45.61/12.51	YES
45.61/12.55	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
45.61/12.55	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
45.61/12.55	
45.61/12.55	
45.61/12.55	Termination w.r.t. Q of the given QTRS could be proven:
45.61/12.55	
45.61/12.55	(0) QTRS
45.61/12.55	(1) DependencyPairsProof [EQUIVALENT, 27 ms]
45.61/12.55	(2) QDP
45.61/12.55	(3) QDPOrderProof [EQUIVALENT, 207 ms]
45.61/12.55	(4) QDP
45.61/12.55	(5) QDPOrderProof [EQUIVALENT, 72 ms]
45.61/12.55	(6) QDP
45.61/12.55	(7) QDPOrderProof [EQUIVALENT, 8 ms]
45.61/12.55	(8) QDP
45.61/12.55	(9) QDPOrderProof [EQUIVALENT, 28 ms]
45.61/12.55	(10) QDP
45.61/12.55	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
45.61/12.55	(12) QDP
45.61/12.55	(13) QDPOrderProof [EQUIVALENT, 0 ms]
45.61/12.55	(14) QDP
45.61/12.55	(15) PisEmptyProof [EQUIVALENT, 0 ms]
45.61/12.55	(16) YES
45.61/12.55	
45.61/12.55	
45.61/12.55	----------------------------------------
45.61/12.55	
45.61/12.55	(0)
45.61/12.55	Obligation:
45.61/12.55	Q restricted rewrite system:
45.61/12.55	The TRS R consists of the following rules:
45.61/12.55	
45.61/12.55	   a(a(x1)) -> b(a(b(x1)))
45.61/12.55	   b(b(x1)) -> a(c(b(x1)))
45.61/12.55	   c(c(x1)) -> c(b(a(x1)))
45.61/12.55	   a(b(x1)) -> b(a(x1))
45.61/12.55	   b(c(x1)) -> c(x1)
45.61/12.55	
45.61/12.55	Q is empty.
45.61/12.55	
45.61/12.55	----------------------------------------
45.61/12.55	
45.61/12.55	(1) DependencyPairsProof (EQUIVALENT)
45.61/12.55	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
45.61/12.55	----------------------------------------
45.61/12.55	
45.61/12.55	(2)
45.61/12.55	Obligation:
45.61/12.55	Q DP problem:
45.61/12.55	The TRS P consists of the following rules:
45.61/12.55	
45.61/12.55	   A(a(x1)) -> B(a(b(x1)))
45.61/12.55	   A(a(x1)) -> A(b(x1))
45.61/12.55	   A(a(x1)) -> B(x1)
45.61/12.55	   B(b(x1)) -> A(c(b(x1)))
45.61/12.55	   B(b(x1)) -> C(b(x1))
45.61/12.55	   C(c(x1)) -> C(b(a(x1)))
45.61/12.55	   C(c(x1)) -> B(a(x1))
45.61/12.55	   C(c(x1)) -> A(x1)
45.61/12.55	   A(b(x1)) -> B(a(x1))
45.61/12.55	   A(b(x1)) -> A(x1)
45.61/12.55	
45.61/12.55	The TRS R consists of the following rules:
45.61/12.55	
45.61/12.55	   a(a(x1)) -> b(a(b(x1)))
45.61/12.55	   b(b(x1)) -> a(c(b(x1)))
45.61/12.55	   c(c(x1)) -> c(b(a(x1)))
45.61/12.55	   a(b(x1)) -> b(a(x1))
45.61/12.55	   b(c(x1)) -> c(x1)
45.61/12.55	
45.61/12.55	Q is empty.
45.61/12.55	We have to consider all minimal (P,Q,R)-chains.
45.61/12.55	----------------------------------------
45.61/12.55	
45.61/12.55	(3) QDPOrderProof (EQUIVALENT)
45.61/12.55	We use the reduction pair processor [LPAR04,JAR06].
45.61/12.55	
45.61/12.55	
45.61/12.55	The following pairs can be oriented strictly and are deleted.
45.61/12.55	
45.61/12.55	   C(c(x1)) -> C(b(a(x1)))
45.61/12.55	The remaining pairs can at least be oriented weakly.
45.61/12.55	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
45.61/12.55	
45.61/12.55	   <<<
45.61/12.55	 POL(A(x_1)) =  	[[1A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
45.61/12.55	>>>
45.61/12.55	
45.61/12.55	   <<<
45.61/12.55	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, -I, -I], [1A, 0A, 0A], [0A, -I, -I]] 	* 	x_1
45.61/12.55	>>>
45.61/12.55	
45.61/12.55	   <<<
45.61/12.55	 POL(B(x_1)) =  	[[1A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
45.61/12.55	>>>
45.61/12.55	
45.61/12.55	   <<<
45.61/12.55	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, -I, -I], [1A, 0A, 0A], [0A, -I, 0A]] 	* 	x_1
45.61/12.55	>>>
45.61/12.55	
45.61/12.55	   <<<
45.61/12.55	 POL(c(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[0A, -I, -I], [0A, 0A, 0A], [1A, 0A, 0A]] 	* 	x_1
45.61/12.55	>>>
45.61/12.55	
45.61/12.55	   <<<
45.61/12.55	 POL(C(x_1)) =  	[[0A]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
45.61/12.55	>>>
45.61/12.55	
45.61/12.55	
45.61/12.55	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
45.61/12.55	
45.61/12.55	   b(b(x1)) -> a(c(b(x1)))
45.61/12.55	   a(a(x1)) -> b(a(b(x1)))
45.61/12.55	   a(b(x1)) -> b(a(x1))
45.61/12.55	   b(c(x1)) -> c(x1)
45.61/12.55	   c(c(x1)) -> c(b(a(x1)))
45.61/12.55	
45.61/12.55	
45.61/12.55	----------------------------------------
45.61/12.55	
45.61/12.55	(4)
45.61/12.55	Obligation:
45.61/12.55	Q DP problem:
45.61/12.55	The TRS P consists of the following rules:
45.61/12.55	
45.61/12.55	   A(a(x1)) -> B(a(b(x1)))
45.61/12.55	   A(a(x1)) -> A(b(x1))
45.61/12.55	   A(a(x1)) -> B(x1)
45.61/12.55	   B(b(x1)) -> A(c(b(x1)))
45.61/12.55	   B(b(x1)) -> C(b(x1))
45.61/12.55	   C(c(x1)) -> B(a(x1))
45.61/12.55	   C(c(x1)) -> A(x1)
45.61/12.55	   A(b(x1)) -> B(a(x1))
45.61/12.55	   A(b(x1)) -> A(x1)
45.61/12.55	
45.61/12.55	The TRS R consists of the following rules:
45.61/12.55	
45.61/12.55	   a(a(x1)) -> b(a(b(x1)))
45.61/12.55	   b(b(x1)) -> a(c(b(x1)))
45.61/12.55	   c(c(x1)) -> c(b(a(x1)))
45.61/12.55	   a(b(x1)) -> b(a(x1))
45.61/12.55	   b(c(x1)) -> c(x1)
45.61/12.55	
45.61/12.55	Q is empty.
45.61/12.55	We have to consider all minimal (P,Q,R)-chains.
45.61/12.55	----------------------------------------
45.61/12.55	
45.61/12.55	(5) QDPOrderProof (EQUIVALENT)
45.61/12.55	We use the reduction pair processor [LPAR04,JAR06].
45.61/12.55	
45.61/12.55	
45.61/12.55	The following pairs can be oriented strictly and are deleted.
45.61/12.55	
45.61/12.55	   A(a(x1)) -> B(a(b(x1)))
45.61/12.55	   C(c(x1)) -> B(a(x1))
45.61/12.55	   A(b(x1)) -> B(a(x1))
45.61/12.55	The remaining pairs can at least be oriented weakly.
45.61/12.55	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
45.61/12.55	
45.61/12.55	   <<<
45.61/12.55	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
45.61/12.55	>>>
45.61/12.55	
45.61/12.55	   <<<
45.61/12.55	 POL(a(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[-I, -I, -I], [0A, 0A, 0A], [-I, -I, -I]] 	* 	x_1
45.61/12.55	>>>
45.61/12.55	
45.61/12.55	   <<<
45.61/12.55	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
45.61/12.55	>>>
45.61/12.55	
45.61/12.55	   <<<
45.61/12.55	 POL(b(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[0A, -I, 0A], [0A, 0A, 0A], [0A, -I, 0A]] 	* 	x_1
45.61/12.55	>>>
45.61/12.55	
45.61/12.55	   <<<
45.61/12.55	 POL(c(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[0A, -I, 0A], [0A, -I, 0A], [0A, 0A, 0A]] 	* 	x_1
45.61/12.55	>>>
45.61/12.55	
45.61/12.55	   <<<
45.61/12.55	 POL(C(x_1)) =  	[[-I]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
45.61/12.55	>>>
45.61/12.55	
45.61/12.55	
45.61/12.55	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
45.61/12.55	
45.61/12.55	   b(b(x1)) -> a(c(b(x1)))
45.61/12.55	   a(a(x1)) -> b(a(b(x1)))
45.61/12.55	   a(b(x1)) -> b(a(x1))
45.61/12.55	   b(c(x1)) -> c(x1)
45.61/12.55	   c(c(x1)) -> c(b(a(x1)))
45.61/12.55	
45.61/12.55	
45.61/12.55	----------------------------------------
45.61/12.55	
45.61/12.55	(6)
45.61/12.55	Obligation:
45.61/12.55	Q DP problem:
45.61/12.55	The TRS P consists of the following rules:
45.61/12.55	
45.61/12.55	   A(a(x1)) -> A(b(x1))
45.61/12.55	   A(a(x1)) -> B(x1)
45.61/12.55	   B(b(x1)) -> A(c(b(x1)))
45.61/12.55	   B(b(x1)) -> C(b(x1))
45.61/12.55	   C(c(x1)) -> A(x1)
45.61/12.55	   A(b(x1)) -> A(x1)
45.61/12.55	
45.61/12.55	The TRS R consists of the following rules:
45.61/12.55	
45.61/12.55	   a(a(x1)) -> b(a(b(x1)))
45.61/12.55	   b(b(x1)) -> a(c(b(x1)))
45.61/12.55	   c(c(x1)) -> c(b(a(x1)))
45.61/12.55	   a(b(x1)) -> b(a(x1))
45.61/12.55	   b(c(x1)) -> c(x1)
45.61/12.55	
45.61/12.55	Q is empty.
45.61/12.55	We have to consider all minimal (P,Q,R)-chains.
45.61/12.55	----------------------------------------
45.61/12.55	
45.61/12.55	(7) QDPOrderProof (EQUIVALENT)
45.61/12.55	We use the reduction pair processor [LPAR04,JAR06].
45.61/12.55	
45.61/12.55	
45.61/12.55	The following pairs can be oriented strictly and are deleted.
45.61/12.55	
45.61/12.55	   B(b(x1)) -> A(c(b(x1)))
45.61/12.55	The remaining pairs can at least be oriented weakly.
45.61/12.55	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
45.61/12.55	
45.61/12.55	   <<<
45.61/12.55	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
45.61/12.55	>>>
45.61/12.55	
45.61/12.55	   <<<
45.61/12.55	 POL(a(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[0A, 0A, 0A], [-I, -I, 0A], [0A, 0A, 0A]] 	* 	x_1
45.61/12.55	>>>
45.61/12.55	
45.61/12.55	   <<<
45.61/12.55	 POL(b(x_1)) =  	[[0A], [-I], [1A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
45.61/12.55	>>>
45.61/12.55	
45.61/12.55	   <<<
45.61/12.55	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
45.61/12.55	>>>
45.61/12.55	
45.61/12.55	   <<<
45.61/12.55	 POL(c(x_1)) =  	[[1A], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [-I, -I, -I], [-I, -I, -I]] 	* 	x_1
45.61/12.55	>>>
45.61/12.55	
45.61/12.55	   <<<
45.61/12.55	 POL(C(x_1)) =  	[[1A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
45.61/12.55	>>>
45.61/12.55	
45.61/12.55	
45.61/12.55	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
45.61/12.55	
45.61/12.55	   b(b(x1)) -> a(c(b(x1)))
45.61/12.55	   a(a(x1)) -> b(a(b(x1)))
45.61/12.55	   a(b(x1)) -> b(a(x1))
45.61/12.55	   b(c(x1)) -> c(x1)
45.61/12.55	   c(c(x1)) -> c(b(a(x1)))
45.61/12.55	
45.61/12.55	
45.61/12.55	----------------------------------------
45.61/12.55	
45.61/12.55	(8)
45.61/12.55	Obligation:
45.61/12.55	Q DP problem:
45.61/12.55	The TRS P consists of the following rules:
45.61/12.55	
45.61/12.55	   A(a(x1)) -> A(b(x1))
45.61/12.55	   A(a(x1)) -> B(x1)
45.61/12.55	   B(b(x1)) -> C(b(x1))
45.61/12.55	   C(c(x1)) -> A(x1)
45.61/12.55	   A(b(x1)) -> A(x1)
45.61/12.55	
45.61/12.55	The TRS R consists of the following rules:
45.61/12.55	
45.61/12.55	   a(a(x1)) -> b(a(b(x1)))
45.61/12.55	   b(b(x1)) -> a(c(b(x1)))
45.61/12.55	   c(c(x1)) -> c(b(a(x1)))
45.61/12.55	   a(b(x1)) -> b(a(x1))
45.61/12.55	   b(c(x1)) -> c(x1)
45.61/12.55	
45.61/12.55	Q is empty.
45.61/12.55	We have to consider all minimal (P,Q,R)-chains.
45.61/12.55	----------------------------------------
45.61/12.55	
45.61/12.55	(9) QDPOrderProof (EQUIVALENT)
45.61/12.55	We use the reduction pair processor [LPAR04,JAR06].
45.61/12.55	
45.61/12.55	
45.61/12.55	The following pairs can be oriented strictly and are deleted.
45.61/12.55	
45.61/12.55	   C(c(x1)) -> A(x1)
45.61/12.55	The remaining pairs can at least be oriented weakly.
45.61/12.55	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
45.61/12.55	
45.61/12.55	   <<<
45.61/12.55	 POL(A(x_1)) =  	[[1A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
45.61/12.55	>>>
45.61/12.55	
45.61/12.55	   <<<
45.61/12.55	 POL(a(x_1)) =  	[[1A], [0A], [-I]] 	 +  	[[1A, 0A, 1A], [0A, -I, 0A], [0A, -I, 0A]] 	* 	x_1
45.61/12.55	>>>
45.61/12.55	
45.61/12.55	   <<<
45.61/12.55	 POL(b(x_1)) =  	[[1A], [0A], [-I]] 	 +  	[[0A, 0A, 1A], [-I, 0A, 0A], [-I, 0A, 0A]] 	* 	x_1
45.61/12.55	>>>
45.61/12.55	
45.61/12.55	   <<<
45.61/12.55	 POL(B(x_1)) =  	[[1A]] 	 +  	[[0A, 0A, 1A]] 	* 	x_1
45.61/12.55	>>>
45.61/12.55	
45.61/12.55	   <<<
45.61/12.55	 POL(C(x_1)) =  	[[-I]] 	 +  	[[0A, 1A, 0A]] 	* 	x_1
45.61/12.55	>>>
45.61/12.55	
45.61/12.55	   <<<
45.61/12.55	 POL(c(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[-I, 0A, 0A], [0A, 1A, 0A], [-I, 0A, 0A]] 	* 	x_1
45.61/12.55	>>>
45.61/12.55	
45.61/12.55	
45.61/12.55	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
45.61/12.55	
45.61/12.55	   b(b(x1)) -> a(c(b(x1)))
45.61/12.55	   a(a(x1)) -> b(a(b(x1)))
45.61/12.55	   a(b(x1)) -> b(a(x1))
45.61/12.55	   b(c(x1)) -> c(x1)
45.61/12.55	   c(c(x1)) -> c(b(a(x1)))
45.61/12.55	
45.61/12.55	
45.61/12.55	----------------------------------------
45.61/12.55	
45.61/12.55	(10)
45.61/12.55	Obligation:
45.61/12.55	Q DP problem:
45.61/12.55	The TRS P consists of the following rules:
45.61/12.55	
45.61/12.55	   A(a(x1)) -> A(b(x1))
45.61/12.55	   A(a(x1)) -> B(x1)
45.61/12.55	   B(b(x1)) -> C(b(x1))
45.61/12.55	   A(b(x1)) -> A(x1)
45.61/12.55	
45.61/12.55	The TRS R consists of the following rules:
45.61/12.55	
45.61/12.55	   a(a(x1)) -> b(a(b(x1)))
45.61/12.55	   b(b(x1)) -> a(c(b(x1)))
45.61/12.55	   c(c(x1)) -> c(b(a(x1)))
45.61/12.55	   a(b(x1)) -> b(a(x1))
45.61/12.55	   b(c(x1)) -> c(x1)
45.61/12.55	
45.61/12.55	Q is empty.
45.61/12.55	We have to consider all minimal (P,Q,R)-chains.
45.61/12.55	----------------------------------------
45.61/12.55	
45.61/12.55	(11) DependencyGraphProof (EQUIVALENT)
45.61/12.55	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
45.61/12.55	----------------------------------------
45.61/12.55	
45.61/12.55	(12)
45.61/12.55	Obligation:
45.61/12.55	Q DP problem:
45.61/12.55	The TRS P consists of the following rules:
45.61/12.55	
45.61/12.55	   A(b(x1)) -> A(x1)
45.61/12.55	   A(a(x1)) -> A(b(x1))
45.61/12.55	
45.61/12.55	The TRS R consists of the following rules:
45.61/12.55	
45.61/12.55	   a(a(x1)) -> b(a(b(x1)))
45.61/12.55	   b(b(x1)) -> a(c(b(x1)))
45.61/12.55	   c(c(x1)) -> c(b(a(x1)))
45.61/12.55	   a(b(x1)) -> b(a(x1))
45.61/12.55	   b(c(x1)) -> c(x1)
45.61/12.55	
45.61/12.55	Q is empty.
45.61/12.55	We have to consider all minimal (P,Q,R)-chains.
45.61/12.55	----------------------------------------
45.61/12.55	
45.61/12.55	(13) QDPOrderProof (EQUIVALENT)
45.61/12.55	We use the reduction pair processor [LPAR04,JAR06].
45.61/12.55	
45.61/12.55	
45.61/12.55	The following pairs can be oriented strictly and are deleted.
45.61/12.55	
45.61/12.55	   A(b(x1)) -> A(x1)
45.61/12.55	   A(a(x1)) -> A(b(x1))
45.61/12.55	The remaining pairs can at least be oriented weakly.
45.61/12.55	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
45.61/12.55	
45.61/12.55	POL( A_1(x_1) ) = 2x_1
45.61/12.55	POL( b_1(x_1) ) = x_1 + 1
45.61/12.55	POL( a_1(x_1) ) = 2x_1 + 2
45.61/12.55	POL( c_1(x_1) ) = max{0, -2}
45.61/12.55	
45.61/12.55	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
45.61/12.55	
45.61/12.55	   b(b(x1)) -> a(c(b(x1)))
45.61/12.55	   a(a(x1)) -> b(a(b(x1)))
45.61/12.55	   a(b(x1)) -> b(a(x1))
45.61/12.55	   b(c(x1)) -> c(x1)
45.61/12.55	   c(c(x1)) -> c(b(a(x1)))
45.61/12.55	
45.61/12.55	
45.61/12.55	----------------------------------------
45.61/12.55	
45.61/12.55	(14)
45.61/12.55	Obligation:
45.61/12.55	Q DP problem:
45.61/12.55	P is empty.
45.61/12.55	The TRS R consists of the following rules:
45.61/12.55	
45.61/12.55	   a(a(x1)) -> b(a(b(x1)))
45.61/12.55	   b(b(x1)) -> a(c(b(x1)))
45.61/12.55	   c(c(x1)) -> c(b(a(x1)))
45.61/12.55	   a(b(x1)) -> b(a(x1))
45.61/12.55	   b(c(x1)) -> c(x1)
45.61/12.55	
45.61/12.55	Q is empty.
45.61/12.55	We have to consider all minimal (P,Q,R)-chains.
45.61/12.55	----------------------------------------
45.61/12.55	
45.61/12.55	(15) PisEmptyProof (EQUIVALENT)
45.61/12.55	The TRS P is empty. Hence, there is no (P,Q,R) chain.
45.61/12.55	----------------------------------------
45.61/12.55	
45.61/12.55	(16)
45.61/12.55	YES
45.99/12.68	EOF