33.72/9.55	YES
33.94/9.64	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
33.94/9.64	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
33.94/9.64	
33.94/9.64	
33.94/9.64	Termination w.r.t. Q of the given QTRS could be proven:
33.94/9.64	
33.94/9.64	(0) QTRS
33.94/9.64	(1) DependencyPairsProof [EQUIVALENT, 22 ms]
33.94/9.64	(2) QDP
33.94/9.64	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
33.94/9.64	(4) QDP
33.94/9.64	(5) QDPOrderProof [EQUIVALENT, 174 ms]
33.94/9.64	(6) QDP
33.94/9.64	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
33.94/9.64	(8) TRUE
33.94/9.64	
33.94/9.64	
33.94/9.64	----------------------------------------
33.94/9.64	
33.94/9.64	(0)
33.94/9.64	Obligation:
33.94/9.64	Q restricted rewrite system:
33.94/9.64	The TRS R consists of the following rules:
33.94/9.64	
33.94/9.64	   a(x1) -> x1
33.94/9.64	   a(b(x1)) -> c(b(b(x1)))
33.94/9.64	   b(x1) -> a(a(c(x1)))
33.94/9.64	   c(c(x1)) -> x1
33.94/9.64	
33.94/9.64	Q is empty.
33.94/9.64	
33.94/9.64	----------------------------------------
33.94/9.64	
33.94/9.64	(1) DependencyPairsProof (EQUIVALENT)
33.94/9.64	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
33.94/9.64	----------------------------------------
33.94/9.64	
33.94/9.64	(2)
33.94/9.64	Obligation:
33.94/9.64	Q DP problem:
33.94/9.64	The TRS P consists of the following rules:
33.94/9.64	
33.94/9.64	   A(b(x1)) -> C(b(b(x1)))
33.94/9.64	   A(b(x1)) -> B(b(x1))
33.94/9.64	   B(x1) -> A(a(c(x1)))
33.94/9.64	   B(x1) -> A(c(x1))
33.94/9.64	   B(x1) -> C(x1)
33.94/9.64	
33.94/9.64	The TRS R consists of the following rules:
33.94/9.64	
33.94/9.64	   a(x1) -> x1
33.94/9.64	   a(b(x1)) -> c(b(b(x1)))
33.94/9.64	   b(x1) -> a(a(c(x1)))
33.94/9.64	   c(c(x1)) -> x1
33.94/9.64	
33.94/9.64	Q is empty.
33.94/9.64	We have to consider all minimal (P,Q,R)-chains.
33.94/9.64	----------------------------------------
33.94/9.64	
33.94/9.64	(3) DependencyGraphProof (EQUIVALENT)
33.94/9.64	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
33.94/9.64	----------------------------------------
33.94/9.64	
33.94/9.64	(4)
33.94/9.64	Obligation:
33.94/9.64	Q DP problem:
33.94/9.64	The TRS P consists of the following rules:
33.94/9.64	
33.94/9.64	   A(b(x1)) -> B(b(x1))
33.94/9.64	   B(x1) -> A(a(c(x1)))
33.94/9.64	   B(x1) -> A(c(x1))
33.94/9.64	
33.94/9.64	The TRS R consists of the following rules:
33.94/9.64	
33.94/9.64	   a(x1) -> x1
33.94/9.64	   a(b(x1)) -> c(b(b(x1)))
33.94/9.64	   b(x1) -> a(a(c(x1)))
33.94/9.64	   c(c(x1)) -> x1
33.94/9.64	
33.94/9.64	Q is empty.
33.94/9.64	We have to consider all minimal (P,Q,R)-chains.
33.94/9.64	----------------------------------------
33.94/9.64	
33.94/9.64	(5) QDPOrderProof (EQUIVALENT)
33.94/9.64	We use the reduction pair processor [LPAR04,JAR06].
33.94/9.64	
33.94/9.64	
33.94/9.64	The following pairs can be oriented strictly and are deleted.
33.94/9.64	
33.94/9.64	   B(x1) -> A(a(c(x1)))
33.94/9.64	   B(x1) -> A(c(x1))
33.94/9.64	The remaining pairs can at least be oriented weakly.
33.94/9.64	Used ordering:  Matrix interpretation [MATRO] with arctic integers [ARCTIC,STERNAGEL_THIEMANN_RTA14]:
33.94/9.64	
33.94/9.64	   <<<
33.94/9.64	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, -1A, -I]] 	* 	x_1
33.94/9.64	>>>
33.94/9.64	
33.94/9.64	   <<<
33.94/9.64	 POL(b(x_1)) =  	[[0A], [2A], [1A]] 	 +  	[[0A, 0A, 0A], [-1A, 1A, 1A], [-1A, 1A, 1A]] 	* 	x_1
33.94/9.64	>>>
33.94/9.64	
33.94/9.64	   <<<
33.94/9.64	 POL(B(x_1)) =  	[[1A]] 	 +  	[[-I, -I, -1A]] 	* 	x_1
33.94/9.64	>>>
33.94/9.64	
33.94/9.64	   <<<
33.94/9.64	 POL(a(x_1)) =  	[[-1A], [-1A], [-1A]] 	 +  	[[0A, 0A, -1A], [-I, 0A, -I], [-I, 2A, 0A]] 	* 	x_1
33.94/9.64	>>>
33.94/9.64	
33.94/9.64	   <<<
33.94/9.64	 POL(c(x_1)) =  	[[-1A], [-1A], [-1A]] 	 +  	[[0A, -1A, -I], [-I, -I, -1A], [-1A, 1A, -1A]] 	* 	x_1
33.94/9.64	>>>
33.94/9.64	
33.94/9.64	
33.94/9.64	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
33.94/9.64	
33.94/9.64	   b(x1) -> a(a(c(x1)))
33.94/9.64	   c(c(x1)) -> x1
33.94/9.64	   a(x1) -> x1
33.94/9.64	   a(b(x1)) -> c(b(b(x1)))
33.94/9.64	
33.94/9.64	
33.94/9.64	----------------------------------------
33.94/9.64	
33.94/9.64	(6)
33.94/9.64	Obligation:
33.94/9.64	Q DP problem:
33.94/9.64	The TRS P consists of the following rules:
33.94/9.64	
33.94/9.64	   A(b(x1)) -> B(b(x1))
33.94/9.64	
33.94/9.64	The TRS R consists of the following rules:
33.94/9.64	
33.94/9.64	   a(x1) -> x1
33.94/9.64	   a(b(x1)) -> c(b(b(x1)))
33.94/9.64	   b(x1) -> a(a(c(x1)))
33.94/9.64	   c(c(x1)) -> x1
33.94/9.64	
33.94/9.64	Q is empty.
33.94/9.64	We have to consider all minimal (P,Q,R)-chains.
33.94/9.64	----------------------------------------
33.94/9.64	
33.94/9.64	(7) DependencyGraphProof (EQUIVALENT)
33.94/9.64	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
33.94/9.64	----------------------------------------
33.94/9.64	
33.94/9.64	(8)
33.94/9.64	TRUE
34.37/9.72	EOF