36.22/10.76	YES
37.02/10.96	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
37.02/10.96	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
37.02/10.96	
37.02/10.96	
37.02/10.96	Termination w.r.t. Q of the given QTRS could be proven:
37.02/10.96	
37.02/10.96	(0) QTRS
37.02/10.96	(1) QTRS Reverse [EQUIVALENT, 0 ms]
37.02/10.96	(2) QTRS
37.02/10.96	(3) DependencyPairsProof [EQUIVALENT, 24 ms]
37.02/10.96	(4) QDP
37.02/10.96	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
37.02/10.96	(6) QDP
37.02/10.96	(7) QDPOrderProof [EQUIVALENT, 123 ms]
37.02/10.96	(8) QDP
37.02/10.96	(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
37.02/10.96	(10) YES
37.02/10.96	
37.02/10.96	
37.02/10.96	----------------------------------------
37.02/10.96	
37.02/10.96	(0)
37.02/10.96	Obligation:
37.02/10.96	Q restricted rewrite system:
37.02/10.96	The TRS R consists of the following rules:
37.02/10.96	
37.02/10.96	   a(x1) -> x1
37.02/10.96	   a(a(x1)) -> a(b(c(a(b(x1)))))
37.02/10.96	   c(b(x1)) -> a(c(x1))
37.02/10.96	
37.02/10.96	Q is empty.
37.02/10.96	
37.02/10.96	----------------------------------------
37.02/10.96	
37.02/10.96	(1) QTRS Reverse (EQUIVALENT)
37.02/10.96	We applied the QTRS Reverse Processor [REVERSE].
37.02/10.96	----------------------------------------
37.02/10.96	
37.02/10.96	(2)
37.02/10.96	Obligation:
37.02/10.96	Q restricted rewrite system:
37.02/10.96	The TRS R consists of the following rules:
37.02/10.96	
37.02/10.96	   a(x1) -> x1
37.02/10.96	   a(a(x1)) -> b(a(c(b(a(x1)))))
37.02/10.96	   b(c(x1)) -> c(a(x1))
37.02/10.96	
37.02/10.96	Q is empty.
37.02/10.96	
37.02/10.96	----------------------------------------
37.02/10.96	
37.02/10.96	(3) DependencyPairsProof (EQUIVALENT)
37.02/10.96	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
37.02/10.96	----------------------------------------
37.02/10.96	
37.02/10.96	(4)
37.02/10.96	Obligation:
37.02/10.96	Q DP problem:
37.02/10.96	The TRS P consists of the following rules:
37.02/10.96	
37.02/10.96	   A(a(x1)) -> B(a(c(b(a(x1)))))
37.02/10.96	   A(a(x1)) -> A(c(b(a(x1))))
37.02/10.96	   A(a(x1)) -> B(a(x1))
37.02/10.96	   B(c(x1)) -> A(x1)
37.02/10.96	
37.02/10.96	The TRS R consists of the following rules:
37.02/10.96	
37.02/10.96	   a(x1) -> x1
37.02/10.96	   a(a(x1)) -> b(a(c(b(a(x1)))))
37.02/10.96	   b(c(x1)) -> c(a(x1))
37.02/10.96	
37.02/10.96	Q is empty.
37.02/10.96	We have to consider all minimal (P,Q,R)-chains.
37.02/10.96	----------------------------------------
37.02/10.96	
37.02/10.96	(5) DependencyGraphProof (EQUIVALENT)
37.02/10.96	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
37.02/10.96	----------------------------------------
37.02/10.96	
37.02/10.96	(6)
37.02/10.96	Obligation:
37.02/10.96	Q DP problem:
37.02/10.96	The TRS P consists of the following rules:
37.02/10.96	
37.02/10.96	   B(c(x1)) -> A(x1)
37.02/10.96	   A(a(x1)) -> B(a(c(b(a(x1)))))
37.02/10.96	   A(a(x1)) -> B(a(x1))
37.02/10.96	
37.02/10.96	The TRS R consists of the following rules:
37.02/10.96	
37.02/10.96	   a(x1) -> x1
37.02/10.96	   a(a(x1)) -> b(a(c(b(a(x1)))))
37.02/10.96	   b(c(x1)) -> c(a(x1))
37.02/10.96	
37.02/10.96	Q is empty.
37.02/10.96	We have to consider all minimal (P,Q,R)-chains.
37.02/10.96	----------------------------------------
37.02/10.96	
37.02/10.96	(7) QDPOrderProof (EQUIVALENT)
37.02/10.96	We use the reduction pair processor [LPAR04,JAR06].
37.02/10.96	
37.02/10.96	
37.02/10.96	The following pairs can be oriented strictly and are deleted.
37.02/10.96	
37.02/10.96	   A(a(x1)) -> B(a(c(b(a(x1)))))
37.02/10.96	The remaining pairs can at least be oriented weakly.
37.02/10.96	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
37.02/10.96	
37.02/10.96	   <<<
37.02/10.96	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
37.02/10.96	>>>
37.02/10.96	
37.02/10.96	   <<<
37.02/10.96	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, -I], [0A, -I, -I], [0A, 0A, 0A]] 	* 	x_1
37.02/10.96	>>>
37.02/10.96	
37.02/10.96	   <<<
37.02/10.96	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, -I, -I]] 	* 	x_1
37.02/10.96	>>>
37.02/10.96	
37.02/10.96	   <<<
37.02/10.96	 POL(a(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, -I], [0A, 0A, 0A]] 	* 	x_1
37.02/10.96	>>>
37.02/10.96	
37.02/10.96	   <<<
37.02/10.96	 POL(b(x_1)) =  	[[0A], [1A], [1A]] 	 +  	[[-I, -I, -I], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
37.02/10.96	>>>
37.02/10.96	
37.02/10.96	
37.02/10.96	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
37.02/10.96	
37.02/10.96	   a(x1) -> x1
37.02/10.96	   a(a(x1)) -> b(a(c(b(a(x1)))))
37.02/10.96	   b(c(x1)) -> c(a(x1))
37.02/10.96	
37.02/10.96	
37.02/10.96	----------------------------------------
37.02/10.96	
37.02/10.96	(8)
37.02/10.96	Obligation:
37.02/10.96	Q DP problem:
37.02/10.96	The TRS P consists of the following rules:
37.02/10.96	
37.02/10.96	   B(c(x1)) -> A(x1)
37.02/10.96	   A(a(x1)) -> B(a(x1))
37.02/10.96	
37.02/10.96	The TRS R consists of the following rules:
37.02/10.96	
37.02/10.96	   a(x1) -> x1
37.02/10.96	   a(a(x1)) -> b(a(c(b(a(x1)))))
37.02/10.96	   b(c(x1)) -> c(a(x1))
37.02/10.96	
37.02/10.96	Q is empty.
37.02/10.96	We have to consider all minimal (P,Q,R)-chains.
37.02/10.96	----------------------------------------
37.02/10.96	
37.02/10.96	(9) QDPSizeChangeProof (EQUIVALENT)
37.02/10.96	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. 
37.02/10.96	
37.02/10.96	From the DPs we obtained the following set of size-change graphs:
37.02/10.96	*A(a(x1)) -> B(a(x1))
37.02/10.96	The graph contains the following edges 1 >= 1
37.02/10.96	
37.02/10.96	
37.02/10.96	*B(c(x1)) -> A(x1)
37.02/10.96	The graph contains the following edges 1 > 1
37.02/10.96	
37.02/10.96	
37.02/10.96	----------------------------------------
37.02/10.96	
37.02/10.96	(10)
37.02/10.96	YES
37.02/11.01	EOF