37.88/10.52	YES
37.88/10.54	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
37.88/10.54	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
37.88/10.54	
37.88/10.54	
37.88/10.54	Termination w.r.t. Q of the given QTRS could be proven:
37.88/10.54	
37.88/10.54	(0) QTRS
37.88/10.54	(1) DependencyPairsProof [EQUIVALENT, 1 ms]
37.88/10.54	(2) QDP
37.88/10.54	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
37.88/10.54	(4) QDP
37.88/10.54	(5) QDPOrderProof [EQUIVALENT, 137 ms]
37.88/10.54	(6) QDP
37.88/10.54	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
37.88/10.54	(8) QDP
37.88/10.54	(9) QDPOrderProof [EQUIVALENT, 55 ms]
37.88/10.54	(10) QDP
37.88/10.54	(11) UsableRulesProof [EQUIVALENT, 0 ms]
37.88/10.54	(12) QDP
37.88/10.54	(13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
37.88/10.54	(14) YES
37.88/10.54	
37.88/10.54	
37.88/10.54	----------------------------------------
37.88/10.54	
37.88/10.54	(0)
37.88/10.54	Obligation:
37.88/10.54	Q restricted rewrite system:
37.88/10.54	The TRS R consists of the following rules:
37.88/10.54	
37.88/10.54	   a(x1) -> b(x1)
37.88/10.54	   b(b(c(x1))) -> c(a(c(b(b(x1)))))
37.88/10.54	   c(c(x1)) -> x1
37.88/10.54	
37.88/10.54	Q is empty.
37.88/10.54	
37.88/10.54	----------------------------------------
37.88/10.54	
37.88/10.54	(1) DependencyPairsProof (EQUIVALENT)
37.88/10.54	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
37.88/10.54	----------------------------------------
37.88/10.54	
37.88/10.54	(2)
37.88/10.54	Obligation:
37.88/10.54	Q DP problem:
37.88/10.54	The TRS P consists of the following rules:
37.88/10.54	
37.88/10.54	   A(x1) -> B(x1)
37.88/10.54	   B(b(c(x1))) -> C(a(c(b(b(x1)))))
37.88/10.54	   B(b(c(x1))) -> A(c(b(b(x1))))
37.88/10.54	   B(b(c(x1))) -> C(b(b(x1)))
37.88/10.54	   B(b(c(x1))) -> B(b(x1))
37.88/10.54	   B(b(c(x1))) -> B(x1)
37.88/10.54	
37.88/10.54	The TRS R consists of the following rules:
37.88/10.54	
37.88/10.54	   a(x1) -> b(x1)
37.88/10.54	   b(b(c(x1))) -> c(a(c(b(b(x1)))))
37.88/10.54	   c(c(x1)) -> x1
37.88/10.54	
37.88/10.54	Q is empty.
37.88/10.54	We have to consider all minimal (P,Q,R)-chains.
37.88/10.54	----------------------------------------
37.88/10.54	
37.88/10.54	(3) DependencyGraphProof (EQUIVALENT)
37.88/10.54	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
37.88/10.54	----------------------------------------
37.88/10.54	
37.88/10.54	(4)
37.88/10.54	Obligation:
37.88/10.54	Q DP problem:
37.88/10.54	The TRS P consists of the following rules:
37.88/10.54	
37.88/10.54	   B(b(c(x1))) -> A(c(b(b(x1))))
37.88/10.54	   A(x1) -> B(x1)
37.88/10.54	   B(b(c(x1))) -> B(b(x1))
37.88/10.54	   B(b(c(x1))) -> B(x1)
37.88/10.54	
37.88/10.54	The TRS R consists of the following rules:
37.88/10.54	
37.88/10.54	   a(x1) -> b(x1)
37.88/10.54	   b(b(c(x1))) -> c(a(c(b(b(x1)))))
37.88/10.54	   c(c(x1)) -> x1
37.88/10.54	
37.88/10.54	Q is empty.
37.88/10.54	We have to consider all minimal (P,Q,R)-chains.
37.88/10.54	----------------------------------------
37.88/10.54	
37.88/10.54	(5) QDPOrderProof (EQUIVALENT)
37.88/10.54	We use the reduction pair processor [LPAR04,JAR06].
37.88/10.54	
37.88/10.54	
37.88/10.54	The following pairs can be oriented strictly and are deleted.
37.88/10.54	
37.88/10.54	   A(x1) -> B(x1)
37.88/10.54	The remaining pairs can at least be oriented weakly.
37.88/10.54	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
37.88/10.54	
37.88/10.54	   <<<
37.88/10.54	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
37.88/10.54	>>>
37.88/10.54	
37.88/10.54	   <<<
37.88/10.54	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, 0A], [-I, -I, 0A], [0A, 0A, -I]] 	* 	x_1
37.88/10.54	>>>
37.88/10.54	
37.88/10.54	   <<<
37.88/10.54	 POL(c(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[1A, 1A, 0A], [0A, 0A, -I], [0A, 0A, -I]] 	* 	x_1
37.88/10.54	>>>
37.88/10.54	
37.88/10.54	   <<<
37.88/10.54	 POL(A(x_1)) =  	[[1A]] 	 +  	[[0A, 1A, 1A]] 	* 	x_1
37.88/10.54	>>>
37.88/10.54	
37.88/10.54	   <<<
37.88/10.54	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, 0A], [-I, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
37.88/10.54	>>>
37.88/10.54	
37.88/10.54	
37.88/10.54	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
37.88/10.54	
37.88/10.54	   b(b(c(x1))) -> c(a(c(b(b(x1)))))
37.88/10.54	   c(c(x1)) -> x1
37.88/10.54	   a(x1) -> b(x1)
37.88/10.54	
37.88/10.54	
37.88/10.54	----------------------------------------
37.88/10.54	
37.88/10.54	(6)
37.88/10.54	Obligation:
37.88/10.54	Q DP problem:
37.88/10.54	The TRS P consists of the following rules:
37.88/10.54	
37.88/10.54	   B(b(c(x1))) -> A(c(b(b(x1))))
37.88/10.54	   B(b(c(x1))) -> B(b(x1))
37.88/10.54	   B(b(c(x1))) -> B(x1)
37.88/10.54	
37.88/10.54	The TRS R consists of the following rules:
37.88/10.54	
37.88/10.54	   a(x1) -> b(x1)
37.88/10.54	   b(b(c(x1))) -> c(a(c(b(b(x1)))))
37.88/10.54	   c(c(x1)) -> x1
37.88/10.54	
37.88/10.54	Q is empty.
37.88/10.54	We have to consider all minimal (P,Q,R)-chains.
37.88/10.54	----------------------------------------
37.88/10.54	
37.88/10.54	(7) DependencyGraphProof (EQUIVALENT)
37.88/10.54	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
37.88/10.54	----------------------------------------
37.88/10.54	
37.88/10.54	(8)
37.88/10.54	Obligation:
37.88/10.54	Q DP problem:
37.88/10.54	The TRS P consists of the following rules:
37.88/10.54	
37.88/10.54	   B(b(c(x1))) -> B(x1)
37.88/10.54	   B(b(c(x1))) -> B(b(x1))
37.88/10.54	
37.88/10.54	The TRS R consists of the following rules:
37.88/10.54	
37.88/10.54	   a(x1) -> b(x1)
37.88/10.54	   b(b(c(x1))) -> c(a(c(b(b(x1)))))
37.88/10.54	   c(c(x1)) -> x1
37.88/10.54	
37.88/10.54	Q is empty.
37.88/10.54	We have to consider all minimal (P,Q,R)-chains.
37.88/10.54	----------------------------------------
37.88/10.54	
37.88/10.54	(9) QDPOrderProof (EQUIVALENT)
37.88/10.54	We use the reduction pair processor [LPAR04,JAR06].
37.88/10.54	
37.88/10.54	
37.88/10.54	The following pairs can be oriented strictly and are deleted.
37.88/10.54	
37.88/10.54	   B(b(c(x1))) -> B(b(x1))
37.88/10.54	The remaining pairs can at least be oriented weakly.
37.88/10.54	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
37.88/10.54	
37.88/10.54	   <<<
37.88/10.54	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
37.88/10.54	>>>
37.88/10.54	
37.88/10.54	   <<<
37.88/10.54	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, -I], [0A, -I, 0A], [-I, 0A, -I]] 	* 	x_1
37.88/10.54	>>>
37.88/10.54	
37.88/10.54	   <<<
37.88/10.54	 POL(c(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[0A, -I, -I], [0A, 1A, 0A], [0A, 0A, -I]] 	* 	x_1
37.88/10.54	>>>
37.88/10.54	
37.88/10.54	   <<<
37.88/10.54	 POL(a(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[0A, -I, -I], [0A, -I, 0A], [0A, 0A, 0A]] 	* 	x_1
37.88/10.54	>>>
37.88/10.54	
37.88/10.54	
37.88/10.54	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
37.88/10.54	
37.88/10.54	   b(b(c(x1))) -> c(a(c(b(b(x1)))))
37.88/10.54	   a(x1) -> b(x1)
37.88/10.54	   c(c(x1)) -> x1
37.88/10.54	
37.88/10.54	
37.88/10.54	----------------------------------------
37.88/10.54	
37.88/10.54	(10)
37.88/10.54	Obligation:
37.88/10.54	Q DP problem:
37.88/10.54	The TRS P consists of the following rules:
37.88/10.54	
37.88/10.54	   B(b(c(x1))) -> B(x1)
37.88/10.54	
37.88/10.54	The TRS R consists of the following rules:
37.88/10.54	
37.88/10.54	   a(x1) -> b(x1)
37.88/10.54	   b(b(c(x1))) -> c(a(c(b(b(x1)))))
37.88/10.54	   c(c(x1)) -> x1
37.88/10.54	
37.88/10.54	Q is empty.
37.88/10.54	We have to consider all minimal (P,Q,R)-chains.
37.88/10.54	----------------------------------------
37.88/10.54	
37.88/10.54	(11) UsableRulesProof (EQUIVALENT)
37.88/10.54	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.
37.88/10.54	----------------------------------------
37.88/10.54	
37.88/10.54	(12)
37.88/10.54	Obligation:
37.88/10.54	Q DP problem:
37.88/10.54	The TRS P consists of the following rules:
37.88/10.54	
37.88/10.54	   B(b(c(x1))) -> B(x1)
37.88/10.54	
37.88/10.54	R is empty.
37.88/10.54	Q is empty.
37.88/10.54	We have to consider all minimal (P,Q,R)-chains.
37.88/10.54	----------------------------------------
37.88/10.54	
37.88/10.54	(13) QDPSizeChangeProof (EQUIVALENT)
37.88/10.54	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. 
37.88/10.54	
37.88/10.54	From the DPs we obtained the following set of size-change graphs:
37.88/10.54	*B(b(c(x1))) -> B(x1)
37.88/10.54	The graph contains the following edges 1 > 1
37.88/10.54	
37.88/10.54	
37.88/10.54	----------------------------------------
37.88/10.54	
37.88/10.54	(14)
37.88/10.54	YES
38.01/10.61	EOF