41.86/11.52	YES
41.86/11.55	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
41.86/11.55	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
41.86/11.55	
41.86/11.55	
41.86/11.55	Termination w.r.t. Q of the given QTRS could be proven:
41.86/11.55	
41.86/11.55	(0) QTRS
41.86/11.55	(1) QTRS Reverse [EQUIVALENT, 0 ms]
41.86/11.55	(2) QTRS
41.86/11.55	(3) DependencyPairsProof [EQUIVALENT, 6 ms]
41.86/11.55	(4) QDP
41.86/11.55	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
41.86/11.55	(6) QDP
41.86/11.55	(7) QDPOrderProof [EQUIVALENT, 161 ms]
41.86/11.55	(8) QDP
41.86/11.55	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
41.86/11.55	(10) AND
41.86/11.55	    (11) QDP
41.86/11.55	        (12) UsableRulesProof [EQUIVALENT, 0 ms]
41.86/11.55	        (13) QDP
41.86/11.55	        (14) QDPSizeChangeProof [EQUIVALENT, 0 ms]
41.86/11.55	        (15) YES
41.86/11.55	    (16) QDP
41.86/11.55	        (17) QDPOrderProof [EQUIVALENT, 0 ms]
41.86/11.55	        (18) QDP
41.86/11.55	        (19) PisEmptyProof [EQUIVALENT, 0 ms]
41.86/11.55	        (20) YES
41.86/11.55	
41.86/11.55	
41.86/11.55	----------------------------------------
41.86/11.55	
41.86/11.55	(0)
41.86/11.55	Obligation:
41.86/11.55	Q restricted rewrite system:
41.86/11.55	The TRS R consists of the following rules:
41.86/11.55	
41.86/11.55	   a(x1) -> x1
41.86/11.55	   a(b(x1)) -> b(c(a(x1)))
41.86/11.55	   c(c(x1)) -> b(a(c(a(x1))))
41.86/11.55	
41.86/11.55	Q is empty.
41.86/11.55	
41.86/11.55	----------------------------------------
41.86/11.55	
41.86/11.55	(1) QTRS Reverse (EQUIVALENT)
41.86/11.55	We applied the QTRS Reverse Processor [REVERSE].
41.86/11.55	----------------------------------------
41.86/11.55	
41.86/11.55	(2)
41.86/11.55	Obligation:
41.86/11.55	Q restricted rewrite system:
41.86/11.55	The TRS R consists of the following rules:
41.86/11.55	
41.86/11.55	   a(x1) -> x1
41.86/11.55	   b(a(x1)) -> a(c(b(x1)))
41.86/11.55	   c(c(x1)) -> a(c(a(b(x1))))
41.86/11.55	
41.86/11.55	Q is empty.
41.86/11.55	
41.86/11.55	----------------------------------------
41.86/11.55	
41.86/11.55	(3) DependencyPairsProof (EQUIVALENT)
41.86/11.55	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
41.86/11.55	----------------------------------------
41.86/11.55	
41.86/11.55	(4)
41.86/11.55	Obligation:
41.86/11.55	Q DP problem:
41.86/11.55	The TRS P consists of the following rules:
41.86/11.55	
41.86/11.55	   B(a(x1)) -> A(c(b(x1)))
41.86/11.55	   B(a(x1)) -> C(b(x1))
41.86/11.55	   B(a(x1)) -> B(x1)
41.86/11.55	   C(c(x1)) -> A(c(a(b(x1))))
41.86/11.55	   C(c(x1)) -> C(a(b(x1)))
41.86/11.55	   C(c(x1)) -> A(b(x1))
41.86/11.55	   C(c(x1)) -> B(x1)
41.86/11.55	
41.86/11.55	The TRS R consists of the following rules:
41.86/11.55	
41.86/11.55	   a(x1) -> x1
41.86/11.55	   b(a(x1)) -> a(c(b(x1)))
41.86/11.55	   c(c(x1)) -> a(c(a(b(x1))))
41.86/11.55	
41.86/11.55	Q is empty.
41.86/11.55	We have to consider all minimal (P,Q,R)-chains.
41.86/11.55	----------------------------------------
41.86/11.55	
41.86/11.55	(5) DependencyGraphProof (EQUIVALENT)
41.86/11.55	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
41.86/11.55	----------------------------------------
41.86/11.55	
41.86/11.55	(6)
41.86/11.55	Obligation:
41.86/11.55	Q DP problem:
41.86/11.55	The TRS P consists of the following rules:
41.86/11.55	
41.86/11.55	   B(a(x1)) -> C(b(x1))
41.86/11.55	   C(c(x1)) -> C(a(b(x1)))
41.86/11.55	   C(c(x1)) -> B(x1)
41.86/11.55	   B(a(x1)) -> B(x1)
41.86/11.55	
41.86/11.55	The TRS R consists of the following rules:
41.86/11.55	
41.86/11.55	   a(x1) -> x1
41.86/11.55	   b(a(x1)) -> a(c(b(x1)))
41.86/11.55	   c(c(x1)) -> a(c(a(b(x1))))
41.86/11.55	
41.86/11.55	Q is empty.
41.86/11.55	We have to consider all minimal (P,Q,R)-chains.
41.86/11.55	----------------------------------------
41.86/11.55	
41.86/11.55	(7) QDPOrderProof (EQUIVALENT)
41.86/11.55	We use the reduction pair processor [LPAR04,JAR06].
41.86/11.55	
41.86/11.55	
41.86/11.55	The following pairs can be oriented strictly and are deleted.
41.86/11.55	
41.86/11.55	   B(a(x1)) -> C(b(x1))
41.86/11.55	The remaining pairs can at least be oriented weakly.
41.86/11.55	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
41.86/11.55	
41.86/11.55	   <<<
41.86/11.55	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
41.86/11.55	>>>
41.86/11.55	
41.86/11.55	   <<<
41.86/11.55	 POL(a(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[0A, 0A, -I], [0A, 1A, -I], [0A, 0A, 0A]] 	* 	x_1
41.86/11.55	>>>
41.86/11.55	
41.86/11.55	   <<<
41.86/11.55	 POL(C(x_1)) =  	[[0A]] 	 +  	[[0A, -I, -I]] 	* 	x_1
41.86/11.55	>>>
41.86/11.55	
41.86/11.55	   <<<
41.86/11.55	 POL(b(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[-I, 0A, -I], [-I, 0A, -I], [-I, 0A, -I]] 	* 	x_1
41.86/11.55	>>>
41.86/11.55	
41.86/11.55	   <<<
41.86/11.55	 POL(c(x_1)) =  	[[-I], [0A], [1A]] 	 +  	[[0A, 0A, 0A], [0A, -I, 0A], [0A, 1A, 1A]] 	* 	x_1
41.86/11.55	>>>
41.86/11.55	
41.86/11.55	
41.86/11.55	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
41.86/11.55	
41.86/11.55	   b(a(x1)) -> a(c(b(x1)))
41.86/11.55	   a(x1) -> x1
41.86/11.55	   c(c(x1)) -> a(c(a(b(x1))))
41.86/11.55	
41.86/11.55	
41.86/11.55	----------------------------------------
41.86/11.55	
41.86/11.55	(8)
41.86/11.55	Obligation:
41.86/11.55	Q DP problem:
41.86/11.55	The TRS P consists of the following rules:
41.86/11.55	
41.86/11.55	   C(c(x1)) -> C(a(b(x1)))
41.86/11.55	   C(c(x1)) -> B(x1)
41.86/11.55	   B(a(x1)) -> B(x1)
41.86/11.55	
41.86/11.55	The TRS R consists of the following rules:
41.86/11.55	
41.86/11.55	   a(x1) -> x1
41.86/11.55	   b(a(x1)) -> a(c(b(x1)))
41.86/11.55	   c(c(x1)) -> a(c(a(b(x1))))
41.86/11.55	
41.86/11.55	Q is empty.
41.86/11.55	We have to consider all minimal (P,Q,R)-chains.
41.86/11.55	----------------------------------------
41.86/11.55	
41.86/11.55	(9) DependencyGraphProof (EQUIVALENT)
41.86/11.55	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
41.86/11.55	----------------------------------------
41.86/11.55	
41.86/11.55	(10)
41.86/11.55	Complex Obligation (AND)
41.86/11.55	
41.86/11.55	----------------------------------------
41.86/11.55	
41.86/11.55	(11)
41.86/11.55	Obligation:
41.86/11.55	Q DP problem:
41.86/11.55	The TRS P consists of the following rules:
41.86/11.55	
41.86/11.55	   B(a(x1)) -> B(x1)
41.86/11.55	
41.86/11.55	The TRS R consists of the following rules:
41.86/11.55	
41.86/11.55	   a(x1) -> x1
41.86/11.55	   b(a(x1)) -> a(c(b(x1)))
41.86/11.55	   c(c(x1)) -> a(c(a(b(x1))))
41.86/11.55	
41.86/11.55	Q is empty.
41.86/11.55	We have to consider all minimal (P,Q,R)-chains.
41.86/11.55	----------------------------------------
41.86/11.55	
41.86/11.55	(12) UsableRulesProof (EQUIVALENT)
41.86/11.55	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.
41.86/11.55	----------------------------------------
41.86/11.55	
41.86/11.55	(13)
41.86/11.55	Obligation:
41.86/11.55	Q DP problem:
41.86/11.55	The TRS P consists of the following rules:
41.86/11.55	
41.86/11.55	   B(a(x1)) -> B(x1)
41.86/11.55	
41.86/11.55	R is empty.
41.86/11.55	Q is empty.
41.86/11.55	We have to consider all minimal (P,Q,R)-chains.
41.86/11.55	----------------------------------------
41.86/11.55	
41.86/11.55	(14) QDPSizeChangeProof (EQUIVALENT)
41.86/11.55	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. 
41.86/11.55	
41.86/11.55	From the DPs we obtained the following set of size-change graphs:
41.86/11.55	*B(a(x1)) -> B(x1)
41.86/11.55	The graph contains the following edges 1 > 1
41.86/11.55	
41.86/11.55	
41.86/11.55	----------------------------------------
41.86/11.55	
41.86/11.55	(15)
41.86/11.55	YES
41.86/11.55	
41.86/11.55	----------------------------------------
41.86/11.55	
41.86/11.55	(16)
41.86/11.55	Obligation:
41.86/11.55	Q DP problem:
41.86/11.55	The TRS P consists of the following rules:
41.86/11.55	
41.86/11.55	   C(c(x1)) -> C(a(b(x1)))
41.86/11.55	
41.86/11.55	The TRS R consists of the following rules:
41.86/11.55	
41.86/11.55	   a(x1) -> x1
41.86/11.55	   b(a(x1)) -> a(c(b(x1)))
41.86/11.55	   c(c(x1)) -> a(c(a(b(x1))))
41.86/11.55	
41.86/11.55	Q is empty.
41.86/11.55	We have to consider all minimal (P,Q,R)-chains.
41.86/11.55	----------------------------------------
41.86/11.55	
41.86/11.55	(17) QDPOrderProof (EQUIVALENT)
41.86/11.55	We use the reduction pair processor [LPAR04,JAR06].
41.86/11.55	
41.86/11.55	
41.86/11.55	The following pairs can be oriented strictly and are deleted.
41.86/11.55	
41.86/11.55	   C(c(x1)) -> C(a(b(x1)))
41.86/11.55	The remaining pairs can at least be oriented weakly.
41.86/11.55	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
41.86/11.55	
41.86/11.55	   <<<
41.86/11.55	 POL(C(x_1)) =  	[[-I]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
41.86/11.55	>>>
41.86/11.55	
41.86/11.55	   <<<
41.86/11.55	 POL(c(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[-I, -I, 0A], [-I, -I, 0A], [1A, 1A, 1A]] 	* 	x_1
41.86/11.55	>>>
41.86/11.55	
41.86/11.55	   <<<
41.86/11.55	 POL(a(x_1)) =  	[[-I], [1A], [-I]] 	 +  	[[0A, 0A, -I], [1A, 1A, -I], [0A, 0A, 0A]] 	* 	x_1
41.86/11.55	>>>
41.86/11.55	
41.86/11.55	   <<<
41.86/11.55	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, -I], [0A, 0A, -I], [-I, 0A, -I]] 	* 	x_1
41.86/11.55	>>>
41.86/11.55	
41.86/11.55	
41.86/11.55	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
41.86/11.55	
41.86/11.55	   b(a(x1)) -> a(c(b(x1)))
41.86/11.55	   a(x1) -> x1
41.86/11.55	   c(c(x1)) -> a(c(a(b(x1))))
41.86/11.55	
41.86/11.55	
41.86/11.55	----------------------------------------
41.86/11.55	
41.86/11.55	(18)
41.86/11.55	Obligation:
41.86/11.55	Q DP problem:
41.86/11.55	P is empty.
41.86/11.55	The TRS R consists of the following rules:
41.86/11.55	
41.86/11.55	   a(x1) -> x1
41.86/11.55	   b(a(x1)) -> a(c(b(x1)))
41.86/11.55	   c(c(x1)) -> a(c(a(b(x1))))
41.86/11.55	
41.86/11.55	Q is empty.
41.86/11.55	We have to consider all minimal (P,Q,R)-chains.
41.86/11.55	----------------------------------------
41.86/11.55	
41.86/11.55	(19) PisEmptyProof (EQUIVALENT)
41.86/11.55	The TRS P is empty. Hence, there is no (P,Q,R) chain.
41.86/11.55	----------------------------------------
41.86/11.55	
41.86/11.55	(20)
41.86/11.55	YES
42.34/11.64	EOF