37.23/10.37	YES
37.48/10.40	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
37.48/10.40	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
37.48/10.40	
37.48/10.40	
37.48/10.40	Termination w.r.t. Q of the given QTRS could be proven:
37.48/10.40	
37.48/10.40	(0) QTRS
37.48/10.40	(1) QTRS Reverse [EQUIVALENT, 0 ms]
37.48/10.40	(2) QTRS
37.48/10.40	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
37.48/10.40	(4) QDP
37.48/10.40	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
37.48/10.40	(6) QDP
37.48/10.40	(7) QDPOrderProof [EQUIVALENT, 113 ms]
37.48/10.40	(8) QDP
37.48/10.40	(9) QDPOrderProof [EQUIVALENT, 55 ms]
37.48/10.40	(10) QDP
37.48/10.40	(11) PisEmptyProof [EQUIVALENT, 0 ms]
37.48/10.40	(12) YES
37.48/10.40	
37.48/10.40	
37.48/10.40	----------------------------------------
37.48/10.40	
37.48/10.40	(0)
37.48/10.40	Obligation:
37.48/10.40	Q restricted rewrite system:
37.48/10.40	The TRS R consists of the following rules:
37.48/10.40	
37.48/10.40	   a(x1) -> x1
37.48/10.40	   a(x1) -> b(x1)
37.48/10.40	   b(a(c(x1))) -> c(c(a(a(x1))))
37.48/10.40	   c(x1) -> b(x1)
37.48/10.40	
37.48/10.40	Q is empty.
37.48/10.40	
37.48/10.40	----------------------------------------
37.48/10.40	
37.48/10.40	(1) QTRS Reverse (EQUIVALENT)
37.48/10.40	We applied the QTRS Reverse Processor [REVERSE].
37.48/10.40	----------------------------------------
37.48/10.40	
37.48/10.40	(2)
37.48/10.40	Obligation:
37.48/10.40	Q restricted rewrite system:
37.48/10.40	The TRS R consists of the following rules:
37.48/10.40	
37.48/10.40	   a(x1) -> x1
37.48/10.40	   a(x1) -> b(x1)
37.48/10.40	   c(a(b(x1))) -> a(a(c(c(x1))))
37.48/10.40	   c(x1) -> b(x1)
37.48/10.40	
37.48/10.40	Q is empty.
37.48/10.40	
37.48/10.40	----------------------------------------
37.48/10.40	
37.48/10.40	(3) DependencyPairsProof (EQUIVALENT)
37.48/10.40	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
37.48/10.40	----------------------------------------
37.48/10.40	
37.48/10.40	(4)
37.48/10.40	Obligation:
37.48/10.40	Q DP problem:
37.48/10.40	The TRS P consists of the following rules:
37.48/10.40	
37.48/10.40	   C(a(b(x1))) -> A(a(c(c(x1))))
37.48/10.40	   C(a(b(x1))) -> A(c(c(x1)))
37.48/10.40	   C(a(b(x1))) -> C(c(x1))
37.48/10.40	   C(a(b(x1))) -> C(x1)
37.48/10.40	
37.48/10.40	The TRS R consists of the following rules:
37.48/10.40	
37.48/10.40	   a(x1) -> x1
37.48/10.40	   a(x1) -> b(x1)
37.48/10.40	   c(a(b(x1))) -> a(a(c(c(x1))))
37.48/10.40	   c(x1) -> b(x1)
37.48/10.40	
37.48/10.40	Q is empty.
37.48/10.40	We have to consider all minimal (P,Q,R)-chains.
37.48/10.40	----------------------------------------
37.48/10.40	
37.48/10.40	(5) DependencyGraphProof (EQUIVALENT)
37.48/10.40	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
37.48/10.40	----------------------------------------
37.48/10.40	
37.48/10.40	(6)
37.48/10.40	Obligation:
37.48/10.40	Q DP problem:
37.48/10.40	The TRS P consists of the following rules:
37.48/10.40	
37.48/10.40	   C(a(b(x1))) -> C(x1)
37.48/10.40	   C(a(b(x1))) -> C(c(x1))
37.48/10.40	
37.48/10.40	The TRS R consists of the following rules:
37.48/10.40	
37.48/10.40	   a(x1) -> x1
37.48/10.40	   a(x1) -> b(x1)
37.48/10.40	   c(a(b(x1))) -> a(a(c(c(x1))))
37.48/10.40	   c(x1) -> b(x1)
37.48/10.40	
37.48/10.40	Q is empty.
37.48/10.40	We have to consider all minimal (P,Q,R)-chains.
37.48/10.40	----------------------------------------
37.48/10.40	
37.48/10.40	(7) QDPOrderProof (EQUIVALENT)
37.48/10.40	We use the reduction pair processor [LPAR04,JAR06].
37.48/10.40	
37.48/10.40	
37.48/10.40	The following pairs can be oriented strictly and are deleted.
37.48/10.40	
37.48/10.40	   C(a(b(x1))) -> C(x1)
37.48/10.40	The remaining pairs can at least be oriented weakly.
37.48/10.40	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
37.48/10.40	
37.48/10.40	   <<<
37.48/10.40	 POL(C(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
37.48/10.40	>>>
37.48/10.40	
37.48/10.40	   <<<
37.48/10.40	 POL(a(x_1)) =  	[[0A], [-I], [1A]] 	 +  	[[0A, 0A, 0A], [1A, 0A, -I], [0A, 0A, 0A]] 	* 	x_1
37.48/10.40	>>>
37.48/10.40	
37.48/10.40	   <<<
37.48/10.40	 POL(b(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, 0A, 0A], [-I, -I, -I], [0A, 0A, 0A]] 	* 	x_1
37.48/10.40	>>>
37.48/10.40	
37.48/10.40	   <<<
37.48/10.40	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
37.48/10.40	>>>
37.48/10.40	
37.48/10.40	
37.48/10.40	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
37.48/10.40	
37.48/10.40	   c(a(b(x1))) -> a(a(c(c(x1))))
37.48/10.40	   c(x1) -> b(x1)
37.48/10.40	   a(x1) -> x1
37.48/10.40	   a(x1) -> b(x1)
37.48/10.40	
37.48/10.40	
37.48/10.40	----------------------------------------
37.48/10.40	
37.48/10.40	(8)
37.48/10.40	Obligation:
37.48/10.40	Q DP problem:
37.48/10.40	The TRS P consists of the following rules:
37.48/10.40	
37.48/10.40	   C(a(b(x1))) -> C(c(x1))
37.48/10.40	
37.48/10.40	The TRS R consists of the following rules:
37.48/10.40	
37.48/10.40	   a(x1) -> x1
37.48/10.40	   a(x1) -> b(x1)
37.48/10.40	   c(a(b(x1))) -> a(a(c(c(x1))))
37.48/10.40	   c(x1) -> b(x1)
37.48/10.40	
37.48/10.40	Q is empty.
37.48/10.40	We have to consider all minimal (P,Q,R)-chains.
37.48/10.40	----------------------------------------
37.48/10.40	
37.48/10.40	(9) QDPOrderProof (EQUIVALENT)
37.48/10.40	We use the reduction pair processor [LPAR04,JAR06].
37.48/10.40	
37.48/10.40	
37.48/10.40	The following pairs can be oriented strictly and are deleted.
37.48/10.40	
37.48/10.40	   C(a(b(x1))) -> C(c(x1))
37.48/10.40	The remaining pairs can at least be oriented weakly.
37.48/10.40	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
37.48/10.40	
37.48/10.40	   <<<
37.48/10.40	 POL(C(x_1)) =  	[[-I]] 	 +  	[[0A, -I, -I]] 	* 	x_1
37.48/10.40	>>>
37.48/10.40	
37.48/10.40	   <<<
37.48/10.40	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, -I], [1A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
37.48/10.40	>>>
37.48/10.40	
37.48/10.40	   <<<
37.48/10.40	 POL(b(x_1)) =  	[[-I], [0A], [0A]] 	 +  	[[0A, -I, -I], [1A, -I, 0A], [-I, -I, -I]] 	* 	x_1
37.48/10.40	>>>
37.48/10.40	
37.48/10.40	   <<<
37.48/10.40	 POL(c(x_1)) =  	[[-I], [0A], [0A]] 	 +  	[[0A, -I, -I], [1A, -I, 0A], [0A, -I, -I]] 	* 	x_1
37.48/10.40	>>>
37.48/10.40	
37.48/10.40	
37.48/10.40	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
37.48/10.40	
37.48/10.40	   c(a(b(x1))) -> a(a(c(c(x1))))
37.48/10.40	   c(x1) -> b(x1)
37.48/10.40	   a(x1) -> x1
37.48/10.40	   a(x1) -> b(x1)
37.48/10.40	
37.48/10.40	
37.48/10.40	----------------------------------------
37.48/10.40	
37.48/10.40	(10)
37.48/10.40	Obligation:
37.48/10.40	Q DP problem:
37.48/10.40	P is empty.
37.48/10.40	The TRS R consists of the following rules:
37.48/10.40	
37.48/10.40	   a(x1) -> x1
37.48/10.40	   a(x1) -> b(x1)
37.48/10.40	   c(a(b(x1))) -> a(a(c(c(x1))))
37.48/10.40	   c(x1) -> b(x1)
37.48/10.40	
37.48/10.40	Q is empty.
37.48/10.40	We have to consider all minimal (P,Q,R)-chains.
37.48/10.40	----------------------------------------
37.48/10.40	
37.48/10.40	(11) PisEmptyProof (EQUIVALENT)
37.48/10.40	The TRS P is empty. Hence, there is no (P,Q,R) chain.
37.48/10.40	----------------------------------------
37.48/10.40	
37.48/10.40	(12)
37.48/10.40	YES
37.65/10.47	EOF