) DependencyPairsProof (EQUIVALENT)
29.34/8.98	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
29.34/8.98	----------------------------------------
29.34/8.98	
29.34/8.98	(2)
29.34/8.98	Obligation:
29.34/8.98	Q DP problem:
29.34/8.98	The TRS P consists of the following rules:
29.34/8.98	
29.34/8.98	   B(b(x1)) -> A(b(c(x1)))
29.34/8.98	   B(b(x1)) -> B(c(x1))
29.34/8.98	   B(b(x1)) -> C(x1)
29.34/8.98	   C(c(x1)) -> B(b(b(x1)))
29.34/8.98	   C(c(x1)) -> B(b(x1))
29.34/8.98	   C(c(x1)) -> B(x1)
29.34/8.98	
29.34/8.98	The TRS R consists of the following rules:
29.34/8.98	
29.34/8.98	   a(a(x1)) -> x1
29.34/8.98	   b(b(x1)) -> a(b(c(x1)))
29.34/8.98	   c(c(x1)) -> b(b(b(x1)))
29.34/8.98	
29.34/8.98	Q is empty.
29.34/8.98	We have to consider all minimal (P,Q,R)-chains.
29.34/8.98	----------------------------------------
29.34/8.98	
29.34/8.98	(3) DependencyGraphProof (EQUIVALENT)
29.34/8.98	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
29.34/8.98	----------------------------------------
29.34/8.98	
29.34/8.98	(4)
29.34/8.98	Obligation:
29.34/8.98	Q DP problem:
29.34/8.98	The TRS P consists of the following rules:
29.34/8.98	
29.34/8.98	   B(b(x1)) -> C(x1)
29.34/8.98	   C(c(x1)) -> B(b(b(x1)))
29.34/8.98	   B(b(x1)) -> B(c(x1))
29.34/8.98	   C(c(x1)) -> B(b(x1))
29.34/8.98	   C(c(x1)) -> B(x1)
29.34/8.98	
29.34/8.98	The TRS R consists of the following rules:
29.34/8.98	
29.34/8.98	   a(a(x1)) -> x1
29.34/8.98	   b(b(x1)) -> a(b(c(x1)))
29.34/8.98	   c(c(x1)) -> b(b(b(x1)))
29.34/8.98	
29.34/8.98	Q is empty.
29.34/8.98	We have to consider all minimal (P,Q,R)-chains.
29.34/8.98	----------------------------------------
29.34/8.98	
29.34/8.98	(5) QDPOrderProof (EQUIVALENT)
29.34/8.98	We use the reduction pair processor [LPAR04,JAR06].
29.34/8.98	
29.34/8.98	
29.34/8.98	The following pairs can be oriented strictly and are deleted.
29.34/8.98	
29.34/8.98	   B(b(x1)) -> C(x1)
29.34/8.98	The remaining pairs can at least be oriented weakly.
29.34/8.98	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
29.34/8.98	
29.34/8.98	   <<<
29.34/8.98	 POL(B(x_1)) =  	[[1A]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
29.34/8.98	>>>
29.34/8.98	
29.34/8.98	   <<<
29.34/8.98	 POL(b(x_1)) =  	[[1A], [-I], [0A]] 	 +  	[[0A, 1A, -I], [-I, -I, -I], [1A, 0A, -I]] 	* 	x_1
29.34/8.98	>>>
29.34/8.98	
29.34/8.98	   <<<
29.34/8.98	 POL(C(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
29.34/8.98	>>>
29.34/8.98	
29.34/8.98	   <<<
29.34/8.98	 POL(c(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[-I, 0A, -I], [0A, 1A, -I], [0A, 1A, 0A]] 	* 	x_1
29.34/8.98	>>>
29.34/8.98	
29.34/8.98	   <<<
29.34/8.98	 POL(a(x_1)) =  	[[1A], [-I], [-I]] 	 +  	[[-I, 0A, 0A], [-I, 0A, -I], [0A, 0A, 0A]] 	* 	x_1
29.34/8.98	>>>
29.34/8.98	
29.34/8.98	
29.34/8.98	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
29.34/8.98	
29.34/8.98	   b(b(x1)) -> a(b(c(x1)))
29.34/8.98	   c(c(x1)) -> b(b(b(x1)))
29.34/8.98	   a(a(x1)) -> x1
29.34/8.98	
29.34/8.98	
29.34/8.98	----------------------------------------
29.34/8.98	
29.34/8.98	(6)
29.34/8.98	Obligation:
29.34/8.98	Q DP problem:
29.34/8.98	The TRS P consists of the following rules:
29.34/8.98	
29.34/8.98	   C(c(x1)) -> B(b(b(x1)))
29.34/8.98	   B(b(x1)) -> B(c(x1))
29.34/8.98	   C(c(x1)) -> B(b(x1))
29.34/8.98	   C(c(x1)) -> B(x1)
29.34/8.98	
29.34/8.98	The TRS R consists of the following rules:
29.34/8.98	
29.34/8.98	   a(a(x1)) -> x1
29.34/8.98	   b(b(x1)) -> a(b(c(x1)))
29.34/8.98	   c(c(x1)) -> b(b(b(x1)))
29.34/8.98	
29.34/8.98	Q is empty.
29.34/8.98	We have to consider all minimal (P,Q,R)-chains.
29.34/8.98	----------------------------------------
29.34/8.98	
29.34/8.98	(7) DependencyGraphProof (EQUIVALENT)
29.34/8.98	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
29.34/8.98	----------------------------------------
29.34/8.98	
29.34/8.98	(8)
29.34/8.98	Obligation:
29.34/8.98	Q DP problem:
29.34/8.98	The TRS P consists of the following rules:
29.34/8.98	
29.34/8.98	   B(b(x1)) -> B(c(x1))
29.34/8.98	
29.34/8.98	The TRS R consists of the following rules:
29.34/8.98	
29.34/8.98	   a(a(x1)) -> x1
29.34/8.98	   b(b(x1)) -> a(b(c(x1)))
29.34/8.98	   c(c(x1)) -> b(b(b(x1)))
29.34/8.98	
29.34/8.98	Q is empty.
29.34/8.98	We have to consider all minimal (P,Q,R)-chains.
29.34/8.98	----------------------------------------
29.34/8.98	
29.34/8.98	(9) QDPOrderProof (EQUIVALENT)
29.34/8.98	We use the reduction pair processor [LPAR04,JAR06].
29.34/8.98	
29.34/8.98	
29.34/8.98	The following pairs can be oriented strictly and are deleted.
29.34/8.98	
29.34/8.98	   B(b(x1)) -> B(c(x1))
29.34/8.98	The remaining pairs can at least be oriented weakly.
29.34/8.98	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
29.34/8.98	
29.34/8.98	   <<<
29.34/8.98	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, -I, -I]] 	* 	x_1
29.34/8.98	>>>
29.34/8.98	
29.34/8.98	   <<<
29.34/8.98	 POL(b(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[0A, 1A, -I], [-I, -I, -I], [1A, 0A, -I]] 	* 	x_1
29.34/8.98	>>>
29.34/8.98	
29.34/8.98	   <<<
29.34/8.98	 POL(c(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[-I, 0A, -I], [0A, 1A, -I], [0A, 1A, 0A]] 	* 	x_1
29.34/8.98	>>>
29.34/8.98	
29.34/8.98	   <<<
29.34/8.98	 POL(a(x_1)) =  	[[1A], [-I], [-I]] 	 +  	[[-I, 0A, 0A], [-I, 0A, -I], [0A, 0A, 0A]] 	* 	x_1
29.34/8.98	>>>
29.34/8.98	
29.34/8.98	
29.34/8.98	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
29.34/8.98	
29.34/8.98	   c(c(x1)) -> b(b(b(x1)))
29.34/8.98	   b(b(x1)) -> a(b(c(x1)))
29.34/8.98	   a(a(x1)) -> x1
29.34/8.98	
29.34/8.98	
29.34/8.98	----------------------------------------
29.34/8.98	
29.34/8.98	(10)
29.34/8.98	Obligation:
29.34/8.98	Q DP problem:
29.34/8.98	P is empty.
29.34/8.98	The TRS R consists of the following rules:
29.34/8.98	
29.34/8.98	   a(a(x1)) -> x1
29.34/8.98	   b(b(x1)) -> a(b(c(x1)))
29.34/8.98	   c(c(x1)) -> b(b(b(x1)))
29.34/8.98	
29.34/8.98	Q is empty.
29.34/8.98	We have to consider all minimal (P,Q,R)-chains.
29.34/8.98	----------------------------------------
29.34/8.98	
29.34/8.98	(11) PisEmptyProof (EQUIVALENT)
29.34/8.98	The TRS P is empty. Hence, there is no (P,Q,R) chain.
29.34/8.98	----------------------------------------
29.34/8.98	
29.34/8.98	(12)
29.34/8.98	YES
29.34/9.80	EOF