16.75/5.28	YES
17.27/5.45	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
17.27/5.45	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
17.27/5.45	
17.27/5.45	
17.27/5.45	Termination w.r.t. Q of the given QTRS could be proven:
17.27/5.45	
17.27/5.45	(0) QTRS
17.27/5.45	(1) DependencyPairsProof [EQUIVALENT, 16 ms]
17.27/5.45	(2) QDP
17.27/5.45	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
17.27/5.45	(4) QDP
17.27/5.45	(5) QDPOrderProof [EQUIVALENT, 71 ms]
17.27/5.45	(6) QDP
17.27/5.45	(7) UsableRulesProof [EQUIVALENT, 0 ms]
17.27/5.45	(8) QDP
17.27/5.45	(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
17.27/5.45	(10) YES
17.27/5.45	
17.27/5.45	
17.27/5.45	----------------------------------------
17.27/5.45	
17.27/5.45	(0)
17.27/5.45	Obligation:
17.27/5.45	Q restricted rewrite system:
17.27/5.45	The TRS R consists of the following rules:
17.27/5.45	
17.27/5.45	   a(b(x1)) -> x1
17.27/5.45	   b(c(x1)) -> a(a(x1))
17.27/5.45	   c(a(x1)) -> b(b(c(c(x1))))
17.27/5.45	
17.27/5.45	Q is empty.
17.27/5.45	
17.27/5.45	----------------------------------------
17.27/5.45	
17.27/5.45	(1) DependencyPairsProof (EQUIVALENT)
17.27/5.45	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
17.27/5.45	----------------------------------------
17.27/5.45	
17.27/5.45	(2)
17.27/5.45	Obligation:
17.27/5.45	Q DP problem:
17.27/5.45	The TRS P consists of the following rules:
17.27/5.45	
17.27/5.45	   B(c(x1)) -> A(a(x1))
17.27/5.45	   B(c(x1)) -> A(x1)
17.27/5.45	   C(a(x1)) -> B(b(c(c(x1))))
17.27/5.45	   C(a(x1)) -> B(c(c(x1)))
17.27/5.45	   C(a(x1)) -> C(c(x1))
17.27/5.45	   C(a(x1)) -> C(x1)
17.27/5.45	
17.27/5.45	The TRS R consists of the following rules:
17.27/5.45	
17.27/5.45	   a(b(x1)) -> x1
17.27/5.45	   b(c(x1)) -> a(a(x1))
17.27/5.45	   c(a(x1)) -> b(b(c(c(x1))))
17.27/5.45	
17.27/5.45	Q is empty.
17.27/5.45	We have to consider all minimal (P,Q,R)-chains.
17.27/5.45	----------------------------------------
17.27/5.45	
17.27/5.45	(3) DependencyGraphProof (EQUIVALENT)
17.27/5.45	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.
17.27/5.45	----------------------------------------
17.27/5.45	
17.27/5.45	(4)
17.27/5.45	Obligation:
17.27/5.45	Q DP problem:
17.27/5.45	The TRS P consists of the following rules:
17.27/5.45	
17.27/5.45	   C(a(x1)) -> C(x1)
17.27/5.45	   C(a(x1)) -> C(c(x1))
17.27/5.45	
17.27/5.45	The TRS R consists of the following rules:
17.27/5.45	
17.27/5.45	   a(b(x1)) -> x1
17.27/5.45	   b(c(x1)) -> a(a(x1))
17.27/5.45	   c(a(x1)) -> b(b(c(c(x1))))
17.27/5.45	
17.27/5.45	Q is empty.
17.27/5.45	We have to consider all minimal (P,Q,R)-chains.
17.27/5.45	----------------------------------------
17.27/5.45	
17.27/5.45	(5) QDPOrderProof (EQUIVALENT)
17.27/5.45	We use the reduction pair processor [LPAR04,JAR06].
17.27/5.45	
17.27/5.45	
17.27/5.45	The following pairs can be oriented strictly and are deleted.
17.27/5.45	
17.27/5.45	   C(a(x1)) -> C(c(x1))
17.27/5.45	The remaining pairs can at least be oriented weakly.
17.27/5.45	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
17.27/5.45	
17.27/5.45	   <<<
17.27/5.45	 POL(C(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
17.27/5.45	>>>
17.27/5.45	
17.27/5.45	   <<<
17.27/5.45	 POL(a(x_1)) =  	[[0A], [1A], [-I]] 	 +  	[[-I, -I, 0A], [0A, 1A, 0A], [-I, 0A, -I]] 	* 	x_1
17.27/5.45	>>>
17.27/5.45	
17.27/5.45	   <<<
17.27/5.45	 POL(c(x_1)) =  	[[1A], [-I], [-I]] 	 +  	[[0A, 1A, 0A], [-I, 0A, -I], [-I, 0A, -I]] 	* 	x_1
17.27/5.45	>>>
17.27/5.45	
17.27/5.45	   <<<
17.27/5.45	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, 0A], [1A, -I, 0A], [0A, -I, 0A]] 	* 	x_1
17.27/5.45	>>>
17.27/5.45	
17.27/5.45	
17.27/5.45	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
17.27/5.45	
17.27/5.45	   c(a(x1)) -> b(b(c(c(x1))))
17.27/5.45	   b(c(x1)) -> a(a(x1))
17.27/5.45	   a(b(x1)) -> x1
17.27/5.45	
17.27/5.45	
17.27/5.45	----------------------------------------
17.27/5.45	
17.27/5.45	(6)
17.27/5.45	Obligation:
17.27/5.45	Q DP problem:
17.27/5.45	The TRS P consists of the following rules:
17.27/5.45	
17.27/5.45	   C(a(x1)) -> C(x1)
17.27/5.45	
17.27/5.45	The TRS R consists of the following rules:
17.27/5.45	
17.27/5.45	   a(b(x1)) -> x1
17.27/5.45	   b(c(x1)) -> a(a(x1))
17.27/5.45	   c(a(x1)) -> b(b(c(c(x1))))
17.27/5.45	
17.27/5.45	Q is empty.
17.27/5.45	We have to consider all minimal (P,Q,R)-chains.
17.27/5.45	----------------------------------------
17.27/5.45	
17.27/5.45	(7) UsableRulesProof (EQUIVALENT)
17.27/5.45	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.
17.27/5.45	----------------------------------------
17.27/5.45	
17.27/5.45	(8)
17.27/5.45	Obligation:
17.27/5.45	Q DP problem:
17.27/5.45	The TRS P consists of the following rules:
17.27/5.45	
17.27/5.45	   C(a(x1)) -> C(x1)
17.27/5.45	
17.27/5.45	R is empty.
17.27/5.45	Q is empty.
17.27/5.45	We have to consider all minimal (P,Q,R)-chains.
17.27/5.45	----------------------------------------
17.27/5.45	
17.27/5.45	(9) QDPSizeChangeProof (EQUIVALENT)
17.27/5.45	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. 
17.27/5.45	
17.27/5.45	From the DPs we obtained the following set of size-change graphs:
17.27/5.45	*C(a(x1)) -> C(x1)
17.27/5.45	The graph contains the following edges 1 > 1
17.27/5.45	
17.27/5.45	
17.27/5.45	----------------------------------------
17.27/5.45	
17.27/5.45	(10)
17.27/5.45	YES
19.77/9.98	EOF