26.92/7.79	YES
26.92/7.80	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
26.92/7.80	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
26.92/7.80	
26.92/7.80	
26.92/7.80	Termination w.r.t. Q of the given QTRS could be proven:
26.92/7.80	
26.92/7.80	(0) QTRS
26.92/7.80	(1) QTRS Reverse [EQUIVALENT, 0 ms]
26.92/7.80	(2) QTRS
26.92/7.80	(3) DependencyPairsProof [EQUIVALENT, 3 ms]
26.92/7.80	(4) QDP
26.92/7.80	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
26.92/7.80	(6) QDP
26.92/7.80	(7) QDPOrderProof [EQUIVALENT, 98 ms]
26.92/7.80	(8) QDP
26.92/7.80	(9) PisEmptyProof [EQUIVALENT, 0 ms]
26.92/7.80	(10) YES
26.92/7.80	
26.92/7.80	
26.92/7.80	----------------------------------------
26.92/7.80	
26.92/7.80	(0)
26.92/7.80	Obligation:
26.92/7.80	Q restricted rewrite system:
26.92/7.80	The TRS R consists of the following rules:
26.92/7.80	
26.92/7.80	   a(x1) -> x1
26.92/7.80	   a(b(x1)) -> b(b(a(c(x1))))
26.92/7.80	   b(b(x1)) -> a(x1)
26.92/7.80	   c(c(x1)) -> x1
26.92/7.80	
26.92/7.80	Q is empty.
26.92/7.80	
26.92/7.80	----------------------------------------
26.92/7.80	
26.92/7.80	(1) QTRS Reverse (EQUIVALENT)
26.92/7.80	We applied the QTRS Reverse Processor [REVERSE].
26.92/7.80	----------------------------------------
26.92/7.80	
26.92/7.80	(2)
26.92/7.80	Obligation:
26.92/7.80	Q restricted rewrite system:
26.92/7.80	The TRS R consists of the following rules:
26.92/7.80	
26.92/7.80	   a(x1) -> x1
26.92/7.80	   b(a(x1)) -> c(a(b(b(x1))))
26.92/7.80	   b(b(x1)) -> a(x1)
26.92/7.80	   c(c(x1)) -> x1
26.92/7.80	
26.92/7.80	Q is empty.
26.92/7.80	
26.92/7.80	----------------------------------------
26.92/7.80	
26.92/7.80	(3) DependencyPairsProof (EQUIVALENT)
26.92/7.80	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
26.92/7.80	----------------------------------------
26.92/7.80	
26.92/7.80	(4)
26.92/7.80	Obligation:
26.92/7.80	Q DP problem:
26.92/7.80	The TRS P consists of the following rules:
26.92/7.80	
26.92/7.80	   B(a(x1)) -> C(a(b(b(x1))))
26.92/7.80	   B(a(x1)) -> A(b(b(x1)))
26.92/7.80	   B(a(x1)) -> B(b(x1))
26.92/7.80	   B(a(x1)) -> B(x1)
26.92/7.80	   B(b(x1)) -> A(x1)
26.92/7.80	
26.92/7.80	The TRS R consists of the following rules:
26.92/7.80	
26.92/7.80	   a(x1) -> x1
26.92/7.80	   b(a(x1)) -> c(a(b(b(x1))))
26.92/7.80	   b(b(x1)) -> a(x1)
26.92/7.80	   c(c(x1)) -> x1
26.92/7.80	
26.92/7.80	Q is empty.
26.92/7.80	We have to consider all minimal (P,Q,R)-chains.
26.92/7.80	----------------------------------------
26.92/7.80	
26.92/7.80	(5) DependencyGraphProof (EQUIVALENT)
26.92/7.80	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
26.92/7.80	----------------------------------------
26.92/7.80	
26.92/7.80	(6)
26.92/7.80	Obligation:
26.92/7.80	Q DP problem:
26.92/7.80	The TRS P consists of the following rules:
26.92/7.80	
26.92/7.80	   B(a(x1)) -> B(x1)
26.92/7.80	   B(a(x1)) -> B(b(x1))
26.92/7.80	
26.92/7.80	The TRS R consists of the following rules:
26.92/7.80	
26.92/7.80	   a(x1) -> x1
26.92/7.80	   b(a(x1)) -> c(a(b(b(x1))))
26.92/7.80	   b(b(x1)) -> a(x1)
26.92/7.80	   c(c(x1)) -> x1
26.92/7.80	
26.92/7.80	Q is empty.
26.92/7.80	We have to consider all minimal (P,Q,R)-chains.
26.92/7.80	----------------------------------------
26.92/7.80	
26.92/7.80	(7) QDPOrderProof (EQUIVALENT)
26.92/7.80	We use the reduction pair processor [LPAR04,JAR06].
26.92/7.80	
26.92/7.80	
26.92/7.80	The following pairs can be oriented strictly and are deleted.
26.92/7.80	
26.92/7.80	   B(a(x1)) -> B(x1)
26.92/7.80	   B(a(x1)) -> B(b(x1))
26.92/7.80	The remaining pairs can at least be oriented weakly.
26.92/7.80	Used ordering:  Matrix interpretation [MATRO] with arctic integers [ARCTIC,STERNAGEL_THIEMANN_RTA14]:
26.92/7.80	
26.92/7.80	   <<<
26.92/7.80	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-1A, -I, -I]] 	* 	x_1
26.92/7.80	>>>
26.92/7.80	
26.92/7.80	   <<<
26.92/7.80	 POL(a(x_1)) =  	[[2A], [2A], [-1A]] 	 +  	[[2A, -I, 0A], [1A, 0A, -1A], [-I, -I, 0A]] 	* 	x_1
26.92/7.80	>>>
26.92/7.80	
26.92/7.80	   <<<
26.92/7.80	 POL(b(x_1)) =  	[[1A], [-I], [-I]] 	 +  	[[1A, -I, -1A], [2A, -I, 1A], [1A, -1A, -I]] 	* 	x_1
26.92/7.80	>>>
26.92/7.80	
26.92/7.80	   <<<
26.92/7.80	 POL(c(x_1)) =  	[[1A], [-I], [-I]] 	 +  	[[-I, -I, 1A], [-I, -I, 1A], [-1A, -1A, -I]] 	* 	x_1
26.92/7.80	>>>
26.92/7.80	
26.92/7.80	
26.92/7.80	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
26.92/7.80	
26.92/7.80	   b(a(x1)) -> c(a(b(b(x1))))
26.92/7.80	   b(b(x1)) -> a(x1)
26.92/7.80	   a(x1) -> x1
26.92/7.80	   c(c(x1)) -> x1
26.92/7.80	
26.92/7.80	
26.92/7.80	----------------------------------------
26.92/7.80	
26.92/7.80	(8)
26.92/7.80	Obligation:
26.92/7.80	Q DP problem:
26.92/7.80	P is empty.
26.92/7.80	The TRS R consists of the following rules:
26.92/7.80	
26.92/7.80	   a(x1) -> x1
26.92/7.80	   b(a(x1)) -> c(a(b(b(x1))))
26.92/7.80	   b(b(x1)) -> a(x1)
26.92/7.80	   c(c(x1)) -> x1
26.92/7.80	
26.92/7.80	Q is empty.
26.92/7.80	We have to consider all minimal (P,Q,R)-chains.
26.92/7.80	----------------------------------------
26.92/7.80	
26.92/7.80	(9) PisEmptyProof (EQUIVALENT)
26.92/7.80	The TRS P is empty. Hence, there is no (P,Q,R) chain.
26.92/7.80	----------------------------------------
26.92/7.80	
26.92/7.80	(10)
26.92/7.80	YES
27.09/7.85	EOF