33.86/9.56	YES
34.12/9.64	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
34.12/9.64	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
34.12/9.64	
34.12/9.64	
34.12/9.64	Termination w.r.t. Q of the given QTRS could be proven:
34.12/9.64	
34.12/9.64	(0) QTRS
34.12/9.64	(1) DependencyPairsProof [EQUIVALENT, 0 ms]
34.12/9.64	(2) QDP
34.12/9.64	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
34.12/9.64	(4) QDP
34.12/9.64	(5) QDPOrderProof [EQUIVALENT, 159 ms]
34.12/9.64	(6) QDP
34.12/9.64	(7) QDPOrderProof [EQUIVALENT, 69 ms]
34.12/9.64	(8) QDP
34.12/9.64	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
34.12/9.64	(10) QDP
34.12/9.64	(11) UsableRulesProof [EQUIVALENT, 0 ms]
34.12/9.64	(12) QDP
34.12/9.64	(13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
34.12/9.64	(14) YES
34.12/9.64	
34.12/9.64	
34.12/9.64	----------------------------------------
34.12/9.64	
34.12/9.64	(0)
34.12/9.64	Obligation:
34.12/9.64	Q restricted rewrite system:
34.12/9.64	The TRS R consists of the following rules:
34.12/9.64	
34.12/9.64	   a(x1) -> b(x1)
34.12/9.64	   b(b(c(x1))) -> c(b(c(a(b(x1)))))
34.12/9.64	   c(c(x1)) -> x1
34.12/9.64	
34.12/9.64	Q is empty.
34.12/9.64	
34.12/9.64	----------------------------------------
34.12/9.64	
34.12/9.64	(1) DependencyPairsProof (EQUIVALENT)
34.12/9.64	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
34.12/9.64	----------------------------------------
34.12/9.64	
34.12/9.64	(2)
34.12/9.64	Obligation:
34.12/9.64	Q DP problem:
34.12/9.64	The TRS P consists of the following rules:
34.12/9.64	
34.12/9.64	   A(x1) -> B(x1)
34.12/9.64	   B(b(c(x1))) -> C(b(c(a(b(x1)))))
34.12/9.64	   B(b(c(x1))) -> B(c(a(b(x1))))
34.12/9.64	   B(b(c(x1))) -> C(a(b(x1)))
34.12/9.64	   B(b(c(x1))) -> A(b(x1))
34.12/9.64	   B(b(c(x1))) -> B(x1)
34.12/9.64	
34.12/9.64	The TRS R consists of the following rules:
34.12/9.64	
34.12/9.64	   a(x1) -> b(x1)
34.12/9.64	   b(b(c(x1))) -> c(b(c(a(b(x1)))))
34.12/9.64	   c(c(x1)) -> x1
34.12/9.64	
34.12/9.64	Q is empty.
34.12/9.64	We have to consider all minimal (P,Q,R)-chains.
34.12/9.64	----------------------------------------
34.12/9.64	
34.12/9.64	(3) DependencyGraphProof (EQUIVALENT)
34.12/9.64	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
34.12/9.64	----------------------------------------
34.12/9.64	
34.12/9.64	(4)
34.12/9.64	Obligation:
34.12/9.64	Q DP problem:
34.12/9.64	The TRS P consists of the following rules:
34.12/9.64	
34.12/9.64	   B(b(c(x1))) -> B(c(a(b(x1))))
34.12/9.64	   B(b(c(x1))) -> A(b(x1))
34.12/9.64	   A(x1) -> B(x1)
34.12/9.64	   B(b(c(x1))) -> B(x1)
34.12/9.64	
34.12/9.64	The TRS R consists of the following rules:
34.12/9.64	
34.12/9.64	   a(x1) -> b(x1)
34.12/9.64	   b(b(c(x1))) -> c(b(c(a(b(x1)))))
34.12/9.64	   c(c(x1)) -> x1
34.12/9.64	
34.12/9.64	Q is empty.
34.12/9.64	We have to consider all minimal (P,Q,R)-chains.
34.12/9.64	----------------------------------------
34.12/9.64	
34.12/9.64	(5) QDPOrderProof (EQUIVALENT)
34.12/9.64	We use the reduction pair processor [LPAR04,JAR06].
34.12/9.64	
34.12/9.64	
34.12/9.64	The following pairs can be oriented strictly and are deleted.
34.12/9.64	
34.12/9.64	   B(b(c(x1))) -> B(c(a(b(x1))))
34.12/9.64	The remaining pairs can at least be oriented weakly.
34.12/9.64	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
34.12/9.64	
34.12/9.64	   <<<
34.12/9.64	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
34.12/9.64	>>>
34.12/9.64	
34.12/9.64	   <<<
34.12/9.64	 POL(b(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[0A, 0A, -I], [-I, -I, 0A], [-I, 0A, -I]] 	* 	x_1
34.12/9.64	>>>
34.12/9.64	
34.12/9.64	   <<<
34.12/9.64	 POL(c(x_1)) =  	[[0A], [-I], [1A]] 	 +  	[[-I, -I, 0A], [-I, -I, 0A], [0A, 0A, 1A]] 	* 	x_1
34.12/9.64	>>>
34.12/9.64	
34.12/9.64	   <<<
34.12/9.64	 POL(a(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [-I, 0A, -I]] 	* 	x_1
34.12/9.64	>>>
34.12/9.64	
34.12/9.64	   <<<
34.12/9.64	 POL(A(x_1)) =  	[[1A]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
34.12/9.64	>>>
34.12/9.64	
34.12/9.64	
34.12/9.64	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
34.12/9.64	
34.12/9.64	   b(b(c(x1))) -> c(b(c(a(b(x1)))))
34.12/9.64	   a(x1) -> b(x1)
34.12/9.64	   c(c(x1)) -> x1
34.12/9.64	
34.12/9.64	
34.12/9.64	----------------------------------------
34.12/9.64	
34.12/9.64	(6)
34.12/9.64	Obligation:
34.12/9.64	Q DP problem:
34.12/9.64	The TRS P consists of the following rules:
34.12/9.64	
34.12/9.64	   B(b(c(x1))) -> A(b(x1))
34.12/9.64	   A(x1) -> B(x1)
34.12/9.64	   B(b(c(x1))) -> B(x1)
34.12/9.64	
34.12/9.64	The TRS R consists of the following rules:
34.12/9.64	
34.12/9.64	   a(x1) -> b(x1)
34.12/9.64	   b(b(c(x1))) -> c(b(c(a(b(x1)))))
34.12/9.64	   c(c(x1)) -> x1
34.12/9.64	
34.12/9.64	Q is empty.
34.12/9.64	We have to consider all minimal (P,Q,R)-chains.
34.12/9.64	----------------------------------------
34.12/9.64	
34.12/9.64	(7) QDPOrderProof (EQUIVALENT)
34.12/9.64	We use the reduction pair processor [LPAR04,JAR06].
34.12/9.64	
34.12/9.64	
34.12/9.64	The following pairs can be oriented strictly and are deleted.
34.12/9.64	
34.12/9.64	   A(x1) -> B(x1)
34.12/9.64	The remaining pairs can at least be oriented weakly.
34.12/9.64	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
34.12/9.64	
34.12/9.64	   <<<
34.12/9.64	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
34.12/9.64	>>>
34.12/9.64	
34.12/9.64	   <<<
34.12/9.64	 POL(b(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[-I, -I, 0A], [-I, -I, -I], [0A, 0A, -I]] 	* 	x_1
34.12/9.64	>>>
34.12/9.64	
34.12/9.64	   <<<
34.12/9.64	 POL(c(x_1)) =  	[[0A], [-I], [1A]] 	 +  	[[-I, 0A, 0A], [-I, 0A, -I], [0A, 0A, 1A]] 	* 	x_1
34.12/9.64	>>>
34.12/9.64	
34.12/9.64	   <<<
34.12/9.64	 POL(A(x_1)) =  	[[-I]] 	 +  	[[1A, 1A, 0A]] 	* 	x_1
34.12/9.64	>>>
34.12/9.64	
34.12/9.64	   <<<
34.12/9.64	 POL(a(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, -I], [0A, 0A, -I]] 	* 	x_1
34.12/9.64	>>>
34.12/9.64	
34.12/9.64	
34.12/9.64	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
34.12/9.64	
34.12/9.64	   b(b(c(x1))) -> c(b(c(a(b(x1)))))
34.12/9.64	   a(x1) -> b(x1)
34.12/9.64	   c(c(x1)) -> x1
34.12/9.64	
34.12/9.64	
34.12/9.64	----------------------------------------
34.12/9.64	
34.12/9.64	(8)
34.12/9.64	Obligation:
34.12/9.64	Q DP problem:
34.12/9.64	The TRS P consists of the following rules:
34.12/9.64	
34.12/9.64	   B(b(c(x1))) -> A(b(x1))
34.12/9.64	   B(b(c(x1))) -> B(x1)
34.12/9.64	
34.12/9.64	The TRS R consists of the following rules:
34.12/9.64	
34.12/9.64	   a(x1) -> b(x1)
34.12/9.64	   b(b(c(x1))) -> c(b(c(a(b(x1)))))
34.12/9.64	   c(c(x1)) -> x1
34.12/9.64	
34.12/9.64	Q is empty.
34.12/9.64	We have to consider all minimal (P,Q,R)-chains.
34.12/9.64	----------------------------------------
34.12/9.64	
34.12/9.64	(9) DependencyGraphProof (EQUIVALENT)
34.12/9.64	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
34.12/9.64	----------------------------------------
34.12/9.64	
34.12/9.64	(10)
34.12/9.64	Obligation:
34.12/9.64	Q DP problem:
34.12/9.64	The TRS P consists of the following rules:
34.12/9.64	
34.12/9.64	   B(b(c(x1))) -> B(x1)
34.12/9.64	
34.12/9.64	The TRS R consists of the following rules:
34.12/9.64	
34.12/9.64	   a(x1) -> b(x1)
34.12/9.64	   b(b(c(x1))) -> c(b(c(a(b(x1)))))
34.12/9.64	   c(c(x1)) -> x1
34.12/9.64	
34.12/9.64	Q is empty.
34.12/9.64	We have to consider all minimal (P,Q,R)-chains.
34.12/9.64	----------------------------------------
34.12/9.64	
34.12/9.64	(11) UsableRulesProof (EQUIVALENT)
34.12/9.64	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.
34.12/9.64	----------------------------------------
34.12/9.64	
34.12/9.64	(12)
34.12/9.64	Obligation:
34.12/9.64	Q DP problem:
34.12/9.64	The TRS P consists of the following rules:
34.12/9.64	
34.12/9.64	   B(b(c(x1))) -> B(x1)
34.12/9.64	
34.12/9.64	R is empty.
34.12/9.64	Q is empty.
34.12/9.64	We have to consider all minimal (P,Q,R)-chains.
34.12/9.64	----------------------------------------
34.12/9.64	
34.12/9.64	(13) QDPSizeChangeProof (EQUIVALENT)
34.12/9.64	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. 
34.12/9.64	
34.12/9.64	From the DPs we obtained the following set of size-change graphs:
34.12/9.64	*B(b(c(x1))) -> B(x1)
34.12/9.64	The graph contains the following edges 1 > 1
34.12/9.64	
34.12/9.64	
34.12/9.64	----------------------------------------
34.12/9.64	
34.12/9.64	(14)
34.12/9.64	YES
34.42/9.71	EOF