38.68/10.84	YES
38.96/10.85	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
38.96/10.85	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
38.96/10.85	
38.96/10.85	
38.96/10.85	Termination w.r.t. Q of the given QTRS could be proven:
38.96/10.85	
38.96/10.85	(0) QTRS
38.96/10.85	(1) DependencyPairsProof [EQUIVALENT, 14 ms]
38.96/10.85	(2) QDP
38.96/10.85	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
38.96/10.85	(4) QDP
38.96/10.85	(5) QDPOrderProof [EQUIVALENT, 0 ms]
38.96/10.85	(6) QDP
38.96/10.85	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
38.96/10.85	(8) TRUE
38.96/10.85	
38.96/10.85	
38.96/10.85	----------------------------------------
38.96/10.85	
38.96/10.85	(0)
38.96/10.85	Obligation:
38.96/10.85	Q restricted rewrite system:
38.96/10.85	The TRS R consists of the following rules:
38.96/10.85	
38.96/10.85	   a(x1) -> x1
38.96/10.85	   a(b(x1)) -> c(b(c(b(a(x1)))))
38.96/10.85	   b(x1) -> a(x1)
38.96/10.85	   c(c(x1)) -> x1
38.96/10.85	
38.96/10.85	Q is empty.
38.96/10.85	
38.96/10.85	----------------------------------------
38.96/10.85	
38.96/10.85	(1) DependencyPairsProof (EQUIVALENT)
38.96/10.85	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
38.96/10.85	----------------------------------------
38.96/10.85	
38.96/10.85	(2)
38.96/10.85	Obligation:
38.96/10.85	Q DP problem:
38.96/10.85	The TRS P consists of the following rules:
38.96/10.85	
38.96/10.85	   A(b(x1)) -> C(b(c(b(a(x1)))))
38.96/10.85	   A(b(x1)) -> B(c(b(a(x1))))
38.96/10.85	   A(b(x1)) -> C(b(a(x1)))
38.96/10.85	   A(b(x1)) -> B(a(x1))
38.96/10.85	   A(b(x1)) -> A(x1)
38.96/10.85	   B(x1) -> A(x1)
38.96/10.85	
38.96/10.85	The TRS R consists of the following rules:
38.96/10.85	
38.96/10.85	   a(x1) -> x1
38.96/10.85	   a(b(x1)) -> c(b(c(b(a(x1)))))
38.96/10.85	   b(x1) -> a(x1)
38.96/10.85	   c(c(x1)) -> x1
38.96/10.85	
38.96/10.85	Q is empty.
38.96/10.85	We have to consider all minimal (P,Q,R)-chains.
38.96/10.85	----------------------------------------
38.96/10.85	
38.96/10.85	(3) DependencyGraphProof (EQUIVALENT)
38.96/10.85	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
38.96/10.85	----------------------------------------
38.96/10.85	
38.96/10.85	(4)
38.96/10.85	Obligation:
38.96/10.85	Q DP problem:
38.96/10.85	The TRS P consists of the following rules:
38.96/10.85	
38.96/10.85	   A(b(x1)) -> B(c(b(a(x1))))
38.96/10.85	   B(x1) -> A(x1)
38.96/10.85	   A(b(x1)) -> B(a(x1))
38.96/10.85	   A(b(x1)) -> A(x1)
38.96/10.85	
38.96/10.85	The TRS R consists of the following rules:
38.96/10.85	
38.96/10.85	   a(x1) -> x1
38.96/10.85	   a(b(x1)) -> c(b(c(b(a(x1)))))
38.96/10.85	   b(x1) -> a(x1)
38.96/10.85	   c(c(x1)) -> x1
38.96/10.85	
38.96/10.85	Q is empty.
38.96/10.85	We have to consider all minimal (P,Q,R)-chains.
38.96/10.85	----------------------------------------
38.96/10.85	
38.96/10.85	(5) QDPOrderProof (EQUIVALENT)
38.96/10.85	We use the reduction pair processor [LPAR04,JAR06].
38.96/10.85	
38.96/10.85	
38.96/10.85	The following pairs can be oriented strictly and are deleted.
38.96/10.85	
38.96/10.85	   B(x1) -> A(x1)
38.96/10.85	   A(b(x1)) -> A(x1)
38.96/10.85	The remaining pairs can at least be oriented weakly.
38.96/10.85	Used ordering:  Matrix interpretation [MATRO] with arctic integers [ARCTIC,STERNAGEL_THIEMANN_RTA14]:
38.96/10.85	
38.96/10.85	   <<<
38.96/10.85	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, -I, -1A]] 	* 	x_1
38.96/10.85	>>>
38.96/10.85	
38.96/10.85	   <<<
38.96/10.85	 POL(b(x_1)) =  	[[-1A], [-1A], [2A]] 	 +  	[[0A, -1A, 0A], [-I, 0A, 2A], [-1A, 0A, 2A]] 	* 	x_1
38.96/10.85	>>>
38.96/10.85	
38.96/10.85	   <<<
38.96/10.85	 POL(B(x_1)) =  	[[1A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
38.96/10.85	>>>
38.96/10.85	
38.96/10.85	   <<<
38.96/10.85	 POL(c(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -1A, -1A], [-1A, -I, 1A], [-I, -1A, -1A]] 	* 	x_1
38.96/10.85	>>>
38.96/10.85	
38.96/10.85	   <<<
38.96/10.85	 POL(a(x_1)) =  	[[-1A], [-1A], [-I]] 	 +  	[[0A, -I, 0A], [-I, 0A, 2A], [-I, -I, 0A]] 	* 	x_1
38.96/10.85	>>>
38.96/10.85	
38.96/10.85	
38.96/10.85	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
38.96/10.85	
38.96/10.85	   a(x1) -> x1
38.96/10.85	   a(b(x1)) -> c(b(c(b(a(x1)))))
38.96/10.85	   b(x1) -> a(x1)
38.96/10.85	   c(c(x1)) -> x1
38.96/10.85	
38.96/10.85	
38.96/10.85	----------------------------------------
38.96/10.85	
38.96/10.85	(6)
38.96/10.85	Obligation:
38.96/10.85	Q DP problem:
38.96/10.85	The TRS P consists of the following rules:
38.96/10.85	
38.96/10.85	   A(b(x1)) -> B(c(b(a(x1))))
38.96/10.85	   A(b(x1)) -> B(a(x1))
38.96/10.85	
38.96/10.85	The TRS R consists of the following rules:
38.96/10.85	
38.96/10.85	   a(x1) -> x1
38.96/10.85	   a(b(x1)) -> c(b(c(b(a(x1)))))
38.96/10.85	   b(x1) -> a(x1)
38.96/10.85	   c(c(x1)) -> x1
38.96/10.85	
38.96/10.85	Q is empty.
38.96/10.85	We have to consider all minimal (P,Q,R)-chains.
38.96/10.85	----------------------------------------
38.96/10.85	
38.96/10.85	(7) DependencyGraphProof (EQUIVALENT)
38.96/10.85	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
38.96/10.85	----------------------------------------
38.96/10.85	
38.96/10.85	(8)
38.96/10.85	TRUE
39.02/10.90	EOF