39.76/10.93	YES
39.76/10.94	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
39.76/10.94	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
39.76/10.94	
39.76/10.94	
39.76/10.94	Termination w.r.t. Q of the given QTRS could be proven:
39.76/10.94	
39.76/10.94	(0) QTRS
39.76/10.94	(1) QTRS Reverse [EQUIVALENT, 0 ms]
39.76/10.94	(2) QTRS
39.76/10.94	(3) DependencyPairsProof [EQUIVALENT, 1 ms]
39.76/10.94	(4) QDP
39.76/10.94	(5) QDPOrderProof [EQUIVALENT, 237 ms]
39.76/10.94	(6) QDP
39.76/10.94	(7) QDPOrderProof [EQUIVALENT, 206 ms]
39.76/10.94	(8) QDP
39.76/10.94	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
39.76/10.94	(10) TRUE
39.76/10.94	
39.76/10.94	
39.76/10.94	----------------------------------------
39.76/10.94	
39.76/10.94	(0)
39.76/10.94	Obligation:
39.76/10.94	Q restricted rewrite system:
39.76/10.94	The TRS R consists of the following rules:
39.76/10.94	
39.76/10.94	   a(a(x1)) -> a(b(c(x1)))
39.76/10.94	   c(a(x1)) -> x1
39.76/10.94	   c(b(x1)) -> a(a(c(x1)))
39.76/10.94	
39.76/10.94	Q is empty.
39.76/10.94	
39.76/10.94	----------------------------------------
39.76/10.94	
39.76/10.94	(1) QTRS Reverse (EQUIVALENT)
39.76/10.94	We applied the QTRS Reverse Processor [REVERSE].
39.76/10.94	----------------------------------------
39.76/10.94	
39.76/10.94	(2)
39.76/10.94	Obligation:
39.76/10.94	Q restricted rewrite system:
39.76/10.94	The TRS R consists of the following rules:
39.76/10.94	
39.76/10.94	   a(a(x1)) -> c(b(a(x1)))
39.76/10.94	   a(c(x1)) -> x1
39.76/10.94	   b(c(x1)) -> c(a(a(x1)))
39.76/10.94	
39.76/10.94	Q is empty.
39.76/10.94	
39.76/10.94	----------------------------------------
39.76/10.94	
39.76/10.94	(3) DependencyPairsProof (EQUIVALENT)
39.76/10.94	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
39.76/10.94	----------------------------------------
39.76/10.94	
39.76/10.94	(4)
39.76/10.94	Obligation:
39.76/10.94	Q DP problem:
39.76/10.94	The TRS P consists of the following rules:
39.76/10.94	
39.76/10.94	   A(a(x1)) -> B(a(x1))
39.76/10.94	   B(c(x1)) -> A(a(x1))
39.76/10.94	   B(c(x1)) -> A(x1)
39.76/10.94	
39.76/10.94	The TRS R consists of the following rules:
39.76/10.94	
39.76/10.94	   a(a(x1)) -> c(b(a(x1)))
39.76/10.94	   a(c(x1)) -> x1
39.76/10.94	   b(c(x1)) -> c(a(a(x1)))
39.76/10.94	
39.76/10.94	Q is empty.
39.76/10.94	We have to consider all minimal (P,Q,R)-chains.
39.76/10.94	----------------------------------------
39.76/10.94	
39.76/10.94	(5) QDPOrderProof (EQUIVALENT)
39.76/10.94	We use the reduction pair processor [LPAR04,JAR06].
39.76/10.94	
39.76/10.94	
39.76/10.94	The following pairs can be oriented strictly and are deleted.
39.76/10.94	
39.76/10.94	   B(c(x1)) -> A(x1)
39.76/10.94	The remaining pairs can at least be oriented weakly.
39.76/10.94	Used ordering:  Matrix interpretation [MATRO] with arctic integers [ARCTIC,STERNAGEL_THIEMANN_RTA14]:
39.76/10.94	
39.76/10.94	   <<<
39.76/10.94	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-1A, -1A, -I]] 	* 	x_1
39.76/10.94	>>>
39.76/10.94	
39.76/10.94	   <<<
39.76/10.94	 POL(a(x_1)) =  	[[0A], [1A], [-1A]] 	 +  	[[-1A, -1A, -1A], [-I, 1A, -1A], [-I, -1A, -I]] 	* 	x_1
39.76/10.94	>>>
39.76/10.94	
39.76/10.94	   <<<
39.76/10.94	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-1A, -1A, 1A]] 	* 	x_1
39.76/10.94	>>>
39.76/10.94	
39.76/10.94	   <<<
39.76/10.94	 POL(c(x_1)) =  	[[0A], [-1A], [0A]] 	 +  	[[-1A, -1A, -1A], [1A, -1A, 1A], [-1A, -1A, -1A]] 	* 	x_1
39.76/10.94	>>>
39.76/10.94	
39.76/10.94	   <<<
39.76/10.94	 POL(b(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[-I, -I, 2A], [-I, -I, 2A], [-I, -1A, 2A]] 	* 	x_1
39.76/10.94	>>>
39.76/10.94	
39.76/10.94	
39.76/10.94	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
39.76/10.94	
39.76/10.94	   a(a(x1)) -> c(b(a(x1)))
39.76/10.94	   a(c(x1)) -> x1
39.76/10.94	   b(c(x1)) -> c(a(a(x1)))
39.76/10.94	
39.76/10.94	
39.76/10.94	----------------------------------------
39.76/10.94	
39.76/10.94	(6)
39.76/10.94	Obligation:
39.76/10.94	Q DP problem:
39.76/10.94	The TRS P consists of the following rules:
39.76/10.94	
39.76/10.94	   A(a(x1)) -> B(a(x1))
39.76/10.94	   B(c(x1)) -> A(a(x1))
39.76/10.94	
39.76/10.94	The TRS R consists of the following rules:
39.76/10.94	
39.76/10.94	   a(a(x1)) -> c(b(a(x1)))
39.76/10.94	   a(c(x1)) -> x1
39.76/10.94	   b(c(x1)) -> c(a(a(x1)))
39.76/10.94	
39.76/10.94	Q is empty.
39.76/10.94	We have to consider all minimal (P,Q,R)-chains.
39.76/10.94	----------------------------------------
39.76/10.94	
39.76/10.94	(7) QDPOrderProof (EQUIVALENT)
39.76/10.94	We use the reduction pair processor [LPAR04,JAR06].
39.76/10.94	
39.76/10.94	
39.76/10.94	The following pairs can be oriented strictly and are deleted.
39.76/10.94	
39.76/10.94	   B(c(x1)) -> A(a(x1))
39.76/10.94	The remaining pairs can at least be oriented weakly.
39.76/10.94	Used ordering:  Matrix interpretation [MATRO] with arctic integers [ARCTIC,STERNAGEL_THIEMANN_RTA14]:
39.76/10.94	
39.76/10.94	   <<<
39.76/10.94	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, -I, -1A]] 	* 	x_1
39.76/10.94	>>>
39.76/10.94	
39.76/10.94	   <<<
39.76/10.94	 POL(a(x_1)) =  	[[0A], [-I], [2A]] 	 +  	[[-I, -I, -1A], [-1A, -1A, 1A], [-I, -1A, 1A]] 	* 	x_1
39.76/10.94	>>>
39.76/10.94	
39.76/10.94	   <<<
39.76/10.94	 POL(B(x_1)) =  	[[-I]] 	 +  	[[1A, -I, -1A]] 	* 	x_1
39.76/10.94	>>>
39.76/10.94	
39.76/10.94	   <<<
39.76/10.94	 POL(c(x_1)) =  	[[1A], [0A], [-I]] 	 +  	[[-1A, -1A, 0A], [-I, -I, -I], [1A, -1A, -1A]] 	* 	x_1
39.76/10.94	>>>
39.76/10.94	
39.76/10.94	   <<<
39.76/10.94	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[2A, -I, -I], [-1A, -I, -I], [1A, -I, -1A]] 	* 	x_1
39.76/10.94	>>>
39.76/10.94	
39.76/10.94	
39.76/10.94	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
39.76/10.94	
39.76/10.94	   a(a(x1)) -> c(b(a(x1)))
39.76/10.94	   a(c(x1)) -> x1
39.76/10.94	   b(c(x1)) -> c(a(a(x1)))
39.76/10.94	
39.76/10.94	
39.76/10.94	----------------------------------------
39.76/10.94	
39.76/10.94	(8)
39.76/10.94	Obligation:
39.76/10.94	Q DP problem:
39.76/10.94	The TRS P consists of the following rules:
39.76/10.94	
39.76/10.94	   A(a(x1)) -> B(a(x1))
39.76/10.94	
39.76/10.94	The TRS R consists of the following rules:
39.76/10.94	
39.76/10.94	   a(a(x1)) -> c(b(a(x1)))
39.76/10.94	   a(c(x1)) -> x1
39.76/10.94	   b(c(x1)) -> c(a(a(x1)))
39.76/10.94	
39.76/10.94	Q is empty.
39.76/10.94	We have to consider all minimal (P,Q,R)-chains.
39.76/10.94	----------------------------------------
39.76/10.94	
39.76/10.94	(9) DependencyGraphProof (EQUIVALENT)
39.76/10.94	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
39.76/10.94	----------------------------------------
39.76/10.94	
39.76/10.94	(10)
39.76/10.94	TRUE
40.10/11.02	EOF