32.88/9.29	YES
34.41/9.67	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
34.41/9.67	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
34.41/9.67	
34.41/9.67	
34.41/9.67	Termination w.r.t. Q of the given QTRS could be proven:
34.41/9.67	
34.41/9.67	(0) QTRS
34.41/9.67	(1) DependencyPairsProof [EQUIVALENT, 5 ms]
34.41/9.67	(2) QDP
34.41/9.67	(3) QDPOrderProof [EQUIVALENT, 28 ms]
34.41/9.67	(4) QDP
34.41/9.67	(5) QDPOrderProof [EQUIVALENT, 0 ms]
34.41/9.67	(6) QDP
34.41/9.67	(7) PisEmptyProof [EQUIVALENT, 0 ms]
34.41/9.67	(8) YES
34.41/9.67	
34.41/9.67	
34.41/9.67	----------------------------------------
34.41/9.67	
34.41/9.67	(0)
34.41/9.67	Obligation:
34.41/9.67	Q restricted rewrite system:
34.41/9.67	The TRS R consists of the following rules:
34.41/9.67	
34.41/9.67	   a(x1) -> x1
34.41/9.67	   a(b(x1)) -> c(b(x1))
34.41/9.67	   a(c(c(x1))) -> c(c(a(a(x1))))
34.41/9.67	
34.41/9.67	Q is empty.
34.41/9.67	
34.41/9.67	----------------------------------------
34.41/9.67	
34.41/9.67	(1) DependencyPairsProof (EQUIVALENT)
34.41/9.67	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
34.41/9.67	----------------------------------------
34.41/9.67	
34.41/9.67	(2)
34.41/9.67	Obligation:
34.41/9.67	Q DP problem:
34.41/9.67	The TRS P consists of the following rules:
34.41/9.67	
34.41/9.67	   A(c(c(x1))) -> A(a(x1))
34.41/9.67	   A(c(c(x1))) -> A(x1)
34.41/9.67	
34.41/9.67	The TRS R consists of the following rules:
34.41/9.67	
34.41/9.67	   a(x1) -> x1
34.41/9.67	   a(b(x1)) -> c(b(x1))
34.41/9.67	   a(c(c(x1))) -> c(c(a(a(x1))))
34.41/9.67	
34.41/9.67	Q is empty.
34.41/9.67	We have to consider all minimal (P,Q,R)-chains.
34.41/9.67	----------------------------------------
34.41/9.67	
34.41/9.67	(3) QDPOrderProof (EQUIVALENT)
34.41/9.67	We use the reduction pair processor [LPAR04,JAR06].
34.41/9.67	
34.41/9.67	
34.41/9.67	The following pairs can be oriented strictly and are deleted.
34.41/9.67	
34.41/9.67	   A(c(c(x1))) -> A(x1)
34.41/9.67	The remaining pairs can at least be oriented weakly.
34.41/9.67	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
34.41/9.67	
34.41/9.67	   <<<
34.41/9.67	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
34.41/9.67	>>>
34.41/9.67	
34.41/9.67	   <<<
34.41/9.67	 POL(c(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[0A, 1A, -I], [0A, 0A, -I], [-I, -I, 0A]] 	* 	x_1
34.41/9.67	>>>
34.41/9.67	
34.41/9.67	   <<<
34.41/9.67	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, -I], [0A, 0A, -I], [0A, -I, 0A]] 	* 	x_1
34.41/9.67	>>>
34.41/9.67	
34.41/9.67	   <<<
34.41/9.67	 POL(b(x_1)) =  	[[1A], [0A], [-I]] 	 +  	[[0A, 0A, -I], [-I, -I, -I], [-I, -I, -I]] 	* 	x_1
34.41/9.67	>>>
34.41/9.67	
34.41/9.67	
34.41/9.67	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
34.41/9.67	
34.41/9.67	   a(x1) -> x1
34.41/9.67	   a(b(x1)) -> c(b(x1))
34.41/9.67	   a(c(c(x1))) -> c(c(a(a(x1))))
34.41/9.67	
34.41/9.67	
34.41/9.67	----------------------------------------
34.41/9.67	
34.41/9.67	(4)
34.41/9.67	Obligation:
34.41/9.67	Q DP problem:
34.41/9.67	The TRS P consists of the following rules:
34.41/9.67	
34.41/9.67	   A(c(c(x1))) -> A(a(x1))
34.41/9.67	
34.41/9.67	The TRS R consists of the following rules:
34.41/9.67	
34.41/9.67	   a(x1) -> x1
34.41/9.67	   a(b(x1)) -> c(b(x1))
34.41/9.67	   a(c(c(x1))) -> c(c(a(a(x1))))
34.41/9.67	
34.41/9.67	Q is empty.
34.41/9.67	We have to consider all minimal (P,Q,R)-chains.
34.41/9.67	----------------------------------------
34.41/9.67	
34.41/9.67	(5) QDPOrderProof (EQUIVALENT)
34.41/9.67	We use the reduction pair processor [LPAR04,JAR06].
34.41/9.67	
34.41/9.67	
34.41/9.67	The following pairs can be oriented strictly and are deleted.
34.41/9.67	
34.41/9.67	   A(c(c(x1))) -> A(a(x1))
34.41/9.67	The remaining pairs can at least be oriented weakly.
34.41/9.67	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
34.41/9.67	
34.41/9.67	   <<<
34.41/9.67	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
34.41/9.67	>>>
34.41/9.67	
34.41/9.67	   <<<
34.41/9.67	 POL(c(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[-I, 1A, 0A], [0A, -I, -I], [-I, -I, -I]] 	* 	x_1
34.41/9.67	>>>
34.41/9.67	
34.41/9.67	   <<<
34.41/9.67	 POL(a(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[0A, -I, -I], [0A, 0A, -I], [1A, 0A, 0A]] 	* 	x_1
34.41/9.67	>>>
34.41/9.67	
34.41/9.67	   <<<
34.41/9.67	 POL(b(x_1)) =  	[[1A], [0A], [-I]] 	 +  	[[0A, 0A, -I], [-I, -I, -I], [0A, 0A, -I]] 	* 	x_1
34.41/9.67	>>>
34.41/9.67	
34.41/9.67	
34.41/9.67	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
34.41/9.67	
34.41/9.67	   a(x1) -> x1
34.41/9.67	   a(b(x1)) -> c(b(x1))
34.41/9.67	   a(c(c(x1))) -> c(c(a(a(x1))))
34.41/9.67	
34.41/9.67	
34.41/9.67	----------------------------------------
34.41/9.67	
34.41/9.67	(6)
34.41/9.67	Obligation:
34.41/9.67	Q DP problem:
34.41/9.67	P is empty.
34.41/9.67	The TRS R consists of the following rules:
34.41/9.67	
34.41/9.67	   a(x1) -> x1
34.41/9.67	   a(b(x1)) -> c(b(x1))
34.41/9.67	   a(c(c(x1))) -> c(c(a(a(x1))))
34.41/9.67	
34.41/9.67	Q is empty.
34.41/9.67	We have to consider all minimal (P,Q,R)-chains.
34.41/9.67	----------------------------------------
34.41/9.67	
34.41/9.67	(7) PisEmptyProof (EQUIVALENT)
34.41/9.67	The TRS P is empty. Hence, there is no (P,Q,R) chain.
34.41/9.67	----------------------------------------
34.41/9.67	
34.41/9.67	(8)
34.41/9.67	YES
34.83/9.83	EOF