33.32/9.48	YES
33.36/9.51	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
33.36/9.51	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
33.36/9.51	
33.36/9.51	
33.36/9.51	Termination w.r.t. Q of the given QTRS could be proven:
33.36/9.51	
33.36/9.51	(0) QTRS
33.36/9.51	(1) DependencyPairsProof [EQUIVALENT, 29 ms]
33.36/9.51	(2) QDP
33.36/9.51	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
33.36/9.51	(4) AND
33.36/9.51	    (5) QDP
33.36/9.51	        (6) UsableRulesProof [EQUIVALENT, 1 ms]
33.36/9.51	        (7) QDP
33.36/9.51	        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
33.36/9.51	        (9) YES
33.36/9.51	    (10) QDP
33.36/9.51	        (11) QDPOrderProof [EQUIVALENT, 138 ms]
33.36/9.51	        (12) QDP
33.36/9.51	        (13) QDPOrderProof [EQUIVALENT, 0 ms]
33.36/9.51	        (14) QDP
33.36/9.51	        (15) PisEmptyProof [EQUIVALENT, 0 ms]
33.36/9.51	        (16) YES
33.36/9.51	
33.36/9.51	
33.36/9.51	----------------------------------------
33.36/9.51	
33.36/9.51	(0)
33.36/9.51	Obligation:
33.36/9.51	Q restricted rewrite system:
33.36/9.51	The TRS R consists of the following rules:
33.36/9.51	
33.36/9.51	   a(x1) -> x1
33.36/9.51	   a(b(x1)) -> c(b(a(a(x1))))
33.36/9.51	   b(x1) -> c(x1)
33.36/9.51	   c(c(x1)) -> b(x1)
33.36/9.51	
33.36/9.51	Q is empty.
33.36/9.51	
33.36/9.51	----------------------------------------
33.36/9.51	
33.36/9.51	(1) DependencyPairsProof (EQUIVALENT)
33.36/9.51	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
33.36/9.51	----------------------------------------
33.36/9.51	
33.36/9.51	(2)
33.36/9.51	Obligation:
33.36/9.51	Q DP problem:
33.36/9.51	The TRS P consists of the following rules:
33.36/9.51	
33.36/9.51	   A(b(x1)) -> C(b(a(a(x1))))
33.36/9.51	   A(b(x1)) -> B(a(a(x1)))
33.36/9.51	   A(b(x1)) -> A(a(x1))
33.36/9.51	   A(b(x1)) -> A(x1)
33.36/9.51	   B(x1) -> C(x1)
33.36/9.51	   C(c(x1)) -> B(x1)
33.36/9.51	
33.36/9.51	The TRS R consists of the following rules:
33.36/9.51	
33.36/9.51	   a(x1) -> x1
33.36/9.51	   a(b(x1)) -> c(b(a(a(x1))))
33.36/9.51	   b(x1) -> c(x1)
33.36/9.51	   c(c(x1)) -> b(x1)
33.36/9.51	
33.36/9.51	Q is empty.
33.36/9.51	We have to consider all minimal (P,Q,R)-chains.
33.36/9.51	----------------------------------------
33.36/9.51	
33.36/9.51	(3) DependencyGraphProof (EQUIVALENT)
33.36/9.51	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 2 less nodes.
33.36/9.51	----------------------------------------
33.36/9.51	
33.36/9.51	(4)
33.36/9.51	Complex Obligation (AND)
33.36/9.51	
33.36/9.51	----------------------------------------
33.36/9.51	
33.36/9.51	(5)
33.36/9.51	Obligation:
33.36/9.51	Q DP problem:
33.36/9.51	The TRS P consists of the following rules:
33.36/9.51	
33.36/9.51	   C(c(x1)) -> B(x1)
33.36/9.51	   B(x1) -> C(x1)
33.36/9.51	
33.36/9.51	The TRS R consists of the following rules:
33.36/9.51	
33.36/9.51	   a(x1) -> x1
33.36/9.51	   a(b(x1)) -> c(b(a(a(x1))))
33.36/9.51	   b(x1) -> c(x1)
33.36/9.51	   c(c(x1)) -> b(x1)
33.36/9.51	
33.36/9.51	Q is empty.
33.36/9.51	We have to consider all minimal (P,Q,R)-chains.
33.36/9.51	----------------------------------------
33.36/9.51	
33.36/9.51	(6) UsableRulesProof (EQUIVALENT)
33.36/9.51	We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
33.36/9.51	----------------------------------------
33.36/9.51	
33.36/9.51	(7)
33.36/9.51	Obligation:
33.36/9.51	Q DP problem:
33.36/9.51	The TRS P consists of the following rules:
33.36/9.51	
33.36/9.51	   C(c(x1)) -> B(x1)
33.36/9.51	   B(x1) -> C(x1)
33.36/9.51	
33.36/9.51	R is empty.
33.36/9.51	Q is empty.
33.36/9.51	We have to consider all minimal (P,Q,R)-chains.
33.36/9.51	----------------------------------------
33.36/9.51	
33.36/9.51	(8) QDPSizeChangeProof (EQUIVALENT)
33.36/9.51	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
33.36/9.51	
33.36/9.51	From the DPs we obtained the following set of size-change graphs:
33.36/9.51	*B(x1) -> C(x1)
33.36/9.51	The graph contains the following edges 1 >= 1
33.36/9.51	
33.36/9.51	
33.36/9.51	*C(c(x1)) -> B(x1)
33.36/9.51	The graph contains the following edges 1 > 1
33.36/9.51	
33.36/9.51	
33.36/9.51	----------------------------------------
33.36/9.51	
33.36/9.51	(9)
33.36/9.51	YES
33.36/9.51	
33.36/9.51	----------------------------------------
33.36/9.51	
33.36/9.51	(10)
33.36/9.51	Obligation:
33.36/9.51	Q DP problem:
33.36/9.51	The TRS P consists of the following rules:
33.36/9.51	
33.36/9.51	   A(b(x1)) -> A(x1)
33.36/9.51	   A(b(x1)) -> A(a(x1))
33.36/9.51	
33.36/9.51	The TRS R consists of the following rules:
33.36/9.51	
33.36/9.51	   a(x1) -> x1
33.36/9.51	   a(b(x1)) -> c(b(a(a(x1))))
33.36/9.51	   b(x1) -> c(x1)
33.36/9.51	   c(c(x1)) -> b(x1)
33.36/9.51	
33.36/9.51	Q is empty.
33.36/9.51	We have to consider all minimal (P,Q,R)-chains.
33.36/9.51	----------------------------------------
33.36/9.51	
33.36/9.51	(11) QDPOrderProof (EQUIVALENT)
33.36/9.51	We use the reduction pair processor [LPAR04,JAR06].
33.36/9.51	
33.36/9.51	
33.36/9.51	The following pairs can be oriented strictly and are deleted.
33.36/9.51	
33.36/9.51	   A(b(x1)) -> A(x1)
33.36/9.51	The remaining pairs can at least be oriented weakly.
33.36/9.51	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
33.36/9.51	
33.36/9.51	   <<<
33.36/9.51	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 1A]] 	* 	x_1
33.36/9.51	>>>
33.36/9.51	
33.36/9.51	   <<<
33.36/9.51	 POL(b(x_1)) =  	[[1A], [1A], [-I]] 	 +  	[[0A, -I, -I], [1A, 0A, 1A], [1A, 0A, 1A]] 	* 	x_1
33.36/9.51	>>>
33.36/9.51	
33.36/9.51	   <<<
33.36/9.51	 POL(a(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[0A, -I, 0A], [1A, 0A, 1A], [0A, -I, 0A]] 	* 	x_1
33.36/9.51	>>>
33.36/9.51	
33.36/9.51	   <<<
33.36/9.51	 POL(c(x_1)) =  	[[1A], [-I], [-I]] 	 +  	[[0A, -I, -I], [1A, 0A, 1A], [0A, 0A, 0A]] 	* 	x_1
33.36/9.51	>>>
33.36/9.51	
33.36/9.51	
33.36/9.51	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
33.36/9.51	
33.36/9.51	   a(x1) -> x1
33.36/9.51	   a(b(x1)) -> c(b(a(a(x1))))
33.36/9.51	   c(c(x1)) -> b(x1)
33.36/9.51	   b(x1) -> c(x1)
33.36/9.51	
33.36/9.51	
33.36/9.51	----------------------------------------
33.36/9.51	
33.36/9.51	(12)
33.36/9.51	Obligation:
33.36/9.51	Q DP problem:
33.36/9.51	The TRS P consists of the following rules:
33.36/9.51	
33.36/9.51	   A(b(x1)) -> A(a(x1))
33.36/9.51	
33.36/9.51	The TRS R consists of the following rules:
33.36/9.51	
33.36/9.51	   a(x1) -> x1
33.36/9.51	   a(b(x1)) -> c(b(a(a(x1))))
33.36/9.51	   b(x1) -> c(x1)
33.36/9.51	   c(c(x1)) -> b(x1)
33.36/9.51	
33.36/9.51	Q is empty.
33.36/9.51	We have to consider all minimal (P,Q,R)-chains.
33.36/9.51	----------------------------------------
33.36/9.51	
33.36/9.51	(13) QDPOrderProof (EQUIVALENT)
33.36/9.51	We use the reduction pair processor [LPAR04,JAR06].
33.36/9.51	
33.36/9.51	
33.36/9.51	The following pairs can be oriented strictly and are deleted.
33.36/9.51	
33.36/9.51	   A(b(x1)) -> A(a(x1))
33.36/9.51	The remaining pairs can at least be oriented weakly.
33.36/9.51	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
33.36/9.51	
33.36/9.51	   <<<
33.36/9.51	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, -I, 1A]] 	* 	x_1
33.36/9.51	>>>
33.36/9.51	
33.36/9.51	   <<<
33.36/9.51	 POL(b(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, 0A, -I], [0A, 0A, 1A], [0A, 0A, 1A]] 	* 	x_1
33.36/9.51	>>>
33.36/9.51	
33.36/9.51	   <<<
33.36/9.51	 POL(a(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, 1A], [-I, 0A, 1A], [-I, -I, 0A]] 	* 	x_1
33.36/9.51	>>>
33.36/9.51	
33.36/9.51	   <<<
33.36/9.51	 POL(c(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, 0A, -I], [0A, -I, 1A], [-I, 0A, 0A]] 	* 	x_1
33.36/9.51	>>>
33.36/9.51	
33.36/9.51	
33.36/9.51	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
33.36/9.51	
33.36/9.51	   a(x1) -> x1
33.36/9.51	   a(b(x1)) -> c(b(a(a(x1))))
33.36/9.51	   c(c(x1)) -> b(x1)
33.36/9.51	   b(x1) -> c(x1)
33.36/9.51	
33.36/9.51	
33.36/9.51	----------------------------------------
33.36/9.51	
33.36/9.51	(14)
33.36/9.51	Obligation:
33.36/9.51	Q DP problem:
33.36/9.51	P is empty.
33.36/9.51	The TRS R consists of the following rules:
33.36/9.51	
33.36/9.51	   a(x1) -> x1
33.36/9.51	   a(b(x1)) -> c(b(a(a(x1))))
33.36/9.51	   b(x1) -> c(x1)
33.36/9.51	   c(c(x1)) -> b(x1)
33.36/9.51	
33.36/9.51	Q is empty.
33.36/9.51	We have to consider all minimal (P,Q,R)-chains.
33.36/9.51	----------------------------------------
33.36/9.51	
33.36/9.51	(15) PisEmptyProof (EQUIVALENT)
33.36/9.51	The TRS P is empty. Hence, there is no (P,Q,R) chain.
33.36/9.51	----------------------------------------
33.36/9.51	
33.36/9.51	(16)
33.36/9.51	YES
33.68/9.63	EOF