23.17/6.84	YES
23.77/6.95	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
23.77/6.95	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
23.77/6.95	
23.77/6.95	
23.77/6.95	Termination w.r.t. Q of the given QTRS could be proven:
23.77/6.95	
23.77/6.95	(0) QTRS
23.77/6.95	(1) DependencyPairsProof [EQUIVALENT, 4 ms]
23.77/6.95	(2) QDP
23.77/6.95	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
23.77/6.95	(4) QDP
23.77/6.95	(5) QDPOrderProof [EQUIVALENT, 101 ms]
23.77/6.95	(6) QDP
23.77/6.95	(7) UsableRulesProof [EQUIVALENT, 1 ms]
23.77/6.95	(8) QDP
23.77/6.95	(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
23.77/6.95	(10) YES
23.77/6.95	
23.77/6.95	
23.77/6.95	----------------------------------------
23.77/6.95	
23.77/6.95	(0)
23.77/6.95	Obligation:
23.77/6.95	Q restricted rewrite system:
23.77/6.95	The TRS R consists of the following rules:
23.77/6.95	
23.77/6.95	   a(x1) -> x1
23.77/6.95	   a(b(x1)) -> b(b(c(a(c(a(x1))))))
23.77/6.95	   b(x1) -> x1
23.77/6.95	   c(c(x1)) -> x1
23.77/6.95	
23.77/6.95	Q is empty.
23.77/6.95	
23.77/6.95	----------------------------------------
23.77/6.95	
23.77/6.95	(1) DependencyPairsProof (EQUIVALENT)
23.77/6.95	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
23.77/6.95	----------------------------------------
23.77/6.95	
23.77/6.95	(2)
23.77/6.95	Obligation:
23.77/6.95	Q DP problem:
23.77/6.95	The TRS P consists of the following rules:
23.77/6.95	
23.77/6.95	   A(b(x1)) -> B(b(c(a(c(a(x1))))))
23.77/6.95	   A(b(x1)) -> B(c(a(c(a(x1)))))
23.77/6.95	   A(b(x1)) -> C(a(c(a(x1))))
23.77/6.95	   A(b(x1)) -> A(c(a(x1)))
23.77/6.95	   A(b(x1)) -> C(a(x1))
23.77/6.95	   A(b(x1)) -> A(x1)
23.77/6.95	
23.77/6.95	The TRS R consists of the following rules:
23.77/6.95	
23.77/6.95	   a(x1) -> x1
23.77/6.95	   a(b(x1)) -> b(b(c(a(c(a(x1))))))
23.77/6.95	   b(x1) -> x1
23.77/6.95	   c(c(x1)) -> x1
23.77/6.95	
23.77/6.95	Q is empty.
23.77/6.95	We have to consider all minimal (P,Q,R)-chains.
23.77/6.95	----------------------------------------
23.77/6.95	
23.77/6.95	(3) DependencyGraphProof (EQUIVALENT)
23.77/6.95	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.
23.77/6.95	----------------------------------------
23.77/6.95	
23.77/6.95	(4)
23.77/6.95	Obligation:
23.77/6.95	Q DP problem:
23.77/6.95	The TRS P consists of the following rules:
23.77/6.95	
23.77/6.95	   A(b(x1)) -> A(x1)
23.77/6.95	   A(b(x1)) -> A(c(a(x1)))
23.77/6.95	
23.77/6.95	The TRS R consists of the following rules:
23.77/6.95	
23.77/6.95	   a(x1) -> x1
23.77/6.95	   a(b(x1)) -> b(b(c(a(c(a(x1))))))
23.77/6.95	   b(x1) -> x1
23.77/6.95	   c(c(x1)) -> x1
23.77/6.95	
23.77/6.95	Q is empty.
23.77/6.95	We have to consider all minimal (P,Q,R)-chains.
23.77/6.95	----------------------------------------
23.77/6.95	
23.77/6.95	(5) QDPOrderProof (EQUIVALENT)
23.77/6.95	We use the reduction pair processor [LPAR04,JAR06].
23.77/6.95	
23.77/6.95	
23.77/6.95	The following pairs can be oriented strictly and are deleted.
23.77/6.95	
23.77/6.95	   A(b(x1)) -> A(c(a(x1)))
23.77/6.95	The remaining pairs can at least be oriented weakly.
23.77/6.95	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
23.77/6.95	
23.77/6.95	   <<<
23.77/6.95	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
23.77/6.95	>>>
23.77/6.95	
23.77/6.95	   <<<
23.77/6.95	 POL(b(x_1)) =  	[[-I], [1A], [-I]] 	 +  	[[0A, -I, -I], [1A, 0A, 1A], [-I, -I, 0A]] 	* 	x_1
23.77/6.95	>>>
23.77/6.95	
23.77/6.95	   <<<
23.77/6.95	 POL(c(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[-I, 0A, -I], [0A, -I, -I], [-I, -I, 0A]] 	* 	x_1
23.77/6.95	>>>
23.77/6.95	
23.77/6.95	   <<<
23.77/6.95	 POL(a(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[0A, -I, -I], [-I, 0A, -I], [-I, -I, 0A]] 	* 	x_1
23.77/6.95	>>>
23.77/6.95	
23.77/6.95	
23.77/6.95	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
23.77/6.95	
23.77/6.95	   a(x1) -> x1
23.77/6.95	   a(b(x1)) -> b(b(c(a(c(a(x1))))))
23.77/6.95	   c(c(x1)) -> x1
23.77/6.95	   b(x1) -> x1
23.77/6.95	
23.77/6.95	
23.77/6.95	----------------------------------------
23.77/6.95	
23.77/6.95	(6)
23.77/6.95	Obligation:
23.77/6.95	Q DP problem:
23.77/6.95	The TRS P consists of the following rules:
23.77/6.95	
23.77/6.95	   A(b(x1)) -> A(x1)
23.77/6.95	
23.77/6.95	The TRS R consists of the following rules:
23.77/6.95	
23.77/6.95	   a(x1) -> x1
23.77/6.95	   a(b(x1)) -> b(b(c(a(c(a(x1))))))
23.77/6.95	   b(x1) -> x1
23.77/6.95	   c(c(x1)) -> x1
23.77/6.95	
23.77/6.95	Q is empty.
23.77/6.95	We have to consider all minimal (P,Q,R)-chains.
23.77/6.95	----------------------------------------
23.77/6.95	
23.77/6.95	(7) UsableRulesProof (EQUIVALENT)
23.77/6.95	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.
23.77/6.95	----------------------------------------
23.77/6.95	
23.77/6.95	(8)
23.77/6.95	Obligation:
23.77/6.95	Q DP problem:
23.77/6.95	The TRS P consists of the following rules:
23.77/6.95	
23.77/6.95	   A(b(x1)) -> A(x1)
23.77/6.95	
23.77/6.95	R is empty.
23.77/6.95	Q is empty.
23.77/6.95	We have to consider all minimal (P,Q,R)-chains.
23.77/6.95	----------------------------------------
23.77/6.95	
23.77/6.95	(9) QDPSizeChangeProof (EQUIVALENT)
23.77/6.95	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. 
23.77/6.95	
23.77/6.95	From the DPs we obtained the following set of size-change graphs:
23.77/6.95	*A(b(x1)) -> A(x1)
23.77/6.95	The graph contains the following edges 1 > 1
23.77/6.95	
23.77/6.95	
23.77/6.95	----------------------------------------
23.77/6.95	
23.77/6.95	(10)
23.77/6.95	YES
23.98/7.04	EOF