27.67/7.87	YES
27.67/7.88	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
27.67/7.88	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
27.67/7.88	
27.67/7.88	
27.67/7.88	Termination w.r.t. Q of the given QTRS could be proven:
27.67/7.88	
27.67/7.88	(0) QTRS
27.67/7.88	(1) DependencyPairsProof [EQUIVALENT, 22 ms]
27.67/7.88	(2) QDP
27.67/7.88	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
27.67/7.88	(4) QDP
27.67/7.88	(5) QDPOrderProof [EQUIVALENT, 110 ms]
27.67/7.88	(6) QDP
27.67/7.88	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
27.67/7.88	(8) QDP
27.67/7.88	(9) UsableRulesProof [EQUIVALENT, 0 ms]
27.67/7.88	(10) QDP
27.67/7.88	(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
27.67/7.88	(12) YES
27.67/7.88	
27.67/7.88	
27.67/7.88	----------------------------------------
27.67/7.88	
27.67/7.88	(0)
27.67/7.88	Obligation:
27.67/7.88	Q restricted rewrite system:
27.67/7.88	The TRS R consists of the following rules:
27.67/7.88	
27.67/7.88	   a(a(x1)) -> a(b(b(c(x1))))
27.67/7.88	   a(b(x1)) -> x1
27.67/7.88	   c(b(x1)) -> a(c(x1))
27.67/7.88	
27.67/7.88	Q is empty.
27.67/7.88	
27.67/7.88	----------------------------------------
27.67/7.88	
27.67/7.88	(1) DependencyPairsProof (EQUIVALENT)
27.67/7.88	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
27.67/7.88	----------------------------------------
27.67/7.88	
27.67/7.88	(2)
27.67/7.88	Obligation:
27.67/7.88	Q DP problem:
27.67/7.88	The TRS P consists of the following rules:
27.67/7.88	
27.67/7.88	   A(a(x1)) -> A(b(b(c(x1))))
27.67/7.88	   A(a(x1)) -> C(x1)
27.67/7.88	   C(b(x1)) -> A(c(x1))
27.67/7.88	   C(b(x1)) -> C(x1)
27.67/7.88	
27.67/7.88	The TRS R consists of the following rules:
27.67/7.88	
27.67/7.88	   a(a(x1)) -> a(b(b(c(x1))))
27.67/7.88	   a(b(x1)) -> x1
27.67/7.88	   c(b(x1)) -> a(c(x1))
27.67/7.88	
27.67/7.88	Q is empty.
27.67/7.88	We have to consider all minimal (P,Q,R)-chains.
27.67/7.88	----------------------------------------
27.67/7.88	
27.67/7.88	(3) DependencyGraphProof (EQUIVALENT)
27.67/7.88	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
27.67/7.88	----------------------------------------
27.67/7.88	
27.67/7.88	(4)
27.67/7.88	Obligation:
27.67/7.88	Q DP problem:
27.67/7.88	The TRS P consists of the following rules:
27.67/7.88	
27.67/7.88	   A(a(x1)) -> C(x1)
27.67/7.88	   C(b(x1)) -> A(c(x1))
27.67/7.88	   C(b(x1)) -> C(x1)
27.67/7.88	
27.67/7.88	The TRS R consists of the following rules:
27.67/7.88	
27.67/7.88	   a(a(x1)) -> a(b(b(c(x1))))
27.67/7.88	   a(b(x1)) -> x1
27.67/7.88	   c(b(x1)) -> a(c(x1))
27.67/7.88	
27.67/7.88	Q is empty.
27.67/7.88	We have to consider all minimal (P,Q,R)-chains.
27.67/7.88	----------------------------------------
27.67/7.88	
27.67/7.88	(5) QDPOrderProof (EQUIVALENT)
27.67/7.88	We use the reduction pair processor [LPAR04,JAR06].
27.67/7.88	
27.67/7.88	
27.67/7.88	The following pairs can be oriented strictly and are deleted.
27.67/7.88	
27.67/7.88	   A(a(x1)) -> C(x1)
27.67/7.88	The remaining pairs can at least be oriented weakly.
27.67/7.88	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
27.67/7.88	
27.67/7.88	   <<<
27.67/7.88	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
27.67/7.88	>>>
27.67/7.88	
27.67/7.88	   <<<
27.67/7.88	 POL(a(x_1)) =  	[[-I], [1A], [-I]] 	 +  	[[0A, 0A, -I], [0A, 1A, 0A], [-I, 0A, -I]] 	* 	x_1
27.67/7.88	>>>
27.67/7.88	
27.67/7.88	   <<<
27.67/7.88	 POL(C(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
27.67/7.88	>>>
27.67/7.88	
27.67/7.88	   <<<
27.67/7.88	 POL(b(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[0A, 0A, -I], [-I, 0A, 0A], [0A, 0A, 1A]] 	* 	x_1
27.67/7.88	>>>
27.67/7.88	
27.67/7.88	   <<<
27.67/7.88	 POL(c(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [-I, -I, 0A], [-I, 0A, -I]] 	* 	x_1
27.67/7.88	>>>
27.67/7.88	
27.67/7.88	
27.67/7.88	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
27.67/7.88	
27.67/7.88	   c(b(x1)) -> a(c(x1))
27.67/7.88	   a(a(x1)) -> a(b(b(c(x1))))
27.67/7.88	   a(b(x1)) -> x1
27.67/7.88	
27.67/7.88	
27.67/7.88	----------------------------------------
27.67/7.88	
27.67/7.88	(6)
27.67/7.88	Obligation:
27.67/7.88	Q DP problem:
27.67/7.88	The TRS P consists of the following rules:
27.67/7.88	
27.67/7.88	   C(b(x1)) -> A(c(x1))
27.67/7.88	   C(b(x1)) -> C(x1)
27.67/7.88	
27.67/7.88	The TRS R consists of the following rules:
27.67/7.88	
27.67/7.88	   a(a(x1)) -> a(b(b(c(x1))))
27.67/7.88	   a(b(x1)) -> x1
27.67/7.88	   c(b(x1)) -> a(c(x1))
27.67/7.88	
27.67/7.88	Q is empty.
27.67/7.88	We have to consider all minimal (P,Q,R)-chains.
27.67/7.88	----------------------------------------
27.67/7.88	
27.67/7.88	(7) DependencyGraphProof (EQUIVALENT)
27.67/7.88	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
27.67/7.88	----------------------------------------
27.67/7.88	
27.67/7.88	(8)
27.67/7.88	Obligation:
27.67/7.88	Q DP problem:
27.67/7.88	The TRS P consists of the following rules:
27.67/7.88	
27.67/7.88	   C(b(x1)) -> C(x1)
27.67/7.88	
27.67/7.88	The TRS R consists of the following rules:
27.67/7.88	
27.67/7.88	   a(a(x1)) -> a(b(b(c(x1))))
27.67/7.88	   a(b(x1)) -> x1
27.67/7.88	   c(b(x1)) -> a(c(x1))
27.67/7.88	
27.67/7.88	Q is empty.
27.67/7.88	We have to consider all minimal (P,Q,R)-chains.
27.67/7.88	----------------------------------------
27.67/7.88	
27.67/7.88	(9) UsableRulesProof (EQUIVALENT)
27.67/7.88	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.
27.67/7.88	----------------------------------------
27.67/7.88	
27.67/7.88	(10)
27.67/7.88	Obligation:
27.67/7.88	Q DP problem:
27.67/7.88	The TRS P consists of the following rules:
27.67/7.88	
27.67/7.88	   C(b(x1)) -> C(x1)
27.67/7.88	
27.67/7.88	R is empty.
27.67/7.88	Q is empty.
27.67/7.88	We have to consider all minimal (P,Q,R)-chains.
27.67/7.88	----------------------------------------
27.67/7.88	
27.67/7.88	(11) QDPSizeChangeProof (EQUIVALENT)
27.67/7.88	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. 
27.67/7.88	
27.67/7.88	From the DPs we obtained the following set of size-change graphs:
27.67/7.88	*C(b(x1)) -> C(x1)
27.67/7.88	The graph contains the following edges 1 > 1
27.67/7.88	
27.67/7.88	
27.67/7.88	----------------------------------------
27.67/7.88	
27.67/7.88	(12)
27.67/7.88	YES
27.78/7.93	EOF