24.75/7.14	YES
24.75/7.16	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
24.75/7.16	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
24.75/7.16	
24.75/7.16	
24.75/7.16	Termination w.r.t. Q of the given QTRS could be proven:
24.75/7.16	
24.75/7.16	(0) QTRS
24.75/7.16	(1) DependencyPairsProof [EQUIVALENT, 0 ms]
24.75/7.16	(2) QDP
24.75/7.16	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
24.75/7.16	(4) QDP
24.75/7.16	(5) QDPOrderProof [EQUIVALENT, 208 ms]
24.75/7.16	(6) QDP
24.75/7.16	(7) QDPOrderProof [EQUIVALENT, 68 ms]
24.75/7.16	(8) QDP
24.75/7.16	(9) UsableRulesProof [EQUIVALENT, 2 ms]
24.75/7.16	(10) QDP
24.75/7.16	(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
24.75/7.16	(12) YES
24.75/7.16	
24.75/7.16	
24.75/7.16	----------------------------------------
24.75/7.16	
24.75/7.16	(0)
24.75/7.16	Obligation:
24.75/7.16	Q restricted rewrite system:
24.75/7.16	The TRS R consists of the following rules:
24.75/7.16	
24.75/7.16	   a(x1) -> b(x1)
24.75/7.16	   b(a(c(x1))) -> c(a(c(a(a(x1)))))
24.75/7.16	   c(c(x1)) -> x1
24.75/7.16	
24.75/7.16	Q is empty.
24.75/7.16	
24.75/7.16	----------------------------------------
24.75/7.16	
24.75/7.16	(1) DependencyPairsProof (EQUIVALENT)
24.75/7.16	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
24.75/7.16	----------------------------------------
24.75/7.16	
24.75/7.16	(2)
24.75/7.16	Obligation:
24.75/7.16	Q DP problem:
24.75/7.16	The TRS P consists of the following rules:
24.75/7.16	
24.75/7.16	   A(x1) -> B(x1)
24.75/7.16	   B(a(c(x1))) -> C(a(c(a(a(x1)))))
24.75/7.16	   B(a(c(x1))) -> A(c(a(a(x1))))
24.75/7.16	   B(a(c(x1))) -> C(a(a(x1)))
24.75/7.16	   B(a(c(x1))) -> A(a(x1))
24.75/7.16	   B(a(c(x1))) -> A(x1)
24.75/7.16	
24.75/7.16	The TRS R consists of the following rules:
24.75/7.16	
24.75/7.16	   a(x1) -> b(x1)
24.75/7.16	   b(a(c(x1))) -> c(a(c(a(a(x1)))))
24.75/7.16	   c(c(x1)) -> x1
24.75/7.16	
24.75/7.16	Q is empty.
24.75/7.16	We have to consider all minimal (P,Q,R)-chains.
24.75/7.16	----------------------------------------
24.75/7.16	
24.75/7.16	(3) DependencyGraphProof (EQUIVALENT)
24.75/7.16	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
24.75/7.16	----------------------------------------
24.75/7.16	
24.75/7.16	(4)
24.75/7.16	Obligation:
24.75/7.16	Q DP problem:
24.75/7.16	The TRS P consists of the following rules:
24.75/7.16	
24.75/7.16	   B(a(c(x1))) -> A(c(a(a(x1))))
24.75/7.16	   A(x1) -> B(x1)
24.75/7.16	   B(a(c(x1))) -> A(a(x1))
24.75/7.16	   B(a(c(x1))) -> A(x1)
24.75/7.16	
24.75/7.16	The TRS R consists of the following rules:
24.75/7.16	
24.75/7.16	   a(x1) -> b(x1)
24.75/7.16	   b(a(c(x1))) -> c(a(c(a(a(x1)))))
24.75/7.16	   c(c(x1)) -> x1
24.75/7.16	
24.75/7.16	Q is empty.
24.75/7.16	We have to consider all minimal (P,Q,R)-chains.
24.75/7.16	----------------------------------------
24.75/7.16	
24.75/7.16	(5) QDPOrderProof (EQUIVALENT)
24.75/7.16	We use the reduction pair processor [LPAR04,JAR06].
24.75/7.16	
24.75/7.16	
24.75/7.16	The following pairs can be oriented strictly and are deleted.
24.75/7.16	
24.75/7.16	   B(a(c(x1))) -> A(c(a(a(x1))))
24.75/7.16	The remaining pairs can at least be oriented weakly.
24.75/7.16	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
24.75/7.16	
24.75/7.16	   <<<
24.75/7.16	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
24.75/7.16	>>>
24.75/7.16	
24.75/7.16	   <<<
24.75/7.16	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, 0A], [0A, 0A, 0A], [0A, -I, -I]] 	* 	x_1
24.75/7.16	>>>
24.75/7.16	
24.75/7.16	   <<<
24.75/7.16	 POL(c(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[1A, 0A, 0A], [0A, -I, -I], [0A, -I, -I]] 	* 	x_1
24.75/7.16	>>>
24.75/7.16	
24.75/7.16	   <<<
24.75/7.16	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
24.75/7.16	>>>
24.75/7.16	
24.75/7.16	   <<<
24.75/7.16	 POL(b(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[-I, -I, 0A], [0A, 0A, 0A], [0A, -I, -I]] 	* 	x_1
24.75/7.16	>>>
24.75/7.16	
24.75/7.16	
24.75/7.16	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
24.75/7.16	
24.75/7.16	   a(x1) -> b(x1)
24.75/7.16	   c(c(x1)) -> x1
24.75/7.16	   b(a(c(x1))) -> c(a(c(a(a(x1)))))
24.75/7.16	
24.75/7.16	
24.75/7.16	----------------------------------------
24.75/7.16	
24.75/7.16	(6)
24.75/7.16	Obligation:
24.75/7.16	Q DP problem:
24.75/7.16	The TRS P consists of the following rules:
24.75/7.16	
24.75/7.16	   A(x1) -> B(x1)
24.75/7.16	   B(a(c(x1))) -> A(a(x1))
24.75/7.16	   B(a(c(x1))) -> A(x1)
24.75/7.16	
24.75/7.16	The TRS R consists of the following rules:
24.75/7.16	
24.75/7.16	   a(x1) -> b(x1)
24.75/7.16	   b(a(c(x1))) -> c(a(c(a(a(x1)))))
24.75/7.16	   c(c(x1)) -> x1
24.75/7.16	
24.75/7.16	Q is empty.
24.75/7.16	We have to consider all minimal (P,Q,R)-chains.
24.75/7.16	----------------------------------------
24.75/7.16	
24.75/7.16	(7) QDPOrderProof (EQUIVALENT)
24.75/7.16	We use the reduction pair processor [LPAR04,JAR06].
24.75/7.16	
24.75/7.16	
24.75/7.16	The following pairs can be oriented strictly and are deleted.
24.75/7.16	
24.75/7.16	   B(a(c(x1))) -> A(a(x1))
24.75/7.16	The remaining pairs can at least be oriented weakly.
24.75/7.16	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
24.75/7.16	
24.75/7.16	   <<<
24.75/7.16	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, -I, -I]] 	* 	x_1
24.75/7.16	>>>
24.75/7.16	
24.75/7.16	   <<<
24.75/7.16	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, -I, -I]] 	* 	x_1
24.75/7.16	>>>
24.75/7.16	
24.75/7.16	   <<<
24.75/7.16	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, 0A], [0A, 0A, 0A], [0A, -I, -I]] 	* 	x_1
24.75/7.16	>>>
24.75/7.16	
24.75/7.16	   <<<
24.75/7.16	 POL(c(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[-I, -I, 0A], [-I, 0A, -I], [0A, -I, 1A]] 	* 	x_1
24.75/7.16	>>>
24.75/7.16	
24.75/7.16	   <<<
24.75/7.16	 POL(b(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[-I, -I, 0A], [0A, 0A, 0A], [0A, -I, -I]] 	* 	x_1
24.75/7.16	>>>
24.75/7.16	
24.75/7.16	
24.75/7.16	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
24.75/7.16	
24.75/7.16	   a(x1) -> b(x1)
24.75/7.16	   b(a(c(x1))) -> c(a(c(a(a(x1)))))
24.75/7.16	   c(c(x1)) -> x1
24.75/7.16	
24.75/7.16	
24.75/7.16	----------------------------------------
24.75/7.16	
24.75/7.16	(8)
24.75/7.16	Obligation:
24.75/7.16	Q DP problem:
24.75/7.16	The TRS P consists of the following rules:
24.75/7.16	
24.75/7.16	   A(x1) -> B(x1)
24.75/7.16	   B(a(c(x1))) -> A(x1)
24.75/7.16	
24.75/7.16	The TRS R consists of the following rules:
24.75/7.16	
24.75/7.16	   a(x1) -> b(x1)
24.75/7.16	   b(a(c(x1))) -> c(a(c(a(a(x1)))))
24.75/7.16	   c(c(x1)) -> x1
24.75/7.16	
24.75/7.16	Q is empty.
24.75/7.16	We have to consider all minimal (P,Q,R)-chains.
24.75/7.16	----------------------------------------
24.75/7.16	
24.75/7.16	(9) UsableRulesProof (EQUIVALENT)
24.75/7.16	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.
24.75/7.16	----------------------------------------
24.75/7.16	
24.75/7.16	(10)
24.75/7.16	Obligation:
24.75/7.16	Q DP problem:
24.75/7.16	The TRS P consists of the following rules:
24.75/7.16	
24.75/7.16	   A(x1) -> B(x1)
24.75/7.16	   B(a(c(x1))) -> A(x1)
24.75/7.16	
24.75/7.16	R is empty.
24.75/7.16	Q is empty.
24.75/7.16	We have to consider all minimal (P,Q,R)-chains.
24.75/7.16	----------------------------------------
24.75/7.16	
24.75/7.16	(11) QDPSizeChangeProof (EQUIVALENT)
24.75/7.16	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. 
24.75/7.16	
24.75/7.16	From the DPs we obtained the following set of size-change graphs:
24.75/7.16	*B(a(c(x1))) -> A(x1)
24.75/7.16	The graph contains the following edges 1 > 1
24.75/7.16	
24.75/7.16	
24.75/7.16	*A(x1) -> B(x1)
24.75/7.16	The graph contains the following edges 1 >= 1
24.75/7.16	
24.75/7.16	
24.75/7.16	----------------------------------------
24.75/7.16	
24.75/7.16	(12)
24.75/7.16	YES
25.06/9.07	EOF