37.51/10.54	YES
37.51/10.60	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
37.51/10.60	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
37.51/10.60	
37.51/10.60	
37.51/10.60	Termination w.r.t. Q of the given QTRS could be proven:
37.51/10.60	
37.51/10.60	(0) QTRS
37.51/10.60	(1) QTRSRRRProof [EQUIVALENT, 39 ms]
37.51/10.60	(2) QTRS
37.51/10.60	(3) DependencyPairsProof [EQUIVALENT, 35 ms]
37.51/10.60	(4) QDP
37.51/10.60	(5) QDPOrderProof [EQUIVALENT, 51 ms]
37.51/10.60	(6) QDP
37.51/10.60	(7) QDPOrderProof [EQUIVALENT, 1 ms]
37.51/10.60	(8) QDP
37.51/10.60	(9) QDPOrderProof [EQUIVALENT, 0 ms]
37.51/10.60	(10) QDP
37.51/10.60	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
37.51/10.60	(12) TRUE
37.51/10.60	
37.51/10.60	
37.51/10.60	----------------------------------------
37.51/10.60	
37.51/10.60	(0)
37.51/10.60	Obligation:
37.51/10.60	Q restricted rewrite system:
37.51/10.60	The TRS R consists of the following rules:
37.51/10.60	
37.51/10.60	   a(b(a(b(x1)))) -> b(b(a(a(x1))))
37.51/10.60	   b(a(b(a(x1)))) -> a(b(a(b(x1))))
37.51/10.60	   b(a(b(b(x1)))) -> a(b(a(a(x1))))
37.51/10.60	
37.51/10.60	Q is empty.
37.51/10.60	
37.51/10.60	----------------------------------------
37.51/10.60	
37.51/10.60	(1) QTRSRRRProof (EQUIVALENT)
37.51/10.60	Used ordering:
37.51/10.60	Polynomial interpretation [POLO]:
37.51/10.60	
37.51/10.60	   POL(a(x_1)) = x_1
37.51/10.60	   POL(b(x_1)) = 1 + x_1
37.51/10.60	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
37.51/10.60	
37.51/10.60	   b(a(b(b(x1)))) -> a(b(a(a(x1))))
37.51/10.60	
37.51/10.60	
37.51/10.60	
37.51/10.60	
37.51/10.60	----------------------------------------
37.51/10.60	
37.51/10.60	(2)
37.51/10.60	Obligation:
37.51/10.60	Q restricted rewrite system:
37.51/10.60	The TRS R consists of the following rules:
37.51/10.60	
37.51/10.60	   a(b(a(b(x1)))) -> b(b(a(a(x1))))
37.51/10.60	   b(a(b(a(x1)))) -> a(b(a(b(x1))))
37.51/10.60	
37.51/10.60	Q is empty.
37.51/10.60	
37.51/10.60	----------------------------------------
37.51/10.60	
37.51/10.60	(3) DependencyPairsProof (EQUIVALENT)
37.51/10.60	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
37.51/10.60	----------------------------------------
37.51/10.60	
37.51/10.60	(4)
37.51/10.60	Obligation:
37.51/10.60	Q DP problem:
37.51/10.60	The TRS P consists of the following rules:
37.51/10.60	
37.51/10.60	   A(b(a(b(x1)))) -> B(b(a(a(x1))))
37.51/10.60	   A(b(a(b(x1)))) -> B(a(a(x1)))
37.51/10.60	   A(b(a(b(x1)))) -> A(a(x1))
37.51/10.60	   A(b(a(b(x1)))) -> A(x1)
37.51/10.60	   B(a(b(a(x1)))) -> A(b(a(b(x1))))
37.51/10.60	   B(a(b(a(x1)))) -> B(a(b(x1)))
37.51/10.60	   B(a(b(a(x1)))) -> A(b(x1))
37.51/10.60	   B(a(b(a(x1)))) -> B(x1)
37.51/10.60	
37.51/10.60	The TRS R consists of the following rules:
37.51/10.60	
37.51/10.60	   a(b(a(b(x1)))) -> b(b(a(a(x1))))
37.51/10.60	   b(a(b(a(x1)))) -> a(b(a(b(x1))))
37.51/10.60	
37.51/10.60	Q is empty.
37.51/10.60	We have to consider all minimal (P,Q,R)-chains.
37.51/10.60	----------------------------------------
37.51/10.60	
37.51/10.60	(5) QDPOrderProof (EQUIVALENT)
37.51/10.60	We use the reduction pair processor [LPAR04,JAR06].
37.51/10.60	
37.51/10.60	
37.51/10.60	The following pairs can be oriented strictly and are deleted.
37.51/10.60	
37.51/10.60	   A(b(a(b(x1)))) -> B(a(a(x1)))
37.51/10.60	   A(b(a(b(x1)))) -> A(a(x1))
37.51/10.60	   A(b(a(b(x1)))) -> A(x1)
37.51/10.60	   B(a(b(a(x1)))) -> A(b(x1))
37.51/10.60	   B(a(b(a(x1)))) -> B(x1)
37.51/10.60	The remaining pairs can at least be oriented weakly.
37.51/10.60	Used ordering:  Polynomial interpretation [POLO]:
37.51/10.60	
37.51/10.60	   POL(A(x_1)) = x_1
37.51/10.60	   POL(B(x_1)) = 1 + x_1
37.51/10.60	   POL(a(x_1)) = x_1
37.51/10.60	   POL(b(x_1)) = 1 + x_1
37.51/10.60	
37.51/10.60	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
37.51/10.60	
37.51/10.60	   b(a(b(a(x1)))) -> a(b(a(b(x1))))
37.51/10.60	   a(b(a(b(x1)))) -> b(b(a(a(x1))))
37.51/10.60	
37.51/10.60	
37.51/10.60	----------------------------------------
37.51/10.60	
37.51/10.60	(6)
37.51/10.60	Obligation:
37.51/10.60	Q DP problem:
37.51/10.60	The TRS P consists of the following rules:
37.51/10.60	
37.51/10.60	   A(b(a(b(x1)))) -> B(b(a(a(x1))))
37.51/10.60	   B(a(b(a(x1)))) -> A(b(a(b(x1))))
37.51/10.60	   B(a(b(a(x1)))) -> B(a(b(x1)))
37.51/10.60	
37.51/10.60	The TRS R consists of the following rules:
37.51/10.60	
37.51/10.60	   a(b(a(b(x1)))) -> b(b(a(a(x1))))
37.51/10.60	   b(a(b(a(x1)))) -> a(b(a(b(x1))))
37.51/10.60	
37.51/10.60	Q is empty.
37.51/10.60	We have to consider all minimal (P,Q,R)-chains.
37.51/10.60	----------------------------------------
37.51/10.60	
37.51/10.60	(7) QDPOrderProof (EQUIVALENT)
37.51/10.60	We use the reduction pair processor [LPAR04,JAR06].
37.51/10.60	
37.51/10.60	
37.51/10.60	The following pairs can be oriented strictly and are deleted.
37.51/10.60	
37.51/10.60	   B(a(b(a(x1)))) -> B(a(b(x1)))
37.51/10.60	The remaining pairs can at least be oriented weakly.
37.51/10.60	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
37.51/10.60	
37.51/10.60	POL( A_1(x_1) ) = max{0, 2x_1 - 2}
37.51/10.60	POL( B_1(x_1) ) = max{0, 2x_1 - 2}
37.51/10.60	POL( b_1(x_1) ) = 2x_1 + 1
37.51/10.60	POL( a_1(x_1) ) = 2x_1 + 1
37.51/10.60	
37.51/10.60	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
37.51/10.60	
37.51/10.60	   b(a(b(a(x1)))) -> a(b(a(b(x1))))
37.51/10.60	   a(b(a(b(x1)))) -> b(b(a(a(x1))))
37.51/10.60	
37.51/10.60	
37.51/10.60	----------------------------------------
37.51/10.60	
37.51/10.60	(8)
37.51/10.60	Obligation:
37.51/10.60	Q DP problem:
37.51/10.60	The TRS P consists of the following rules:
37.51/10.60	
37.51/10.60	   A(b(a(b(x1)))) -> B(b(a(a(x1))))
37.51/10.60	   B(a(b(a(x1)))) -> A(b(a(b(x1))))
37.51/10.60	
37.51/10.60	The TRS R consists of the following rules:
37.51/10.60	
37.51/10.60	   a(b(a(b(x1)))) -> b(b(a(a(x1))))
37.51/10.60	   b(a(b(a(x1)))) -> a(b(a(b(x1))))
37.51/10.60	
37.51/10.60	Q is empty.
37.51/10.60	We have to consider all minimal (P,Q,R)-chains.
37.51/10.60	----------------------------------------
37.51/10.60	
37.51/10.60	(9) QDPOrderProof (EQUIVALENT)
37.51/10.60	We use the reduction pair processor [LPAR04,JAR06].
37.51/10.60	
37.51/10.60	
37.51/10.60	The following pairs can be oriented strictly and are deleted.
37.51/10.60	
37.51/10.60	   A(b(a(b(x1)))) -> B(b(a(a(x1))))
37.51/10.60	The remaining pairs can at least be oriented weakly.
37.51/10.60	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
37.51/10.60	
37.51/10.60	   <<<
37.51/10.60	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
37.51/10.60	>>>
37.51/10.60	
37.51/10.60	   <<<
37.51/10.60	 POL(b(x_1)) =  	[[1A], [-I], [-I]] 	 +  	[[-I, 0A, 1A], [-I, 0A, -I], [-I, 0A, -I]] 	* 	x_1
37.51/10.60	>>>
37.51/10.60	
37.51/10.60	   <<<
37.51/10.60	 POL(a(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[-I, -I, 0A], [0A, -I, -I], [-I, 0A, 0A]] 	* 	x_1
37.51/10.60	>>>
37.51/10.60	
37.51/10.60	   <<<
37.51/10.60	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
37.51/10.60	>>>
37.51/10.60	
37.51/10.60	
37.51/10.60	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
37.51/10.60	
37.51/10.60	   b(a(b(a(x1)))) -> a(b(a(b(x1))))
37.51/10.60	   a(b(a(b(x1)))) -> b(b(a(a(x1))))
37.51/10.60	
37.51/10.60	
37.51/10.60	----------------------------------------
37.51/10.60	
37.51/10.60	(10)
37.51/10.60	Obligation:
37.51/10.60	Q DP problem:
37.51/10.60	The TRS P consists of the following rules:
37.51/10.60	
37.51/10.60	   B(a(b(a(x1)))) -> A(b(a(b(x1))))
37.51/10.60	
37.51/10.60	The TRS R consists of the following rules:
37.51/10.60	
37.51/10.60	   a(b(a(b(x1)))) -> b(b(a(a(x1))))
37.51/10.60	   b(a(b(a(x1)))) -> a(b(a(b(x1))))
37.51/10.60	
37.51/10.60	Q is empty.
37.51/10.60	We have to consider all minimal (P,Q,R)-chains.
37.51/10.60	----------------------------------------
37.51/10.60	
37.51/10.60	(11) DependencyGraphProof (EQUIVALENT)
37.51/10.60	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
37.51/10.60	----------------------------------------
37.51/10.60	
37.51/10.60	(12)
37.51/10.60	TRUE
37.83/10.66	EOF