31.01/8.75	YES
31.21/8.81	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
31.21/8.81	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
31.21/8.81	
31.21/8.81	
31.21/8.81	Termination w.r.t. Q of the given QTRS could be proven:
31.21/8.81	
31.21/8.81	(0) QTRS
31.21/8.81	(1) DependencyPairsProof [EQUIVALENT, 4 ms]
31.21/8.81	(2) QDP
31.21/8.81	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
31.21/8.81	(4) AND
31.21/8.81	    (5) QDP
31.21/8.81	        (6) UsableRulesProof [EQUIVALENT, 0 ms]
31.21/8.81	        (7) QDP
31.21/8.81	        (8) QDPSizeChangeProof [EQUIVALENT, 2 ms]
31.21/8.81	        (9) YES
31.21/8.81	    (10) QDP
31.21/8.81	        (11) QDPOrderProof [EQUIVALENT, 108 ms]
31.21/8.81	        (12) QDP
31.21/8.81	        (13) QDPOrderProof [EQUIVALENT, 0 ms]
31.21/8.81	        (14) QDP
31.21/8.81	        (15) PisEmptyProof [EQUIVALENT, 0 ms]
31.21/8.81	        (16) YES
31.21/8.81	
31.21/8.81	
31.21/8.81	----------------------------------------
31.21/8.81	
31.21/8.81	(0)
31.21/8.81	Obligation:
31.21/8.81	Q restricted rewrite system:
31.21/8.81	The TRS R consists of the following rules:
31.21/8.81	
31.21/8.81	   a(a(x1)) -> b(x1)
31.21/8.81	   b(c(x1)) -> a(x1)
31.21/8.81	   c(b(x1)) -> b(c(c(a(x1))))
31.21/8.81	
31.21/8.81	Q is empty.
31.21/8.81	
31.21/8.81	----------------------------------------
31.21/8.81	
31.21/8.81	(1) DependencyPairsProof (EQUIVALENT)
31.21/8.81	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
31.21/8.81	----------------------------------------
31.21/8.81	
31.21/8.81	(2)
31.21/8.81	Obligation:
31.21/8.81	Q DP problem:
31.21/8.81	The TRS P consists of the following rules:
31.21/8.81	
31.21/8.81	   A(a(x1)) -> B(x1)
31.21/8.81	   B(c(x1)) -> A(x1)
31.21/8.81	   C(b(x1)) -> B(c(c(a(x1))))
31.21/8.81	   C(b(x1)) -> C(c(a(x1)))
31.21/8.81	   C(b(x1)) -> C(a(x1))
31.21/8.81	   C(b(x1)) -> A(x1)
31.21/8.81	
31.21/8.81	The TRS R consists of the following rules:
31.21/8.81	
31.21/8.81	   a(a(x1)) -> b(x1)
31.21/8.81	   b(c(x1)) -> a(x1)
31.21/8.81	   c(b(x1)) -> b(c(c(a(x1))))
31.21/8.81	
31.21/8.81	Q is empty.
31.21/8.81	We have to consider all minimal (P,Q,R)-chains.
31.21/8.81	----------------------------------------
31.21/8.81	
31.21/8.81	(3) DependencyGraphProof (EQUIVALENT)
31.21/8.81	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 2 less nodes.
31.21/8.81	----------------------------------------
31.21/8.81	
31.21/8.81	(4)
31.21/8.81	Complex Obligation (AND)
31.21/8.81	
31.21/8.81	----------------------------------------
31.21/8.81	
31.21/8.81	(5)
31.21/8.81	Obligation:
31.21/8.81	Q DP problem:
31.21/8.81	The TRS P consists of the following rules:
31.21/8.81	
31.21/8.81	   B(c(x1)) -> A(x1)
31.21/8.81	   A(a(x1)) -> B(x1)
31.21/8.81	
31.21/8.81	The TRS R consists of the following rules:
31.21/8.81	
31.21/8.81	   a(a(x1)) -> b(x1)
31.21/8.81	   b(c(x1)) -> a(x1)
31.21/8.81	   c(b(x1)) -> b(c(c(a(x1))))
31.21/8.81	
31.21/8.81	Q is empty.
31.21/8.81	We have to consider all minimal (P,Q,R)-chains.
31.21/8.81	----------------------------------------
31.21/8.81	
31.21/8.81	(6) UsableRulesProof (EQUIVALENT)
31.21/8.81	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.
31.21/8.81	----------------------------------------
31.21/8.81	
31.21/8.81	(7)
31.21/8.81	Obligation:
31.21/8.81	Q DP problem:
31.21/8.81	The TRS P consists of the following rules:
31.21/8.81	
31.21/8.81	   B(c(x1)) -> A(x1)
31.21/8.81	   A(a(x1)) -> B(x1)
31.21/8.81	
31.21/8.81	R is empty.
31.21/8.81	Q is empty.
31.21/8.81	We have to consider all minimal (P,Q,R)-chains.
31.21/8.81	----------------------------------------
31.21/8.81	
31.21/8.81	(8) QDPSizeChangeProof (EQUIVALENT)
31.21/8.81	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. 
31.21/8.81	
31.21/8.81	From the DPs we obtained the following set of size-change graphs:
31.21/8.81	*A(a(x1)) -> B(x1)
31.21/8.81	The graph contains the following edges 1 > 1
31.21/8.81	
31.21/8.81	
31.21/8.81	*B(c(x1)) -> A(x1)
31.21/8.81	The graph contains the following edges 1 > 1
31.21/8.81	
31.21/8.81	
31.21/8.81	----------------------------------------
31.21/8.81	
31.21/8.81	(9)
31.21/8.81	YES
31.21/8.81	
31.21/8.81	----------------------------------------
31.21/8.81	
31.21/8.81	(10)
31.21/8.81	Obligation:
31.21/8.81	Q DP problem:
31.21/8.81	The TRS P consists of the following rules:
31.21/8.81	
31.21/8.81	   C(b(x1)) -> C(a(x1))
31.21/8.81	   C(b(x1)) -> C(c(a(x1)))
31.21/8.81	
31.21/8.81	The TRS R consists of the following rules:
31.21/8.81	
31.21/8.81	   a(a(x1)) -> b(x1)
31.21/8.81	   b(c(x1)) -> a(x1)
31.21/8.81	   c(b(x1)) -> b(c(c(a(x1))))
31.21/8.81	
31.21/8.81	Q is empty.
31.21/8.81	We have to consider all minimal (P,Q,R)-chains.
31.21/8.81	----------------------------------------
31.21/8.81	
31.21/8.81	(11) QDPOrderProof (EQUIVALENT)
31.21/8.81	We use the reduction pair processor [LPAR04,JAR06].
31.21/8.81	
31.21/8.81	
31.21/8.81	The following pairs can be oriented strictly and are deleted.
31.21/8.81	
31.21/8.81	   C(b(x1)) -> C(a(x1))
31.21/8.81	The remaining pairs can at least be oriented weakly.
31.21/8.81	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
31.21/8.81	
31.21/8.81	   <<<
31.21/8.81	 POL(C(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 1A]] 	* 	x_1
31.21/8.81	>>>
31.21/8.81	
31.21/8.81	   <<<
31.21/8.81	 POL(b(x_1)) =  	[[-I], [1A], [1A]] 	 +  	[[0A, -I, -I], [1A, 0A, -I], [0A, 1A, 0A]] 	* 	x_1
31.21/8.81	>>>
31.21/8.81	
31.21/8.81	   <<<
31.21/8.81	 POL(a(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[-I, 0A, -I], [0A, 1A, 0A], [-I, 0A, -I]] 	* 	x_1
31.21/8.81	>>>
31.21/8.81	
31.21/8.81	   <<<
31.21/8.81	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, -I], [0A, -I, 0A], [-I, -I, 1A]] 	* 	x_1
31.21/8.81	>>>
31.21/8.81	
31.21/8.81	
31.21/8.81	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
31.21/8.81	
31.21/8.81	   b(c(x1)) -> a(x1)
31.21/8.81	   a(a(x1)) -> b(x1)
31.21/8.81	   c(b(x1)) -> b(c(c(a(x1))))
31.21/8.81	
31.21/8.81	
31.21/8.81	----------------------------------------
31.21/8.81	
31.21/8.81	(12)
31.21/8.81	Obligation:
31.21/8.81	Q DP problem:
31.21/8.81	The TRS P consists of the following rules:
31.21/8.81	
31.21/8.81	   C(b(x1)) -> C(c(a(x1)))
31.21/8.81	
31.21/8.81	The TRS R consists of the following rules:
31.21/8.81	
31.21/8.81	   a(a(x1)) -> b(x1)
31.21/8.81	   b(c(x1)) -> a(x1)
31.21/8.81	   c(b(x1)) -> b(c(c(a(x1))))
31.21/8.81	
31.21/8.81	Q is empty.
31.21/8.81	We have to consider all minimal (P,Q,R)-chains.
31.21/8.81	----------------------------------------
31.21/8.81	
31.21/8.81	(13) QDPOrderProof (EQUIVALENT)
31.21/8.81	We use the reduction pair processor [LPAR04,JAR06].
31.21/8.81	
31.21/8.81	
31.21/8.81	The following pairs can be oriented strictly and are deleted.
31.21/8.81	
31.21/8.81	   C(b(x1)) -> C(c(a(x1)))
31.21/8.81	The remaining pairs can at least be oriented weakly.
31.21/8.81	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
31.21/8.81	
31.21/8.81	   <<<
31.21/8.81	 POL(C(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
31.21/8.81	>>>
31.21/8.81	
31.21/8.81	   <<<
31.21/8.81	 POL(b(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[0A, 0A, 0A], [1A, 0A, 0A], [0A, 1A, 0A]] 	* 	x_1
31.21/8.81	>>>
31.21/8.81	
31.21/8.81	   <<<
31.21/8.81	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, 0A], [-I, -I, 0A], [0A, -I, 0A]] 	* 	x_1
31.21/8.81	>>>
31.21/8.81	
31.21/8.81	   <<<
31.21/8.81	 POL(a(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[-I, 0A, -I], [0A, 1A, 0A], [-I, 0A, -I]] 	* 	x_1
31.21/8.81	>>>
31.21/8.81	
31.21/8.81	
31.21/8.81	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
31.21/8.81	
31.21/8.81	   b(c(x1)) -> a(x1)
31.21/8.81	   a(a(x1)) -> b(x1)
31.21/8.81	   c(b(x1)) -> b(c(c(a(x1))))
31.21/8.81	
31.21/8.81	
31.21/8.81	----------------------------------------
31.21/8.81	
31.21/8.81	(14)
31.21/8.81	Obligation:
31.21/8.81	Q DP problem:
31.21/8.81	P is empty.
31.21/8.81	The TRS R consists of the following rules:
31.21/8.81	
31.21/8.81	   a(a(x1)) -> b(x1)
31.21/8.81	   b(c(x1)) -> a(x1)
31.21/8.81	   c(b(x1)) -> b(c(c(a(x1))))
31.21/8.81	
31.21/8.81	Q is empty.
31.21/8.81	We have to consider all minimal (P,Q,R)-chains.
31.21/8.81	----------------------------------------
31.21/8.81	
31.21/8.81	(15) PisEmptyProof (EQUIVALENT)
31.21/8.81	The TRS P is empty. Hence, there is no (P,Q,R) chain.
31.21/8.81	----------------------------------------
31.21/8.81	
31.21/8.81	(16)
31.21/8.81	YES
31.57/8.89	EOF