27.20/7.76	YES
27.20/7.79	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
27.20/7.79	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
27.20/7.79	
27.20/7.79	
27.20/7.79	Termination w.r.t. Q of the given QTRS could be proven:
27.20/7.79	
27.20/7.79	(0) QTRS
27.20/7.79	(1) DependencyPairsProof [EQUIVALENT, 0 ms]
27.20/7.79	(2) QDP
27.20/7.79	(3) MRRProof [EQUIVALENT, 62 ms]
27.20/7.79	(4) QDP
27.20/7.79	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
27.20/7.79	(6) AND
27.20/7.79	    (7) QDP
27.20/7.79	        (8) QDPOrderProof [EQUIVALENT, 0 ms]
27.20/7.79	        (9) QDP
27.20/7.79	        (10) PisEmptyProof [EQUIVALENT, 0 ms]
27.20/7.79	        (11) YES
27.20/7.79	    (12) QDP
27.20/7.79	        (13) QDPOrderProof [EQUIVALENT, 0 ms]
27.20/7.79	        (14) QDP
27.20/7.79	        (15) QDPOrderProof [EQUIVALENT, 32 ms]
27.20/7.79	        (16) QDP
27.20/7.79	        (17) PisEmptyProof [EQUIVALENT, 0 ms]
27.20/7.79	        (18) YES
27.20/7.79	
27.20/7.79	
27.20/7.79	----------------------------------------
27.20/7.79	
27.20/7.79	(0)
27.20/7.79	Obligation:
27.20/7.79	Q restricted rewrite system:
27.20/7.79	The TRS R consists of the following rules:
27.20/7.79	
27.20/7.79	   b(a(x1)) -> b(b(x1))
27.20/7.79	   b(a(b(x1))) -> b(a(a(x1)))
27.20/7.79	   a(a(a(x1))) -> a(b(b(x1)))
27.20/7.79	
27.20/7.79	Q is empty.
27.20/7.79	
27.20/7.79	----------------------------------------
27.20/7.79	
27.20/7.79	(1) DependencyPairsProof (EQUIVALENT)
27.20/7.79	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
27.20/7.79	----------------------------------------
27.20/7.79	
27.20/7.79	(2)
27.20/7.79	Obligation:
27.20/7.79	Q DP problem:
27.20/7.79	The TRS P consists of the following rules:
27.20/7.79	
27.20/7.79	   B(a(x1)) -> B(b(x1))
27.20/7.79	   B(a(x1)) -> B(x1)
27.20/7.79	   B(a(b(x1))) -> B(a(a(x1)))
27.20/7.79	   B(a(b(x1))) -> A(a(x1))
27.20/7.79	   B(a(b(x1))) -> A(x1)
27.20/7.79	   A(a(a(x1))) -> A(b(b(x1)))
27.20/7.79	   A(a(a(x1))) -> B(b(x1))
27.20/7.79	   A(a(a(x1))) -> B(x1)
27.20/7.79	
27.20/7.79	The TRS R consists of the following rules:
27.20/7.79	
27.20/7.79	   b(a(x1)) -> b(b(x1))
27.20/7.79	   b(a(b(x1))) -> b(a(a(x1)))
27.20/7.79	   a(a(a(x1))) -> a(b(b(x1)))
27.20/7.79	
27.20/7.79	Q is empty.
27.20/7.79	We have to consider all minimal (P,Q,R)-chains.
27.20/7.79	----------------------------------------
27.20/7.79	
27.20/7.79	(3) MRRProof (EQUIVALENT)
27.20/7.79	By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
27.20/7.79	
27.20/7.79	Strictly oriented dependency pairs:
27.20/7.79	
27.20/7.79	   B(a(x1)) -> B(x1)
27.20/7.79	   B(a(b(x1))) -> A(x1)
27.20/7.79	   A(a(a(x1))) -> B(b(x1))
27.20/7.79	   A(a(a(x1))) -> B(x1)
27.20/7.79	
27.20/7.79	
27.20/7.79	Used ordering: Polynomial interpretation [POLO]:
27.20/7.79	
27.20/7.79	   POL(A(x_1)) = 3 + 2*x_1
27.20/7.79	   POL(B(x_1)) = 2 + x_1
27.20/7.79	   POL(a(x_1)) = 1 + 2*x_1
27.20/7.79	   POL(b(x_1)) = 1 + 2*x_1
27.20/7.79	
27.20/7.79	
27.20/7.79	----------------------------------------
27.20/7.79	
27.20/7.79	(4)
27.20/7.79	Obligation:
27.20/7.79	Q DP problem:
27.20/7.79	The TRS P consists of the following rules:
27.20/7.79	
27.20/7.79	   B(a(x1)) -> B(b(x1))
27.20/7.79	   B(a(b(x1))) -> B(a(a(x1)))
27.20/7.79	   B(a(b(x1))) -> A(a(x1))
27.20/7.79	   A(a(a(x1))) -> A(b(b(x1)))
27.20/7.79	
27.20/7.79	The TRS R consists of the following rules:
27.20/7.79	
27.20/7.79	   b(a(x1)) -> b(b(x1))
27.20/7.79	   b(a(b(x1))) -> b(a(a(x1)))
27.20/7.79	   a(a(a(x1))) -> a(b(b(x1)))
27.20/7.79	
27.20/7.79	Q is empty.
27.20/7.79	We have to consider all minimal (P,Q,R)-chains.
27.20/7.79	----------------------------------------
27.20/7.79	
27.20/7.79	(5) DependencyGraphProof (EQUIVALENT)
27.20/7.79	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
27.20/7.79	----------------------------------------
27.20/7.79	
27.20/7.79	(6)
27.20/7.79	Complex Obligation (AND)
27.20/7.79	
27.20/7.79	----------------------------------------
27.20/7.79	
27.20/7.79	(7)
27.20/7.79	Obligation:
27.20/7.79	Q DP problem:
27.20/7.79	The TRS P consists of the following rules:
27.20/7.79	
27.20/7.79	   A(a(a(x1))) -> A(b(b(x1)))
27.20/7.79	
27.20/7.79	The TRS R consists of the following rules:
27.20/7.79	
27.20/7.79	   b(a(x1)) -> b(b(x1))
27.20/7.79	   b(a(b(x1))) -> b(a(a(x1)))
27.20/7.79	   a(a(a(x1))) -> a(b(b(x1)))
27.20/7.79	
27.20/7.79	Q is empty.
27.20/7.79	We have to consider all minimal (P,Q,R)-chains.
27.20/7.79	----------------------------------------
27.20/7.79	
27.20/7.79	(8) QDPOrderProof (EQUIVALENT)
27.20/7.79	We use the reduction pair processor [LPAR04,JAR06].
27.20/7.79	
27.20/7.79	
27.20/7.79	The following pairs can be oriented strictly and are deleted.
27.20/7.79	
27.20/7.79	   A(a(a(x1))) -> A(b(b(x1)))
27.20/7.79	The remaining pairs can at least be oriented weakly.
27.20/7.79	Used ordering:  Polynomial interpretation [POLO]:
27.20/7.79	
27.20/7.79	   POL(A(x_1)) = x_1
27.20/7.79	   POL(a(x_1)) = 1 + x_1
27.20/7.79	   POL(b(x_1)) = 1
27.20/7.79	
27.20/7.79	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
27.20/7.79	
27.20/7.79	   b(a(b(x1))) -> b(a(a(x1)))
27.20/7.79	   b(a(x1)) -> b(b(x1))
27.20/7.79	
27.20/7.79	
27.20/7.79	----------------------------------------
27.20/7.79	
27.20/7.79	(9)
27.20/7.79	Obligation:
27.20/7.79	Q DP problem:
27.20/7.79	P is empty.
27.20/7.79	The TRS R consists of the following rules:
27.20/7.79	
27.20/7.79	   b(a(x1)) -> b(b(x1))
27.20/7.79	   b(a(b(x1))) -> b(a(a(x1)))
27.20/7.79	   a(a(a(x1))) -> a(b(b(x1)))
27.20/7.79	
27.20/7.79	Q is empty.
27.20/7.79	We have to consider all minimal (P,Q,R)-chains.
27.20/7.79	----------------------------------------
27.20/7.79	
27.20/7.79	(10) PisEmptyProof (EQUIVALENT)
27.20/7.79	The TRS P is empty. Hence, there is no (P,Q,R) chain.
27.20/7.79	----------------------------------------
27.20/7.79	
27.20/7.79	(11)
27.20/7.79	YES
27.20/7.79	
27.20/7.79	----------------------------------------
27.20/7.79	
27.20/7.79	(12)
27.20/7.79	Obligation:
27.20/7.79	Q DP problem:
27.20/7.79	The TRS P consists of the following rules:
27.20/7.79	
27.20/7.79	   B(a(b(x1))) -> B(a(a(x1)))
27.20/7.79	   B(a(x1)) -> B(b(x1))
27.20/7.79	
27.20/7.79	The TRS R consists of the following rules:
27.20/7.79	
27.20/7.79	   b(a(x1)) -> b(b(x1))
27.20/7.79	   b(a(b(x1))) -> b(a(a(x1)))
27.20/7.79	   a(a(a(x1))) -> a(b(b(x1)))
27.20/7.79	
27.20/7.79	Q is empty.
27.20/7.79	We have to consider all minimal (P,Q,R)-chains.
27.20/7.79	----------------------------------------
27.20/7.79	
27.20/7.79	(13) QDPOrderProof (EQUIVALENT)
27.20/7.79	We use the reduction pair processor [LPAR04,JAR06].
27.20/7.79	
27.20/7.79	
27.20/7.79	The following pairs can be oriented strictly and are deleted.
27.20/7.79	
27.20/7.79	   B(a(x1)) -> B(b(x1))
27.20/7.79	The remaining pairs can at least be oriented weakly.
27.20/7.79	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
27.20/7.79	
27.20/7.79	POL( B_1(x_1) ) = max{0, 2x_1 - 2}
27.20/7.79	POL( a_1(x_1) ) = 2
27.20/7.79	POL( b_1(x_1) ) = 0
27.20/7.79	
27.20/7.79	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
27.20/7.79	
27.20/7.79	   a(a(a(x1))) -> a(b(b(x1)))
27.20/7.79	   b(a(b(x1))) -> b(a(a(x1)))
27.20/7.79	   b(a(x1)) -> b(b(x1))
27.20/7.79	
27.20/7.79	
27.20/7.79	----------------------------------------
27.20/7.79	
27.20/7.79	(14)
27.20/7.79	Obligation:
27.20/7.79	Q DP problem:
27.20/7.79	The TRS P consists of the following rules:
27.20/7.79	
27.20/7.79	   B(a(b(x1))) -> B(a(a(x1)))
27.20/7.79	
27.20/7.79	The TRS R consists of the following rules:
27.20/7.79	
27.20/7.79	   b(a(x1)) -> b(b(x1))
27.20/7.79	   b(a(b(x1))) -> b(a(a(x1)))
27.20/7.79	   a(a(a(x1))) -> a(b(b(x1)))
27.20/7.79	
27.20/7.79	Q is empty.
27.20/7.79	We have to consider all minimal (P,Q,R)-chains.
27.20/7.79	----------------------------------------
27.20/7.79	
27.20/7.79	(15) QDPOrderProof (EQUIVALENT)
27.20/7.79	We use the reduction pair processor [LPAR04,JAR06].
27.20/7.79	
27.20/7.79	
27.20/7.79	The following pairs can be oriented strictly and are deleted.
27.20/7.79	
27.20/7.79	   B(a(b(x1))) -> B(a(a(x1)))
27.20/7.79	The remaining pairs can at least be oriented weakly.
27.20/7.79	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
27.20/7.79	
27.20/7.79	   <<<
27.20/7.79	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
27.20/7.79	>>>
27.20/7.79	
27.20/7.79	   <<<
27.20/7.79	 POL(a(x_1)) =  	[[1A], [-I], [-I]] 	 +  	[[1A, 0A, 1A], [0A, -I, -I], [-I, 0A, -I]] 	* 	x_1
27.20/7.79	>>>
27.20/7.79	
27.20/7.79	   <<<
27.20/7.79	 POL(b(x_1)) =  	[[0A], [1A], [1A]] 	 +  	[[0A, -I, -I], [1A, 0A, 0A], [1A, 0A, 1A]] 	* 	x_1
27.20/7.79	>>>
27.20/7.79	
27.20/7.79	
27.20/7.79	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
27.20/7.79	
27.20/7.79	   a(a(a(x1))) -> a(b(b(x1)))
27.20/7.79	   b(a(b(x1))) -> b(a(a(x1)))
27.20/7.79	   b(a(x1)) -> b(b(x1))
27.20/7.79	
27.20/7.79	
27.20/7.79	----------------------------------------
27.20/7.79	
27.20/7.79	(16)
27.20/7.79	Obligation:
27.20/7.79	Q DP problem:
27.20/7.79	P is empty.
27.20/7.79	The TRS R consists of the following rules:
27.20/7.79	
27.20/7.79	   b(a(x1)) -> b(b(x1))
27.20/7.79	   b(a(b(x1))) -> b(a(a(x1)))
27.20/7.79	   a(a(a(x1))) -> a(b(b(x1)))
27.20/7.79	
27.20/7.79	Q is empty.
27.20/7.79	We have to consider all minimal (P,Q,R)-chains.
27.20/7.79	----------------------------------------
27.20/7.79	
27.20/7.79	(17) PisEmptyProof (EQUIVALENT)
27.20/7.79	The TRS P is empty. Hence, there is no (P,Q,R) chain.
27.20/7.79	----------------------------------------
27.20/7.79	
27.20/7.79	(18)
27.20/7.79	YES
27.49/10.58	EOF