27.94/7.95	YES
28.40/8.00	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
28.40/8.00	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
28.40/8.00	
28.40/8.00	
28.40/8.00	Termination w.r.t. Q of the given QTRS could be proven:
28.40/8.00	
28.40/8.00	(0) QTRS
28.40/8.00	(1) DependencyPairsProof [EQUIVALENT, 0 ms]
28.40/8.00	(2) QDP
28.40/8.00	(3) DependencyGraphProof [EQUIVALENT, 6 ms]
28.40/8.00	(4) QDP
28.40/8.00	(5) QDPOrderProof [EQUIVALENT, 123 ms]
28.40/8.00	(6) QDP
28.40/8.00	(7) UsableRulesProof [EQUIVALENT, 0 ms]
28.40/8.00	(8) QDP
28.40/8.00	(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
28.40/8.00	(10) YES
28.40/8.00	
28.40/8.00	
28.40/8.00	----------------------------------------
28.40/8.00	
28.40/8.00	(0)
28.40/8.00	Obligation:
28.40/8.00	Q restricted rewrite system:
28.40/8.00	The TRS R consists of the following rules:
28.40/8.00	
28.40/8.00	   a(x1) -> b(x1)
28.40/8.00	   a(b(x1)) -> x1
28.40/8.00	   c(c(b(x1))) -> a(c(a(c(c(x1)))))
28.40/8.00	
28.40/8.00	Q is empty.
28.40/8.00	
28.40/8.00	----------------------------------------
28.40/8.00	
28.40/8.00	(1) DependencyPairsProof (EQUIVALENT)
28.40/8.00	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
28.40/8.00	----------------------------------------
28.40/8.00	
28.40/8.00	(2)
28.40/8.00	Obligation:
28.40/8.00	Q DP problem:
28.40/8.00	The TRS P consists of the following rules:
28.40/8.00	
28.40/8.00	   C(c(b(x1))) -> A(c(a(c(c(x1)))))
28.40/8.00	   C(c(b(x1))) -> C(a(c(c(x1))))
28.40/8.00	   C(c(b(x1))) -> A(c(c(x1)))
28.40/8.00	   C(c(b(x1))) -> C(c(x1))
28.40/8.00	   C(c(b(x1))) -> C(x1)
28.40/8.00	
28.40/8.00	The TRS R consists of the following rules:
28.40/8.00	
28.40/8.00	   a(x1) -> b(x1)
28.40/8.00	   a(b(x1)) -> x1
28.40/8.00	   c(c(b(x1))) -> a(c(a(c(c(x1)))))
28.40/8.00	
28.40/8.00	Q is empty.
28.40/8.00	We have to consider all minimal (P,Q,R)-chains.
28.40/8.00	----------------------------------------
28.40/8.00	
28.40/8.00	(3) DependencyGraphProof (EQUIVALENT)
28.40/8.00	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
28.40/8.00	----------------------------------------
28.40/8.00	
28.40/8.00	(4)
28.40/8.00	Obligation:
28.40/8.00	Q DP problem:
28.40/8.00	The TRS P consists of the following rules:
28.40/8.00	
28.40/8.00	   C(c(b(x1))) -> C(c(x1))
28.40/8.00	   C(c(b(x1))) -> C(a(c(c(x1))))
28.40/8.00	   C(c(b(x1))) -> C(x1)
28.40/8.00	
28.40/8.00	The TRS R consists of the following rules:
28.40/8.00	
28.40/8.00	   a(x1) -> b(x1)
28.40/8.00	   a(b(x1)) -> x1
28.40/8.00	   c(c(b(x1))) -> a(c(a(c(c(x1)))))
28.40/8.00	
28.40/8.00	Q is empty.
28.40/8.00	We have to consider all minimal (P,Q,R)-chains.
28.40/8.00	----------------------------------------
28.40/8.00	
28.40/8.00	(5) QDPOrderProof (EQUIVALENT)
28.40/8.00	We use the reduction pair processor [LPAR04,JAR06].
28.40/8.00	
28.40/8.00	
28.40/8.00	The following pairs can be oriented strictly and are deleted.
28.40/8.00	
28.40/8.00	   C(c(b(x1))) -> C(c(x1))
28.40/8.00	   C(c(b(x1))) -> C(a(c(c(x1))))
28.40/8.00	The remaining pairs can at least be oriented weakly.
28.40/8.00	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
28.40/8.00	
28.40/8.00	   <<<
28.40/8.00	 POL(C(x_1)) =  	[[-I]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
28.40/8.00	>>>
28.40/8.00	
28.40/8.00	   <<<
28.40/8.00	 POL(c(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, -I], [0A, -I, 0A], [0A, 0A, -I]] 	* 	x_1
28.40/8.00	>>>
28.40/8.00	
28.40/8.00	   <<<
28.40/8.00	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, -I], [1A, 1A, 0A], [-I, 0A, -I]] 	* 	x_1
28.40/8.00	>>>
28.40/8.00	
28.40/8.00	   <<<
28.40/8.00	 POL(a(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, -I], [1A, 1A, 0A], [0A, 0A, -I]] 	* 	x_1
28.40/8.00	>>>
28.40/8.00	
28.40/8.00	
28.40/8.00	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
28.40/8.00	
28.40/8.00	   c(c(b(x1))) -> a(c(a(c(c(x1)))))
28.40/8.00	   a(x1) -> b(x1)
28.40/8.00	   a(b(x1)) -> x1
28.40/8.00	
28.40/8.00	
28.40/8.00	----------------------------------------
28.40/8.00	
28.40/8.00	(6)
28.40/8.00	Obligation:
28.40/8.00	Q DP problem:
28.40/8.00	The TRS P consists of the following rules:
28.40/8.00	
28.40/8.00	   C(c(b(x1))) -> C(x1)
28.40/8.00	
28.40/8.00	The TRS R consists of the following rules:
28.40/8.00	
28.40/8.00	   a(x1) -> b(x1)
28.40/8.00	   a(b(x1)) -> x1
28.40/8.00	   c(c(b(x1))) -> a(c(a(c(c(x1)))))
28.40/8.00	
28.40/8.00	Q is empty.
28.40/8.00	We have to consider all minimal (P,Q,R)-chains.
28.40/8.00	----------------------------------------
28.40/8.00	
28.40/8.00	(7) UsableRulesProof (EQUIVALENT)
28.40/8.00	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.
28.40/8.00	----------------------------------------
28.40/8.00	
28.40/8.00	(8)
28.40/8.00	Obligation:
28.40/8.00	Q DP problem:
28.40/8.00	The TRS P consists of the following rules:
28.40/8.00	
28.40/8.00	   C(c(b(x1))) -> C(x1)
28.40/8.00	
28.40/8.00	R is empty.
28.40/8.00	Q is empty.
28.40/8.00	We have to consider all minimal (P,Q,R)-chains.
28.40/8.00	----------------------------------------
28.40/8.00	
28.40/8.00	(9) QDPSizeChangeProof (EQUIVALENT)
28.40/8.00	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. 
28.40/8.00	
28.40/8.00	From the DPs we obtained the following set of size-change graphs:
28.40/8.00	*C(c(b(x1))) -> C(x1)
28.40/8.00	The graph contains the following edges 1 > 1
28.40/8.00	
28.40/8.00	
28.40/8.00	----------------------------------------
28.40/8.00	
28.40/8.00	(10)
28.40/8.00	YES
28.60/8.10	EOF