383.92/98.28	YES
384.31/98.34	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
384.31/98.34	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
384.31/98.34	
384.31/98.34	
384.31/98.34	Termination w.r.t. Q of the given QTRS could be proven:
384.31/98.34	
384.31/98.34	(0) QTRS
384.31/98.34	(1) DependencyPairsProof [EQUIVALENT, 0 ms]
384.31/98.34	(2) QDP
384.31/98.34	(3) QDPOrderProof [EQUIVALENT, 469 ms]
384.31/98.34	(4) QDP
384.31/98.34	(5) QDPOrderProof [EQUIVALENT, 51 ms]
384.31/98.34	(6) QDP
384.31/98.34	(7) QDPOrderProof [EQUIVALENT, 223 ms]
384.31/98.34	(8) QDP
384.31/98.34	(9) QDPOrderProof [EQUIVALENT, 2868 ms]
384.31/98.34	(10) QDP
384.31/98.34	(11) QDPOrderProof [EQUIVALENT, 1517 ms]
384.31/98.34	(12) QDP
384.31/98.34	(13) DependencyGraphProof [EQUIVALENT, 0 ms]
384.31/98.34	(14) QDP
384.31/98.34	(15) UsableRulesProof [EQUIVALENT, 0 ms]
384.31/98.34	(16) QDP
384.31/98.34	(17) QDPSizeChangeProof [EQUIVALENT, 0 ms]
384.31/98.34	(18) YES
384.31/98.34	
384.31/98.34	
384.31/98.34	----------------------------------------
384.31/98.34	
384.31/98.34	(0)
384.31/98.34	Obligation:
384.31/98.34	Q restricted rewrite system:
384.31/98.34	The TRS R consists of the following rules:
384.31/98.34	
384.31/98.34	   a(a(a(b(x1)))) -> b(a(b(a(x1))))
384.31/98.34	   b(b(a(x1))) -> a(a(a(b(x1))))
384.31/98.34	
384.31/98.34	Q is empty.
384.31/98.34	
384.31/98.34	----------------------------------------
384.31/98.34	
384.31/98.34	(1) DependencyPairsProof (EQUIVALENT)
384.31/98.34	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
384.31/98.34	----------------------------------------
384.31/98.34	
384.31/98.34	(2)
384.31/98.34	Obligation:
384.31/98.34	Q DP problem:
384.31/98.34	The TRS P consists of the following rules:
384.31/98.34	
384.31/98.34	   A(a(a(b(x1)))) -> B(a(b(a(x1))))
384.31/98.34	   A(a(a(b(x1)))) -> A(b(a(x1)))
384.31/98.34	   A(a(a(b(x1)))) -> B(a(x1))
384.31/98.34	   A(a(a(b(x1)))) -> A(x1)
384.31/98.34	   B(b(a(x1))) -> A(a(a(b(x1))))
384.31/98.34	   B(b(a(x1))) -> A(a(b(x1)))
384.31/98.34	   B(b(a(x1))) -> A(b(x1))
384.31/98.34	   B(b(a(x1))) -> B(x1)
384.31/98.34	
384.31/98.34	The TRS R consists of the following rules:
384.31/98.34	
384.31/98.34	   a(a(a(b(x1)))) -> b(a(b(a(x1))))
384.31/98.34	   b(b(a(x1))) -> a(a(a(b(x1))))
384.31/98.34	
384.31/98.34	Q is empty.
384.31/98.34	We have to consider all minimal (P,Q,R)-chains.
384.31/98.34	----------------------------------------
384.31/98.34	
384.31/98.34	(3) QDPOrderProof (EQUIVALENT)
384.31/98.34	We use the reduction pair processor [LPAR04,JAR06].
384.31/98.34	
384.31/98.34	
384.31/98.34	The following pairs can be oriented strictly and are deleted.
384.31/98.34	
384.31/98.34	   A(a(a(b(x1)))) -> B(a(b(a(x1))))
384.31/98.34	The remaining pairs can at least be oriented weakly.
384.31/98.34	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
384.31/98.34	
384.31/98.34	   <<<
384.31/98.34	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
384.31/98.34	>>>
384.31/98.34	
384.31/98.34	   <<<
384.31/98.34	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, -I], [-I, -I, 0A], [0A, -I, -I]] 	* 	x_1
384.31/98.34	>>>
384.31/98.34	
384.31/98.34	   <<<
384.31/98.34	 POL(b(x_1)) =  	[[0A], [1A], [1A]] 	 +  	[[-I, -I, -I], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
384.31/98.34	>>>
384.31/98.34	
384.31/98.34	   <<<
384.31/98.34	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
384.31/98.34	>>>
384.31/98.34	
384.31/98.34	
384.31/98.34	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
384.31/98.34	
384.31/98.34	   b(b(a(x1))) -> a(a(a(b(x1))))
384.31/98.34	   a(a(a(b(x1)))) -> b(a(b(a(x1))))
384.31/98.34	
384.31/98.34	
384.31/98.34	----------------------------------------
384.31/98.34	
384.31/98.34	(4)
384.31/98.34	Obligation:
384.31/98.34	Q DP problem:
384.31/98.34	The TRS P consists of the following rules:
384.31/98.34	
384.31/98.34	   A(a(a(b(x1)))) -> A(b(a(x1)))
384.31/98.34	   A(a(a(b(x1)))) -> B(a(x1))
384.31/98.34	   A(a(a(b(x1)))) -> A(x1)
384.31/98.34	   B(b(a(x1))) -> A(a(a(b(x1))))
384.31/98.34	   B(b(a(x1))) -> A(a(b(x1)))
384.31/98.34	   B(b(a(x1))) -> A(b(x1))
384.31/98.34	   B(b(a(x1))) -> B(x1)
384.31/98.34	
384.31/98.34	The TRS R consists of the following rules:
384.31/98.34	
384.31/98.34	   a(a(a(b(x1)))) -> b(a(b(a(x1))))
384.31/98.34	   b(b(a(x1))) -> a(a(a(b(x1))))
384.31/98.34	
384.31/98.34	Q is empty.
384.31/98.34	We have to consider all minimal (P,Q,R)-chains.
384.31/98.34	----------------------------------------
384.31/98.34	
384.31/98.34	(5) QDPOrderProof (EQUIVALENT)
384.31/98.34	We use the reduction pair processor [LPAR04,JAR06].
384.31/98.34	
384.31/98.34	
384.31/98.34	The following pairs can be oriented strictly and are deleted.
384.31/98.34	
384.31/98.34	   A(a(a(b(x1)))) -> A(b(a(x1)))
384.31/98.34	   B(b(a(x1))) -> A(b(x1))
384.31/98.34	The remaining pairs can at least be oriented weakly.
384.31/98.34	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
384.31/98.34	
384.31/98.34	   <<<
384.31/98.34	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, -I, -I]] 	* 	x_1
384.31/98.34	>>>
384.31/98.34	
384.31/98.34	   <<<
384.31/98.34	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, -I], [-I, -I, 0A], [0A, -I, -I]] 	* 	x_1
384.31/98.34	>>>
384.31/98.34	
384.31/98.34	   <<<
384.31/98.34	 POL(b(x_1)) =  	[[0A], [1A], [1A]] 	 +  	[[-I, -I, -I], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
384.31/98.34	>>>
384.31/98.34	
384.31/98.34	   <<<
384.31/98.34	 POL(B(x_1)) =  	[[1A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
384.31/98.34	>>>
384.31/98.34	
384.31/98.34	
384.31/98.34	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
384.31/98.34	
384.31/98.34	   b(b(a(x1))) -> a(a(a(b(x1))))
384.31/98.34	   a(a(a(b(x1)))) -> b(a(b(a(x1))))
384.31/98.34	
384.31/98.34	
384.31/98.34	----------------------------------------
384.31/98.34	
384.31/98.34	(6)
384.31/98.34	Obligation:
384.31/98.34	Q DP problem:
384.31/98.34	The TRS P consists of the following rules:
384.31/98.34	
384.31/98.34	   A(a(a(b(x1)))) -> B(a(x1))
384.31/98.34	   A(a(a(b(x1)))) -> A(x1)
384.31/98.34	   B(b(a(x1))) -> A(a(a(b(x1))))
384.31/98.34	   B(b(a(x1))) -> A(a(b(x1)))
384.31/98.34	   B(b(a(x1))) -> B(x1)
384.31/98.34	
384.31/98.34	The TRS R consists of the following rules:
384.31/98.34	
384.31/98.34	   a(a(a(b(x1)))) -> b(a(b(a(x1))))
384.31/98.34	   b(b(a(x1))) -> a(a(a(b(x1))))
384.31/98.34	
384.31/98.34	Q is empty.
384.31/98.34	We have to consider all minimal (P,Q,R)-chains.
384.31/98.34	----------------------------------------
384.31/98.34	
384.31/98.34	(7) QDPOrderProof (EQUIVALENT)
384.31/98.34	We use the reduction pair processor [LPAR04,JAR06].
384.31/98.34	
384.31/98.34	
384.31/98.34	The following pairs can be oriented strictly and are deleted.
384.31/98.34	
384.31/98.34	   B(b(a(x1))) -> A(a(b(x1)))
384.31/98.34	The remaining pairs can at least be oriented weakly.
384.31/98.34	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
384.31/98.34	
384.31/98.34	   <<<
384.31/98.34	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
384.31/98.34	>>>
384.31/98.34	
384.31/98.34	   <<<
384.31/98.34	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, 0A], [0A, -I, -I], [-I, 0A, -I]] 	* 	x_1
384.31/98.34	>>>
384.31/98.34	
384.31/98.34	   <<<
384.31/98.34	 POL(b(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [-I, -I, -I], [-I, -I, -I]] 	* 	x_1
384.31/98.34	>>>
384.31/98.34	
384.31/98.34	   <<<
384.31/98.34	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
384.31/98.34	>>>
384.31/98.34	
384.31/98.34	
384.31/98.34	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
384.31/98.34	
384.31/98.34	   b(b(a(x1))) -> a(a(a(b(x1))))
384.31/98.34	   a(a(a(b(x1)))) -> b(a(b(a(x1))))
384.31/98.34	
384.31/98.34	
384.31/98.34	----------------------------------------
384.31/98.34	
384.31/98.34	(8)
384.31/98.34	Obligation:
384.31/98.34	Q DP problem:
384.31/98.34	The TRS P consists of the following rules:
384.31/98.34	
384.31/98.34	   A(a(a(b(x1)))) -> B(a(x1))
384.31/98.34	   A(a(a(b(x1)))) -> A(x1)
384.31/98.34	   B(b(a(x1))) -> A(a(a(b(x1))))
384.31/98.34	   B(b(a(x1))) -> B(x1)
384.31/98.34	
384.31/98.34	The TRS R consists of the following rules:
384.31/98.34	
384.31/98.34	   a(a(a(b(x1)))) -> b(a(b(a(x1))))
384.31/98.34	   b(b(a(x1))) -> a(a(a(b(x1))))
384.31/98.34	
384.31/98.34	Q is empty.
384.31/98.34	We have to consider all minimal (P,Q,R)-chains.
384.31/98.34	----------------------------------------
384.31/98.34	
384.31/98.34	(9) QDPOrderProof (EQUIVALENT)
384.31/98.34	We use the reduction pair processor [LPAR04,JAR06].
384.31/98.34	
384.31/98.34	
384.31/98.34	The following pairs can be oriented strictly and are deleted.
384.31/98.34	
384.31/98.34	   A(a(a(b(x1)))) -> A(x1)
384.31/98.34	The remaining pairs can at least be oriented weakly.
384.31/98.34	Used ordering:  Matrix interpretation [MATRO] to (N^6, +, *, >=, >) :
384.31/98.34	
384.31/98.34	   <<<
384.31/98.34	 POL(A(x_1)) =  	[[0]] 	 +  	[[0, 0, 0, 0, 1, 0]] 	* 	x_1
384.31/98.34	>>>
384.31/98.34	
384.31/98.34	   <<<
384.31/98.34	 POL(a(x_1)) =  	[[0], [0], [0], [0], [0], [0]] 	 +  	[[0, 0, 0, 0, 1, 1], [0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 1, 1], [0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0]] 	* 	x_1
384.31/98.34	>>>
384.31/98.34	
384.31/98.34	   <<<
384.31/98.34	 POL(b(x_1)) =  	[[1], [0], [1], [0], [0], [0]] 	 +  	[[0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 1], [0, 0, 1, 1, 0, 0], [0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0]] 	* 	x_1
384.31/98.34	>>>
384.31/98.34	
384.31/98.34	   <<<
384.31/98.34	 POL(B(x_1)) =  	[[1]] 	 +  	[[0, 0, 0, 0, 0, 1]] 	* 	x_1
384.31/98.34	>>>
384.31/98.34	
384.31/98.34	
384.31/98.34	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
384.31/98.34	
384.31/98.34	   b(b(a(x1))) -> a(a(a(b(x1))))
384.31/98.34	   a(a(a(b(x1)))) -> b(a(b(a(x1))))
384.31/98.34	
384.31/98.34	
384.31/98.34	----------------------------------------
384.31/98.34	
384.31/98.34	(10)
384.31/98.34	Obligation:
384.31/98.34	Q DP problem:
384.31/98.34	The TRS P consists of the following rules:
384.31/98.34	
384.31/98.34	   A(a(a(b(x1)))) -> B(a(x1))
384.31/98.34	   B(b(a(x1))) -> A(a(a(b(x1))))
384.31/98.34	   B(b(a(x1))) -> B(x1)
384.31/98.34	
384.31/98.34	The TRS R consists of the following rules:
384.31/98.34	
384.31/98.34	   a(a(a(b(x1)))) -> b(a(b(a(x1))))
384.31/98.34	   b(b(a(x1))) -> a(a(a(b(x1))))
384.31/98.34	
384.31/98.34	Q is empty.
384.31/98.34	We have to consider all minimal (P,Q,R)-chains.
384.31/98.34	----------------------------------------
384.31/98.34	
384.31/98.34	(11) QDPOrderProof (EQUIVALENT)
384.31/98.34	We use the reduction pair processor [LPAR04,JAR06].
384.31/98.34	
384.31/98.34	
384.31/98.34	The following pairs can be oriented strictly and are deleted.
384.31/98.34	
384.31/98.34	   A(a(a(b(x1)))) -> B(a(x1))
384.31/98.34	The remaining pairs can at least be oriented weakly.
384.31/98.34	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
384.31/98.34	
384.31/98.34	   <<<
384.31/98.34	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A, 0A, 0A]] 	* 	x_1
384.31/98.34	>>>
384.31/98.34	
384.31/98.34	   <<<
384.31/98.34	 POL(a(x_1)) =  	[[0A], [0A], [0A], [-I], [1A]] 	 +  	[[-I, -I, -I, 0A, -I], [-I, 0A, -I, 0A, 0A], [0A, -I, -I, 0A, -I], [-I, -I, -I, -I, 0A], [-I, 0A, -I, 0A, 0A]] 	* 	x_1
384.31/98.34	>>>
384.31/98.34	
384.31/98.34	   <<<
384.31/98.34	 POL(b(x_1)) =  	[[0A], [0A], [0A], [0A], [0A]] 	 +  	[[1A, -I, 0A, -I, -I], [0A, -I, -I, 1A, -I], [0A, -I, 0A, 0A, -I], [0A, -I, -I, -I, -I], [0A, -I, -I, -I, -I]] 	* 	x_1
384.31/98.34	>>>
384.31/98.34	
384.31/98.34	   <<<
384.31/98.34	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, -I, -I, -I, -I]] 	* 	x_1
384.31/98.34	>>>
384.31/98.34	
384.31/98.34	
384.31/98.34	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
384.31/98.34	
384.31/98.34	   b(b(a(x1))) -> a(a(a(b(x1))))
384.31/98.34	   a(a(a(b(x1)))) -> b(a(b(a(x1))))
384.31/98.34	
384.31/98.34	
384.31/98.34	----------------------------------------
384.31/98.34	
384.31/98.34	(12)
384.31/98.34	Obligation:
384.31/98.34	Q DP problem:
384.31/98.34	The TRS P consists of the following rules:
384.31/98.34	
384.31/98.34	   B(b(a(x1))) -> A(a(a(b(x1))))
384.31/98.34	   B(b(a(x1))) -> B(x1)
384.31/98.34	
384.31/98.34	The TRS R consists of the following rules:
384.31/98.34	
384.31/98.34	   a(a(a(b(x1)))) -> b(a(b(a(x1))))
384.31/98.34	   b(b(a(x1))) -> a(a(a(b(x1))))
384.31/98.34	
384.31/98.34	Q is empty.
384.31/98.34	We have to consider all minimal (P,Q,R)-chains.
384.31/98.34	----------------------------------------
384.31/98.34	
384.31/98.34	(13) DependencyGraphProof (EQUIVALENT)
384.31/98.34	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
384.31/98.34	----------------------------------------
384.31/98.34	
384.31/98.34	(14)
384.31/98.34	Obligation:
384.31/98.34	Q DP problem:
384.31/98.34	The TRS P consists of the following rules:
384.31/98.34	
384.31/98.34	   B(b(a(x1))) -> B(x1)
384.31/98.34	
384.31/98.34	The TRS R consists of the following rules:
384.31/98.34	
384.31/98.34	   a(a(a(b(x1)))) -> b(a(b(a(x1))))
384.31/98.34	   b(b(a(x1))) -> a(a(a(b(x1))))
384.31/98.34	
384.31/98.34	Q is empty.
384.31/98.34	We have to consider all minimal (P,Q,R)-chains.
384.31/98.34	----------------------------------------
384.31/98.34	
384.31/98.34	(15) UsableRulesProof (EQUIVALENT)
384.31/98.34	We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
384.31/98.34	----------------------------------------
384.31/98.34	
384.31/98.34	(16)
384.31/98.34	Obligation:
384.31/98.34	Q DP problem:
384.31/98.34	The TRS P consists of the following rules:
384.31/98.34	
384.31/98.34	   B(b(a(x1))) -> B(x1)
384.31/98.34	
384.31/98.34	R is empty.
384.31/98.34	Q is empty.
384.31/98.34	We have to consider all minimal (P,Q,R)-chains.
384.31/98.34	----------------------------------------
384.31/98.34	
384.31/98.34	(17) QDPSizeChangeProof (EQUIVALENT)
384.31/98.34	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
384.31/98.34	
384.31/98.34	From the DPs we obtained the following set of size-change graphs:
384.31/98.34	*B(b(a(x1))) -> B(x1)
384.31/98.34	The graph contains the following edges 1 > 1
384.31/98.34	
384.31/98.34	
384.31/98.34	----------------------------------------
384.31/98.34	
384.31/98.34	(18)
384.31/98.34	YES
384.37/98.40	EOF