34.59/9.69	YES
34.59/9.71	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
34.59/9.71	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
34.59/9.71	
34.59/9.71	
34.59/9.71	Termination w.r.t. Q of the given QTRS could be proven:
34.59/9.71	
34.59/9.71	(0) QTRS
34.59/9.71	(1) DependencyPairsProof [EQUIVALENT, 28 ms]
34.59/9.71	(2) QDP
34.59/9.71	(3) DependencyGraphProof [EQUIVALENT, 2 ms]
34.59/9.71	(4) QDP
34.59/9.71	(5) QDPOrderProof [EQUIVALENT, 137 ms]
34.59/9.71	(6) QDP
34.59/9.71	(7) QDPOrderProof [EQUIVALENT, 70 ms]
34.59/9.71	(8) QDP
34.59/9.71	(9) UsableRulesProof [EQUIVALENT, 0 ms]
34.59/9.71	(10) QDP
34.59/9.71	(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
34.59/9.71	(12) YES
34.59/9.71	
34.59/9.71	
34.59/9.71	----------------------------------------
34.59/9.71	
34.59/9.71	(0)
34.59/9.71	Obligation:
34.59/9.71	Q restricted rewrite system:
34.59/9.71	The TRS R consists of the following rules:
34.59/9.71	
34.59/9.71	   a(x1) -> b(x1)
34.59/9.71	   b(b(c(x1))) -> c(b(c(a(a(x1)))))
34.59/9.71	   c(c(x1)) -> x1
34.59/9.71	
34.59/9.71	Q is empty.
34.59/9.71	
34.59/9.71	----------------------------------------
34.59/9.71	
34.59/9.71	(1) DependencyPairsProof (EQUIVALENT)
34.59/9.71	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
34.59/9.71	----------------------------------------
34.59/9.71	
34.59/9.71	(2)
34.59/9.71	Obligation:
34.59/9.71	Q DP problem:
34.59/9.71	The TRS P consists of the following rules:
34.59/9.71	
34.59/9.71	   A(x1) -> B(x1)
34.59/9.71	   B(b(c(x1))) -> C(b(c(a(a(x1)))))
34.59/9.71	   B(b(c(x1))) -> B(c(a(a(x1))))
34.59/9.71	   B(b(c(x1))) -> C(a(a(x1)))
34.59/9.71	   B(b(c(x1))) -> A(a(x1))
34.59/9.71	   B(b(c(x1))) -> A(x1)
34.59/9.71	
34.59/9.71	The TRS R consists of the following rules:
34.59/9.71	
34.59/9.71	   a(x1) -> b(x1)
34.59/9.71	   b(b(c(x1))) -> c(b(c(a(a(x1)))))
34.59/9.71	   c(c(x1)) -> x1
34.59/9.71	
34.59/9.71	Q is empty.
34.59/9.71	We have to consider all minimal (P,Q,R)-chains.
34.59/9.71	----------------------------------------
34.59/9.71	
34.59/9.71	(3) DependencyGraphProof (EQUIVALENT)
34.59/9.71	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
34.59/9.71	----------------------------------------
34.59/9.71	
34.59/9.71	(4)
34.59/9.71	Obligation:
34.59/9.71	Q DP problem:
34.59/9.71	The TRS P consists of the following rules:
34.59/9.71	
34.59/9.71	   B(b(c(x1))) -> B(c(a(a(x1))))
34.59/9.71	   B(b(c(x1))) -> A(a(x1))
34.59/9.71	   A(x1) -> B(x1)
34.59/9.71	   B(b(c(x1))) -> A(x1)
34.59/9.71	
34.59/9.71	The TRS R consists of the following rules:
34.59/9.71	
34.59/9.71	   a(x1) -> b(x1)
34.59/9.71	   b(b(c(x1))) -> c(b(c(a(a(x1)))))
34.59/9.71	   c(c(x1)) -> x1
34.59/9.71	
34.59/9.71	Q is empty.
34.59/9.71	We have to consider all minimal (P,Q,R)-chains.
34.59/9.71	----------------------------------------
34.59/9.71	
34.59/9.71	(5) QDPOrderProof (EQUIVALENT)
34.59/9.71	We use the reduction pair processor [LPAR04,JAR06].
34.59/9.71	
34.59/9.71	
34.59/9.71	The following pairs can be oriented strictly and are deleted.
34.59/9.71	
34.59/9.71	   B(b(c(x1))) -> B(c(a(a(x1))))
34.59/9.71	The remaining pairs can at least be oriented weakly.
34.59/9.71	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
34.59/9.71	
34.59/9.71	   <<<
34.59/9.71	 POL(B(x_1)) =  	[[0A]] 	 +  	[[1A, 0A, 0A]] 	* 	x_1
34.59/9.71	>>>
34.59/9.71	
34.59/9.71	   <<<
34.59/9.71	 POL(b(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [-I, -I, 0A], [-I, 0A, -I]] 	* 	x_1
34.59/9.71	>>>
34.59/9.71	
34.59/9.71	   <<<
34.59/9.71	 POL(c(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[-I, -I, 0A], [-I, -I, 0A], [0A, 0A, 1A]] 	* 	x_1
34.59/9.71	>>>
34.59/9.71	
34.59/9.71	   <<<
34.59/9.71	 POL(a(x_1)) =  	[[1A], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [-I, -I, 0A], [-I, 0A, -I]] 	* 	x_1
34.59/9.71	>>>
34.59/9.71	
34.59/9.71	   <<<
34.59/9.71	 POL(A(x_1)) =  	[[0A]] 	 +  	[[1A, 0A, 0A]] 	* 	x_1
34.59/9.71	>>>
34.59/9.71	
34.59/9.71	
34.59/9.71	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
34.59/9.71	
34.59/9.71	   a(x1) -> b(x1)
34.59/9.71	   c(c(x1)) -> x1
34.59/9.71	   b(b(c(x1))) -> c(b(c(a(a(x1)))))
34.59/9.71	
34.59/9.71	
34.59/9.71	----------------------------------------
34.59/9.71	
34.59/9.71	(6)
34.59/9.71	Obligation:
34.59/9.71	Q DP problem:
34.59/9.71	The TRS P consists of the following rules:
34.59/9.71	
34.59/9.71	   B(b(c(x1))) -> A(a(x1))
34.59/9.71	   A(x1) -> B(x1)
34.59/9.71	   B(b(c(x1))) -> A(x1)
34.59/9.71	
34.59/9.71	The TRS R consists of the following rules:
34.59/9.71	
34.59/9.71	   a(x1) -> b(x1)
34.59/9.71	   b(b(c(x1))) -> c(b(c(a(a(x1)))))
34.59/9.71	   c(c(x1)) -> x1
34.59/9.71	
34.59/9.71	Q is empty.
34.59/9.71	We have to consider all minimal (P,Q,R)-chains.
34.59/9.71	----------------------------------------
34.59/9.71	
34.59/9.71	(7) QDPOrderProof (EQUIVALENT)
34.59/9.71	We use the reduction pair processor [LPAR04,JAR06].
34.59/9.71	
34.59/9.71	
34.59/9.71	The following pairs can be oriented strictly and are deleted.
34.59/9.71	
34.59/9.71	   B(b(c(x1))) -> A(a(x1))
34.59/9.71	The remaining pairs can at least be oriented weakly.
34.59/9.71	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
34.59/9.71	
34.59/9.71	   <<<
34.59/9.71	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, -I, -I]] 	* 	x_1
34.59/9.71	>>>
34.59/9.71	
34.59/9.71	   <<<
34.59/9.71	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, 0A], [0A, 0A, -I], [0A, -I, -I]] 	* 	x_1
34.59/9.71	>>>
34.59/9.71	
34.59/9.71	   <<<
34.59/9.71	 POL(c(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[-I, -I, 0A], [0A, 0A, 0A], [0A, 0A, 1A]] 	* 	x_1
34.59/9.71	>>>
34.59/9.71	
34.59/9.71	   <<<
34.59/9.71	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, -I, -I]] 	* 	x_1
34.59/9.71	>>>
34.59/9.71	
34.59/9.71	   <<<
34.59/9.71	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, 0A], [0A, 0A, 0A], [0A, -I, -I]] 	* 	x_1
34.59/9.71	>>>
34.59/9.71	
34.59/9.71	
34.59/9.71	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
34.59/9.71	
34.59/9.71	   a(x1) -> b(x1)
34.59/9.71	   b(b(c(x1))) -> c(b(c(a(a(x1)))))
34.59/9.71	   c(c(x1)) -> x1
34.59/9.71	
34.59/9.71	
34.59/9.71	----------------------------------------
34.59/9.71	
34.59/9.71	(8)
34.59/9.71	Obligation:
34.59/9.71	Q DP problem:
34.59/9.71	The TRS P consists of the following rules:
34.59/9.71	
34.59/9.71	   A(x1) -> B(x1)
34.59/9.71	   B(b(c(x1))) -> A(x1)
34.59/9.71	
34.59/9.71	The TRS R consists of the following rules:
34.59/9.71	
34.59/9.71	   a(x1) -> b(x1)
34.59/9.71	   b(b(c(x1))) -> c(b(c(a(a(x1)))))
34.59/9.71	   c(c(x1)) -> x1
34.59/9.71	
34.59/9.71	Q is empty.
34.59/9.71	We have to consider all minimal (P,Q,R)-chains.
34.59/9.71	----------------------------------------
34.59/9.71	
34.59/9.71	(9) UsableRulesProof (EQUIVALENT)
34.59/9.71	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.
34.59/9.71	----------------------------------------
34.59/9.71	
34.59/9.71	(10)
34.59/9.71	Obligation:
34.59/9.71	Q DP problem:
34.59/9.71	The TRS P consists of the following rules:
34.59/9.71	
34.59/9.71	   A(x1) -> B(x1)
34.59/9.71	   B(b(c(x1))) -> A(x1)
34.59/9.71	
34.59/9.71	R is empty.
34.59/9.71	Q is empty.
34.59/9.71	We have to consider all minimal (P,Q,R)-chains.
34.59/9.71	----------------------------------------
34.59/9.71	
34.59/9.71	(11) QDPSizeChangeProof (EQUIVALENT)
34.59/9.71	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. 
34.59/9.71	
34.59/9.71	From the DPs we obtained the following set of size-change graphs:
34.59/9.71	*B(b(c(x1))) -> A(x1)
34.59/9.71	The graph contains the following edges 1 > 1
34.59/9.71	
34.59/9.71	
34.59/9.71	*A(x1) -> B(x1)
34.59/9.71	The graph contains the following edges 1 >= 1
34.59/9.71	
34.59/9.71	
34.59/9.71	----------------------------------------
34.59/9.71	
34.59/9.71	(12)
34.59/9.71	YES
34.94/9.77	EOF