28.00/8.08	YES
30.05/9.65	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
30.05/9.65	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
30.05/9.65	
30.05/9.65	
30.05/9.65	Termination w.r.t. Q of the given QTRS could be proven:
30.05/9.65	
30.05/9.65	(0) QTRS
30.05/9.65	(1) QTRS Reverse [EQUIVALENT, 0 ms]
30.05/9.65	(2) QTRS
30.05/9.65	(3) DependencyPairsProof [EQUIVALENT, 26 ms]
30.05/9.65	(4) QDP
30.05/9.65	(5) QDPOrderProof [EQUIVALENT, 72 ms]
30.05/9.65	(6) QDP
30.05/9.65	(7) QDPOrderProof [EQUIVALENT, 0 ms]
30.05/9.65	(8) QDP
30.05/9.65	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
30.05/9.65	(10) TRUE
30.05/9.65	
30.05/9.65	
30.05/9.65	----------------------------------------
30.05/9.65	
30.05/9.65	(0)
30.05/9.65	Obligation:
30.05/9.65	Q restricted rewrite system:
30.05/9.65	The TRS R consists of the following rules:
30.05/9.65	
30.05/9.65	   b(a(b(x1))) -> a(b(a(x1)))
30.05/9.65	   b(b(a(x1))) -> b(b(b(x1)))
30.05/9.65	
30.05/9.65	Q is empty.
30.05/9.65	
30.05/9.65	----------------------------------------
30.05/9.65	
30.05/9.65	(1) QTRS Reverse (EQUIVALENT)
30.05/9.65	We applied the QTRS Reverse Processor [REVERSE].
30.05/9.65	----------------------------------------
30.05/9.65	
30.05/9.65	(2)
30.05/9.65	Obligation:
30.05/9.65	Q restricted rewrite system:
30.05/9.65	The TRS R consists of the following rules:
30.05/9.65	
30.05/9.65	   b(a(b(x1))) -> a(b(a(x1)))
30.05/9.65	   a(b(b(x1))) -> b(b(b(x1)))
30.05/9.65	
30.05/9.65	Q is empty.
30.05/9.65	
30.05/9.65	----------------------------------------
30.05/9.65	
30.05/9.65	(3) DependencyPairsProof (EQUIVALENT)
30.05/9.65	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
30.05/9.65	----------------------------------------
30.05/9.65	
30.05/9.65	(4)
30.05/9.65	Obligation:
30.05/9.65	Q DP problem:
30.05/9.65	The TRS P consists of the following rules:
30.05/9.65	
30.05/9.65	   B(a(b(x1))) -> A(b(a(x1)))
30.05/9.65	   B(a(b(x1))) -> B(a(x1))
30.05/9.65	   B(a(b(x1))) -> A(x1)
30.05/9.65	   A(b(b(x1))) -> B(b(b(x1)))
30.05/9.65	
30.05/9.65	The TRS R consists of the following rules:
30.05/9.65	
30.05/9.65	   b(a(b(x1))) -> a(b(a(x1)))
30.05/9.65	   a(b(b(x1))) -> b(b(b(x1)))
30.05/9.65	
30.05/9.65	Q is empty.
30.05/9.65	We have to consider all minimal (P,Q,R)-chains.
30.05/9.65	----------------------------------------
30.05/9.65	
30.05/9.65	(5) QDPOrderProof (EQUIVALENT)
30.05/9.65	We use the reduction pair processor [LPAR04,JAR06].
30.05/9.65	
30.05/9.65	
30.05/9.65	The following pairs can be oriented strictly and are deleted.
30.05/9.65	
30.05/9.65	   B(a(b(x1))) -> B(a(x1))
30.05/9.65	   B(a(b(x1))) -> A(x1)
30.05/9.65	The remaining pairs can at least be oriented weakly.
30.05/9.65	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
30.05/9.65	
30.05/9.65	POL( A_1(x_1) ) = x_1 + 2
30.05/9.65	POL( B_1(x_1) ) = x_1 + 2
30.05/9.65	POL( a_1(x_1) ) = 2x_1 + 1
30.05/9.65	POL( b_1(x_1) ) = 2x_1 + 1
30.05/9.65	
30.05/9.65	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
30.05/9.65	
30.05/9.65	   a(b(b(x1))) -> b(b(b(x1)))
30.05/9.65	   b(a(b(x1))) -> a(b(a(x1)))
30.05/9.65	
30.05/9.65	
30.05/9.65	----------------------------------------
30.05/9.65	
30.05/9.65	(6)
30.05/9.65	Obligation:
30.05/9.65	Q DP problem:
30.05/9.65	The TRS P consists of the following rules:
30.05/9.65	
30.05/9.65	   B(a(b(x1))) -> A(b(a(x1)))
30.05/9.65	   A(b(b(x1))) -> B(b(b(x1)))
30.05/9.65	
30.05/9.65	The TRS R consists of the following rules:
30.05/9.65	
30.05/9.65	   b(a(b(x1))) -> a(b(a(x1)))
30.05/9.65	   a(b(b(x1))) -> b(b(b(x1)))
30.05/9.65	
30.05/9.65	Q is empty.
30.05/9.65	We have to consider all minimal (P,Q,R)-chains.
30.05/9.65	----------------------------------------
30.05/9.65	
30.05/9.65	(7) QDPOrderProof (EQUIVALENT)
30.05/9.65	We use the reduction pair processor [LPAR04,JAR06].
30.05/9.65	
30.05/9.65	
30.05/9.65	The following pairs can be oriented strictly and are deleted.
30.05/9.65	
30.05/9.65	   A(b(b(x1))) -> B(b(b(x1)))
30.05/9.65	The remaining pairs can at least be oriented weakly.
30.05/9.65	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
30.05/9.65	
30.05/9.65	   <<<
30.05/9.65	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
30.05/9.65	>>>
30.05/9.65	
30.05/9.65	   <<<
30.05/9.65	 POL(a(x_1)) =  	[[1A], [1A], [0A]] 	 +  	[[0A, 0A, -I], [0A, 0A, -I], [0A, 0A, 0A]] 	* 	x_1
30.05/9.65	>>>
30.05/9.65	
30.05/9.65	   <<<
30.05/9.65	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [-I, 0A, -I], [0A, 1A, 0A]] 	* 	x_1
30.05/9.65	>>>
30.05/9.65	
30.05/9.65	   <<<
30.05/9.65	 POL(A(x_1)) =  	[[1A]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
30.05/9.65	>>>
30.05/9.65	
30.05/9.65	
30.05/9.65	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
30.05/9.65	
30.05/9.65	   a(b(b(x1))) -> b(b(b(x1)))
30.05/9.65	   b(a(b(x1))) -> a(b(a(x1)))
30.05/9.65	
30.05/9.65	
30.05/9.65	----------------------------------------
30.05/9.65	
30.05/9.65	(8)
30.05/9.65	Obligation:
30.05/9.65	Q DP problem:
30.05/9.65	The TRS P consists of the following rules:
30.05/9.65	
30.05/9.65	   B(a(b(x1))) -> A(b(a(x1)))
30.05/9.65	
30.05/9.65	The TRS R consists of the following rules:
30.05/9.65	
30.05/9.65	   b(a(b(x1))) -> a(b(a(x1)))
30.05/9.65	   a(b(b(x1))) -> b(b(b(x1)))
30.05/9.65	
30.05/9.65	Q is empty.
30.05/9.65	We have to consider all minimal (P,Q,R)-chains.
30.05/9.65	----------------------------------------
30.05/9.65	
30.05/9.65	(9) DependencyGraphProof (EQUIVALENT)
30.05/9.65	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
30.05/9.65	----------------------------------------
30.05/9.65	
30.05/9.65	(10)
30.05/9.65	TRUE
30.07/9.67	EOF