19.30/5.93	YES
19.30/5.94	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
19.30/5.94	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
19.30/5.94	
19.30/5.94	
19.30/5.94	Termination w.r.t. Q of the given QTRS could be proven:
19.30/5.94	
19.30/5.94	(0) QTRS
19.30/5.94	(1) QTRS Reverse [EQUIVALENT, 0 ms]
19.30/5.94	(2) QTRS
19.30/5.94	(3) DependencyPairsProof [EQUIVALENT, 1 ms]
19.30/5.94	(4) QDP
19.30/5.94	(5) QDPOrderProof [EQUIVALENT, 25 ms]
19.30/5.94	(6) QDP
19.30/5.94	(7) QDPOrderProof [EQUIVALENT, 39 ms]
19.30/5.94	(8) QDP
19.30/5.94	(9) PisEmptyProof [EQUIVALENT, 0 ms]
19.30/5.94	(10) YES
19.30/5.94	
19.30/5.94	
19.30/5.94	----------------------------------------
19.30/5.94	
19.30/5.94	(0)
19.30/5.94	Obligation:
19.30/5.94	Q restricted rewrite system:
19.30/5.94	The TRS R consists of the following rules:
19.30/5.94	
19.30/5.94	   a(a(x1)) -> b(a(c(x1)))
19.30/5.94	   b(b(x1)) -> a(a(x1))
19.30/5.94	   c(b(x1)) -> a(x1)
19.30/5.94	
19.30/5.94	Q is empty.
19.30/5.94	
19.30/5.94	----------------------------------------
19.30/5.94	
19.30/5.94	(1) QTRS Reverse (EQUIVALENT)
19.30/5.94	We applied the QTRS Reverse Processor [REVERSE].
19.30/5.94	----------------------------------------
19.30/5.94	
19.30/5.94	(2)
19.30/5.94	Obligation:
19.30/5.94	Q restricted rewrite system:
19.30/5.94	The TRS R consists of the following rules:
19.30/5.94	
19.30/5.94	   a(a(x1)) -> c(a(b(x1)))
19.30/5.94	   b(b(x1)) -> a(a(x1))
19.30/5.94	   b(c(x1)) -> a(x1)
19.30/5.94	
19.30/5.94	Q is empty.
19.30/5.94	
19.30/5.94	----------------------------------------
19.30/5.94	
19.30/5.94	(3) DependencyPairsProof (EQUIVALENT)
19.30/5.94	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
19.30/5.94	----------------------------------------
19.30/5.94	
19.30/5.94	(4)
19.30/5.94	Obligation:
19.30/5.94	Q DP problem:
19.30/5.94	The TRS P consists of the following rules:
19.30/5.94	
19.30/5.94	   A(a(x1)) -> A(b(x1))
19.30/5.94	   A(a(x1)) -> B(x1)
19.30/5.94	   B(b(x1)) -> A(a(x1))
19.30/5.94	   B(b(x1)) -> A(x1)
19.30/5.94	   B(c(x1)) -> A(x1)
19.30/5.94	
19.30/5.94	The TRS R consists of the following rules:
19.30/5.94	
19.30/5.94	   a(a(x1)) -> c(a(b(x1)))
19.30/5.94	   b(b(x1)) -> a(a(x1))
19.30/5.94	   b(c(x1)) -> a(x1)
19.30/5.94	
19.30/5.94	Q is empty.
19.30/5.94	We have to consider all minimal (P,Q,R)-chains.
19.30/5.94	----------------------------------------
19.30/5.94	
19.30/5.94	(5) QDPOrderProof (EQUIVALENT)
19.30/5.94	We use the reduction pair processor [LPAR04,JAR06].
19.30/5.94	
19.30/5.94	
19.30/5.94	The following pairs can be oriented strictly and are deleted.
19.30/5.94	
19.30/5.94	   A(a(x1)) -> B(x1)
19.30/5.94	   B(b(x1)) -> A(a(x1))
19.30/5.94	   B(b(x1)) -> A(x1)
19.30/5.94	   B(c(x1)) -> A(x1)
19.30/5.94	The remaining pairs can at least be oriented weakly.
19.30/5.94	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
19.30/5.94	
19.30/5.94	POL( A_1(x_1) ) = x_1 + 1
19.30/5.94	POL( b_1(x_1) ) = x_1 + 2
19.30/5.94	POL( a_1(x_1) ) = x_1 + 2
19.30/5.94	POL( c_1(x_1) ) = x_1
19.30/5.94	POL( B_1(x_1) ) = x_1 + 2
19.30/5.94	
19.30/5.94	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
19.30/5.94	
19.30/5.94	   b(b(x1)) -> a(a(x1))
19.30/5.94	   b(c(x1)) -> a(x1)
19.30/5.94	   a(a(x1)) -> c(a(b(x1)))
19.30/5.94	
19.30/5.94	
19.30/5.94	----------------------------------------
19.30/5.94	
19.30/5.94	(6)
19.30/5.94	Obligation:
19.30/5.94	Q DP problem:
19.30/5.94	The TRS P consists of the following rules:
19.30/5.94	
19.30/5.94	   A(a(x1)) -> A(b(x1))
19.30/5.94	
19.30/5.94	The TRS R consists of the following rules:
19.30/5.94	
19.30/5.94	   a(a(x1)) -> c(a(b(x1)))
19.30/5.94	   b(b(x1)) -> a(a(x1))
19.30/5.94	   b(c(x1)) -> a(x1)
19.30/5.94	
19.30/5.94	Q is empty.
19.30/5.94	We have to consider all minimal (P,Q,R)-chains.
19.30/5.94	----------------------------------------
19.30/5.94	
19.30/5.94	(7) QDPOrderProof (EQUIVALENT)
19.30/5.94	We use the reduction pair processor [LPAR04,JAR06].
19.30/5.94	
19.30/5.94	
19.30/5.94	The following pairs can be oriented strictly and are deleted.
19.30/5.94	
19.30/5.94	   A(a(x1)) -> A(b(x1))
19.30/5.94	The remaining pairs can at least be oriented weakly.
19.30/5.94	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
19.30/5.94	
19.30/5.94	   <<<
19.30/5.94	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, -I, -I]] 	* 	x_1
19.30/5.94	>>>
19.30/5.94	
19.30/5.94	   <<<
19.30/5.94	 POL(a(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[-I, 1A, -I], [0A, -I, -I], [1A, 0A, 1A]] 	* 	x_1
19.30/5.94	>>>
19.30/5.94	
19.30/5.94	   <<<
19.30/5.94	 POL(b(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[-I, 0A, -I], [1A, 0A, 0A], [0A, 1A, 1A]] 	* 	x_1
19.30/5.94	>>>
19.30/5.94	
19.30/5.94	   <<<
19.30/5.94	 POL(c(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[-I, -I, -I], [-I, 1A, -I], [0A, 0A, 0A]] 	* 	x_1
19.30/5.94	>>>
19.30/5.94	
19.30/5.94	
19.30/5.94	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
19.30/5.94	
19.30/5.94	   b(b(x1)) -> a(a(x1))
19.30/5.94	   b(c(x1)) -> a(x1)
19.30/5.94	   a(a(x1)) -> c(a(b(x1)))
19.30/5.94	
19.30/5.94	
19.30/5.94	----------------------------------------
19.30/5.94	
19.30/5.94	(8)
19.30/5.94	Obligation:
19.30/5.94	Q DP problem:
19.30/5.94	P is empty.
19.30/5.94	The TRS R consists of the following rules:
19.30/5.94	
19.30/5.94	   a(a(x1)) -> c(a(b(x1)))
19.30/5.94	   b(b(x1)) -> a(a(x1))
19.30/5.94	   b(c(x1)) -> a(x1)
19.30/5.94	
19.30/5.94	Q is empty.
19.30/5.94	We have to consider all minimal (P,Q,R)-chains.
19.30/5.94	----------------------------------------
19.30/5.94	
19.30/5.94	(9) PisEmptyProof (EQUIVALENT)
19.30/5.94	The TRS P is empty. Hence, there is no (P,Q,R) chain.
19.30/5.94	----------------------------------------
19.30/5.94	
19.30/5.94	(10)
19.30/5.94	YES
19.76/6.03	EOF