12.73/4.13	YES
12.73/4.15	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
12.73/4.15	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.73/4.15	
12.73/4.15	
12.73/4.15	Termination w.r.t. Q of the given QTRS could be proven:
12.73/4.15	
12.73/4.15	(0) QTRS
12.73/4.15	(1) DependencyPairsProof [EQUIVALENT, 0 ms]
12.73/4.15	(2) QDP
12.73/4.15	(3) QDPOrderProof [EQUIVALENT, 14 ms]
12.73/4.15	(4) QDP
12.73/4.15	(5) QDPOrderProof [EQUIVALENT, 182 ms]
12.73/4.15	(6) QDP
12.73/4.15	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
12.73/4.15	(8) TRUE
12.73/4.15	
12.73/4.15	
12.73/4.15	----------------------------------------
12.73/4.15	
12.73/4.15	(0)
12.73/4.15	Obligation:
12.73/4.15	Q restricted rewrite system:
12.73/4.15	The TRS R consists of the following rules:
12.73/4.15	
12.73/4.15	   b(b(b(x1))) -> a(a(a(x1)))
12.73/4.15	   a(a(a(x1))) -> b(a(b(x1)))
12.73/4.15	
12.73/4.15	Q is empty.
12.73/4.15	
12.73/4.15	----------------------------------------
12.73/4.15	
12.73/4.15	(1) DependencyPairsProof (EQUIVALENT)
12.73/4.15	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
12.73/4.15	----------------------------------------
12.73/4.15	
12.73/4.15	(2)
12.73/4.15	Obligation:
12.73/4.15	Q DP problem:
12.73/4.15	The TRS P consists of the following rules:
12.73/4.15	
12.73/4.15	   B(b(b(x1))) -> A(a(a(x1)))
12.73/4.15	   B(b(b(x1))) -> A(a(x1))
12.73/4.15	   B(b(b(x1))) -> A(x1)
12.73/4.15	   A(a(a(x1))) -> B(a(b(x1)))
12.73/4.15	   A(a(a(x1))) -> A(b(x1))
12.73/4.15	   A(a(a(x1))) -> B(x1)
12.73/4.15	
12.73/4.15	The TRS R consists of the following rules:
12.73/4.15	
12.73/4.15	   b(b(b(x1))) -> a(a(a(x1)))
12.73/4.15	   a(a(a(x1))) -> b(a(b(x1)))
12.73/4.15	
12.73/4.15	Q is empty.
12.73/4.15	We have to consider all minimal (P,Q,R)-chains.
12.73/4.15	----------------------------------------
12.73/4.15	
12.73/4.15	(3) QDPOrderProof (EQUIVALENT)
12.73/4.15	We use the reduction pair processor [LPAR04,JAR06].
12.73/4.15	
12.73/4.15	
12.73/4.15	The following pairs can be oriented strictly and are deleted.
12.73/4.15	
12.73/4.15	   B(b(b(x1))) -> A(a(x1))
12.73/4.15	   B(b(b(x1))) -> A(x1)
12.73/4.15	   A(a(a(x1))) -> A(b(x1))
12.73/4.15	   A(a(a(x1))) -> B(x1)
12.73/4.15	The remaining pairs can at least be oriented weakly.
12.73/4.15	Used ordering:  Polynomial interpretation [POLO]:
12.73/4.15	
12.73/4.15	   POL(A(x_1)) = x_1
12.73/4.15	   POL(B(x_1)) = x_1
12.73/4.15	   POL(a(x_1)) = 1 + x_1
12.73/4.15	   POL(b(x_1)) = 1 + x_1
12.73/4.15	
12.73/4.15	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
12.73/4.15	
12.73/4.15	   a(a(a(x1))) -> b(a(b(x1)))
12.73/4.15	   b(b(b(x1))) -> a(a(a(x1)))
12.73/4.15	
12.73/4.15	
12.73/4.15	----------------------------------------
12.73/4.15	
12.73/4.15	(4)
12.73/4.15	Obligation:
12.73/4.15	Q DP problem:
12.73/4.15	The TRS P consists of the following rules:
12.73/4.15	
12.73/4.15	   B(b(b(x1))) -> A(a(a(x1)))
12.73/4.15	   A(a(a(x1))) -> B(a(b(x1)))
12.73/4.15	
12.73/4.15	The TRS R consists of the following rules:
12.73/4.15	
12.73/4.15	   b(b(b(x1))) -> a(a(a(x1)))
12.73/4.15	   a(a(a(x1))) -> b(a(b(x1)))
12.73/4.15	
12.73/4.15	Q is empty.
12.73/4.15	We have to consider all minimal (P,Q,R)-chains.
12.73/4.15	----------------------------------------
12.73/4.15	
12.73/4.15	(5) QDPOrderProof (EQUIVALENT)
12.73/4.15	We use the reduction pair processor [LPAR04,JAR06].
12.73/4.15	
12.73/4.15	
12.73/4.15	The following pairs can be oriented strictly and are deleted.
12.73/4.15	
12.73/4.15	   B(b(b(x1))) -> A(a(a(x1)))
12.73/4.15	The remaining pairs can at least be oriented weakly.
12.73/4.15	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
12.73/4.15	
12.73/4.15	   <<<
12.73/4.15	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, -I, -I]] 	* 	x_1
12.73/4.15	>>>
12.73/4.15	
12.73/4.15	   <<<
12.73/4.15	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[1A, 0A, 0A], [0A, -I, -I], [0A, 0A, -I]] 	* 	x_1
12.73/4.15	>>>
12.73/4.15	
12.73/4.15	   <<<
12.73/4.15	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
12.73/4.15	>>>
12.73/4.15	
12.73/4.15	   <<<
12.73/4.15	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, 0A], [0A, 0A, -I], [0A, 1A, -I]] 	* 	x_1
12.73/4.15	>>>
12.73/4.15	
12.73/4.15	
12.73/4.15	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
12.73/4.15	
12.73/4.15	   a(a(a(x1))) -> b(a(b(x1)))
12.73/4.15	   b(b(b(x1))) -> a(a(a(x1)))
12.73/4.15	
12.73/4.15	
12.73/4.15	----------------------------------------
12.73/4.15	
12.73/4.15	(6)
12.73/4.15	Obligation:
12.73/4.15	Q DP problem:
12.73/4.15	The TRS P consists of the following rules:
12.73/4.15	
12.73/4.15	   A(a(a(x1))) -> B(a(b(x1)))
12.73/4.15	
12.73/4.15	The TRS R consists of the following rules:
12.73/4.15	
12.73/4.15	   b(b(b(x1))) -> a(a(a(x1)))
12.73/4.15	   a(a(a(x1))) -> b(a(b(x1)))
12.73/4.15	
12.73/4.15	Q is empty.
12.73/4.15	We have to consider all minimal (P,Q,R)-chains.
12.73/4.15	----------------------------------------
12.73/4.15	
12.73/4.15	(7) DependencyGraphProof (EQUIVALENT)
12.73/4.15	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
12.73/4.15	----------------------------------------
12.73/4.15	
12.73/4.15	(8)
12.73/4.15	TRUE
13.03/4.23	EOF