20.38/6.17	YES
20.95/6.34	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
20.95/6.34	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
20.95/6.34	
20.95/6.34	
20.95/6.34	Termination w.r.t. Q of the given QTRS could be proven:
20.95/6.34	
20.95/6.34	(0) QTRS
20.95/6.34	(1) DependencyPairsProof [EQUIVALENT, 17 ms]
20.95/6.34	(2) QDP
20.95/6.34	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
20.95/6.34	(4) QDP
20.95/6.34	(5) QDPOrderProof [EQUIVALENT, 191 ms]
20.95/6.34	(6) QDP
20.95/6.34	(7) QDPOrderProof [EQUIVALENT, 33 ms]
20.95/6.34	(8) QDP
20.95/6.34	(9) UsableRulesProof [EQUIVALENT, 0 ms]
20.95/6.34	(10) QDP
20.95/6.34	(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
20.95/6.34	(12) YES
20.95/6.34	
20.95/6.34	
20.95/6.34	----------------------------------------
20.95/6.34	
20.95/6.34	(0)
20.95/6.34	Obligation:
20.95/6.34	Q restricted rewrite system:
20.95/6.34	The TRS R consists of the following rules:
20.95/6.34	
20.95/6.34	   a(x1) -> b(x1)
20.95/6.34	   b(b(c(x1))) -> c(a(c(b(a(x1)))))
20.95/6.34	   c(c(x1)) -> x1
20.95/6.34	
20.95/6.34	Q is empty.
20.95/6.34	
20.95/6.34	----------------------------------------
20.95/6.34	
20.95/6.34	(1) DependencyPairsProof (EQUIVALENT)
20.95/6.34	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
20.95/6.34	----------------------------------------
20.95/6.34	
20.95/6.34	(2)
20.95/6.34	Obligation:
20.95/6.34	Q DP problem:
20.95/6.34	The TRS P consists of the following rules:
20.95/6.34	
20.95/6.34	   A(x1) -> B(x1)
20.95/6.34	   B(b(c(x1))) -> C(a(c(b(a(x1)))))
20.95/6.34	   B(b(c(x1))) -> A(c(b(a(x1))))
20.95/6.34	   B(b(c(x1))) -> C(b(a(x1)))
20.95/6.34	   B(b(c(x1))) -> B(a(x1))
20.95/6.34	   B(b(c(x1))) -> A(x1)
20.95/6.34	
20.95/6.34	The TRS R consists of the following rules:
20.95/6.34	
20.95/6.34	   a(x1) -> b(x1)
20.95/6.34	   b(b(c(x1))) -> c(a(c(b(a(x1)))))
20.95/6.34	   c(c(x1)) -> x1
20.95/6.34	
20.95/6.34	Q is empty.
20.95/6.34	We have to consider all minimal (P,Q,R)-chains.
20.95/6.34	----------------------------------------
20.95/6.34	
20.95/6.34	(3) DependencyGraphProof (EQUIVALENT)
20.95/6.34	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
20.95/6.34	----------------------------------------
20.95/6.34	
20.95/6.34	(4)
20.95/6.34	Obligation:
20.95/6.34	Q DP problem:
20.95/6.34	The TRS P consists of the following rules:
20.95/6.34	
20.95/6.34	   B(b(c(x1))) -> A(c(b(a(x1))))
20.95/6.34	   A(x1) -> B(x1)
20.95/6.34	   B(b(c(x1))) -> B(a(x1))
20.95/6.34	   B(b(c(x1))) -> A(x1)
20.95/6.34	
20.95/6.34	The TRS R consists of the following rules:
20.95/6.34	
20.95/6.34	   a(x1) -> b(x1)
20.95/6.34	   b(b(c(x1))) -> c(a(c(b(a(x1)))))
20.95/6.34	   c(c(x1)) -> x1
20.95/6.34	
20.95/6.34	Q is empty.
20.95/6.34	We have to consider all minimal (P,Q,R)-chains.
20.95/6.34	----------------------------------------
20.95/6.34	
20.95/6.34	(5) QDPOrderProof (EQUIVALENT)
20.95/6.34	We use the reduction pair processor [LPAR04,JAR06].
20.95/6.34	
20.95/6.34	
20.95/6.34	The following pairs can be oriented strictly and are deleted.
20.95/6.34	
20.95/6.34	   B(b(c(x1))) -> A(c(b(a(x1))))
20.95/6.34	The remaining pairs can at least be oriented weakly.
20.95/6.34	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
20.95/6.34	
20.95/6.34	   <<<
20.95/6.34	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
20.95/6.34	>>>
20.95/6.34	
20.95/6.34	   <<<
20.95/6.34	 POL(b(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[0A, -I, 0A], [-I, -I, 0A], [-I, 0A, -I]] 	* 	x_1
20.95/6.34	>>>
20.95/6.34	
20.95/6.34	   <<<
20.95/6.34	 POL(c(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[-I, 0A, -I], [0A, 1A, 0A], [-I, 0A, -I]] 	* 	x_1
20.95/6.34	>>>
20.95/6.34	
20.95/6.34	   <<<
20.95/6.34	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
20.95/6.34	>>>
20.95/6.34	
20.95/6.34	   <<<
20.95/6.34	 POL(a(x_1)) =  	[[1A], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [-I, -I, 0A], [-I, 0A, -I]] 	* 	x_1
20.95/6.34	>>>
20.95/6.34	
20.95/6.34	
20.95/6.34	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
20.95/6.34	
20.95/6.34	   a(x1) -> b(x1)
20.95/6.34	   b(b(c(x1))) -> c(a(c(b(a(x1)))))
20.95/6.34	   c(c(x1)) -> x1
20.95/6.34	
20.95/6.34	
20.95/6.34	----------------------------------------
20.95/6.34	
20.95/6.34	(6)
20.95/6.34	Obligation:
20.95/6.34	Q DP problem:
20.95/6.34	The TRS P consists of the following rules:
20.95/6.34	
20.95/6.34	   A(x1) -> B(x1)
20.95/6.34	   B(b(c(x1))) -> B(a(x1))
20.95/6.34	   B(b(c(x1))) -> A(x1)
20.95/6.34	
20.95/6.34	The TRS R consists of the following rules:
20.95/6.34	
20.95/6.34	   a(x1) -> b(x1)
20.95/6.34	   b(b(c(x1))) -> c(a(c(b(a(x1)))))
20.95/6.34	   c(c(x1)) -> x1
20.95/6.34	
20.95/6.34	Q is empty.
20.95/6.34	We have to consider all minimal (P,Q,R)-chains.
20.95/6.34	----------------------------------------
20.95/6.34	
20.95/6.34	(7) QDPOrderProof (EQUIVALENT)
20.95/6.34	We use the reduction pair processor [LPAR04,JAR06].
20.95/6.34	
20.95/6.34	
20.95/6.34	The following pairs can be oriented strictly and are deleted.
20.95/6.34	
20.95/6.34	   B(b(c(x1))) -> B(a(x1))
20.95/6.34	The remaining pairs can at least be oriented weakly.
20.95/6.34	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
20.95/6.34	
20.95/6.34	   <<<
20.95/6.34	 POL(A(x_1)) =  	[[1A]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
20.95/6.34	>>>
20.95/6.34	
20.95/6.34	   <<<
20.95/6.34	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
20.95/6.34	>>>
20.95/6.34	
20.95/6.34	   <<<
20.95/6.34	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, -I], [0A, -I, -I], [-I, 0A, 0A]] 	* 	x_1
20.95/6.34	>>>
20.95/6.34	
20.95/6.34	   <<<
20.95/6.34	 POL(c(x_1)) =  	[[1A], [0A], [-I]] 	 +  	[[1A, 0A, -I], [0A, -I, -I], [0A, -I, 0A]] 	* 	x_1
20.95/6.34	>>>
20.95/6.34	
20.95/6.34	   <<<
20.95/6.34	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, -I], [0A, -I, -I], [0A, 0A, 0A]] 	* 	x_1
20.95/6.34	>>>
20.95/6.34	
20.95/6.34	
20.95/6.34	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
20.95/6.34	
20.95/6.34	   a(x1) -> b(x1)
20.95/6.34	   b(b(c(x1))) -> c(a(c(b(a(x1)))))
20.95/6.34	   c(c(x1)) -> x1
20.95/6.34	
20.95/6.34	
20.95/6.34	----------------------------------------
20.95/6.34	
20.95/6.34	(8)
20.95/6.34	Obligation:
20.95/6.34	Q DP problem:
20.95/6.34	The TRS P consists of the following rules:
20.95/6.34	
20.95/6.34	   A(x1) -> B(x1)
20.95/6.34	   B(b(c(x1))) -> A(x1)
20.95/6.34	
20.95/6.34	The TRS R consists of the following rules:
20.95/6.34	
20.95/6.34	   a(x1) -> b(x1)
20.95/6.34	   b(b(c(x1))) -> c(a(c(b(a(x1)))))
20.95/6.34	   c(c(x1)) -> x1
20.95/6.34	
20.95/6.34	Q is empty.
20.95/6.34	We have to consider all minimal (P,Q,R)-chains.
20.95/6.34	----------------------------------------
20.95/6.34	
20.95/6.34	(9) UsableRulesProof (EQUIVALENT)
20.95/6.34	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.
20.95/6.34	----------------------------------------
20.95/6.34	
20.95/6.34	(10)
20.95/6.34	Obligation:
20.95/6.34	Q DP problem:
20.95/6.34	The TRS P consists of the following rules:
20.95/6.34	
20.95/6.34	   A(x1) -> B(x1)
20.95/6.34	   B(b(c(x1))) -> A(x1)
20.95/6.34	
20.95/6.34	R is empty.
20.95/6.34	Q is empty.
20.95/6.34	We have to consider all minimal (P,Q,R)-chains.
20.95/6.34	----------------------------------------
20.95/6.34	
20.95/6.34	(11) QDPSizeChangeProof (EQUIVALENT)
20.95/6.34	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. 
20.95/6.34	
20.95/6.34	From the DPs we obtained the following set of size-change graphs:
20.95/6.34	*B(b(c(x1))) -> A(x1)
20.95/6.34	The graph contains the following edges 1 > 1
20.95/6.34	
20.95/6.34	
20.95/6.34	*A(x1) -> B(x1)
20.95/6.34	The graph contains the following edges 1 >= 1
20.95/6.34	
20.95/6.34	
20.95/6.34	----------------------------------------
20.95/6.34	
20.95/6.34	(12)
20.95/6.34	YES
21.34/6.41	EOF