24.80/7.27	YES
24.80/7.28	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
24.80/7.28	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
24.80/7.28	
24.80/7.28	
24.80/7.28	Termination w.r.t. Q of the given QTRS could be proven:
24.80/7.28	
24.80/7.28	(0) QTRS
24.80/7.28	(1) QTRS Reverse [EQUIVALENT, 0 ms]
24.80/7.28	(2) QTRS
24.80/7.28	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
24.80/7.28	(4) QDP
24.80/7.28	(5) DependencyGraphProof [EQUIVALENT, 7 ms]
24.80/7.28	(6) QDP
24.80/7.28	(7) QDPOrderProof [EQUIVALENT, 133 ms]
24.80/7.28	(8) QDP
24.80/7.28	(9) QDPOrderProof [EQUIVALENT, 0 ms]
24.80/7.28	(10) QDP
24.80/7.28	(11) UsableRulesProof [EQUIVALENT, 0 ms]
24.80/7.28	(12) QDP
24.80/7.28	(13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
24.80/7.28	(14) YES
24.80/7.28	
24.80/7.28	
24.80/7.28	----------------------------------------
24.80/7.28	
24.80/7.28	(0)
24.80/7.28	Obligation:
24.80/7.28	Q restricted rewrite system:
24.80/7.28	The TRS R consists of the following rules:
24.80/7.28	
24.80/7.28	   a(x1) -> b(b(c(b(c(x1)))))
24.80/7.28	   c(b(b(x1))) -> a(x1)
24.80/7.28	   c(c(x1)) -> x1
24.80/7.28	
24.80/7.28	Q is empty.
24.80/7.28	
24.80/7.28	----------------------------------------
24.80/7.28	
24.80/7.28	(1) QTRS Reverse (EQUIVALENT)
24.80/7.28	We applied the QTRS Reverse Processor [REVERSE].
24.80/7.28	----------------------------------------
24.80/7.28	
24.80/7.28	(2)
24.80/7.28	Obligation:
24.80/7.28	Q restricted rewrite system:
24.80/7.28	The TRS R consists of the following rules:
24.80/7.28	
24.80/7.28	   a(x1) -> c(b(c(b(b(x1)))))
24.80/7.28	   b(b(c(x1))) -> a(x1)
24.80/7.28	   c(c(x1)) -> x1
24.80/7.28	
24.80/7.28	Q is empty.
24.80/7.28	
24.80/7.28	----------------------------------------
24.80/7.28	
24.80/7.28	(3) DependencyPairsProof (EQUIVALENT)
24.80/7.28	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
24.80/7.28	----------------------------------------
24.80/7.28	
24.80/7.28	(4)
24.80/7.28	Obligation:
24.80/7.28	Q DP problem:
24.80/7.28	The TRS P consists of the following rules:
24.80/7.28	
24.80/7.28	   A(x1) -> C(b(c(b(b(x1)))))
24.80/7.28	   A(x1) -> B(c(b(b(x1))))
24.80/7.28	   A(x1) -> C(b(b(x1)))
24.80/7.28	   A(x1) -> B(b(x1))
24.80/7.28	   A(x1) -> B(x1)
24.80/7.28	   B(b(c(x1))) -> A(x1)
24.80/7.28	
24.80/7.28	The TRS R consists of the following rules:
24.80/7.28	
24.80/7.28	   a(x1) -> c(b(c(b(b(x1)))))
24.80/7.28	   b(b(c(x1))) -> a(x1)
24.80/7.28	   c(c(x1)) -> x1
24.80/7.28	
24.80/7.28	Q is empty.
24.80/7.28	We have to consider all minimal (P,Q,R)-chains.
24.80/7.28	----------------------------------------
24.80/7.28	
24.80/7.28	(5) DependencyGraphProof (EQUIVALENT)
24.80/7.28	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
24.80/7.28	----------------------------------------
24.80/7.28	
24.80/7.28	(6)
24.80/7.28	Obligation:
24.80/7.28	Q DP problem:
24.80/7.28	The TRS P consists of the following rules:
24.80/7.28	
24.80/7.28	   A(x1) -> B(c(b(b(x1))))
24.80/7.28	   B(b(c(x1))) -> A(x1)
24.80/7.28	   A(x1) -> B(b(x1))
24.80/7.28	   A(x1) -> B(x1)
24.80/7.28	
24.80/7.28	The TRS R consists of the following rules:
24.80/7.28	
24.80/7.28	   a(x1) -> c(b(c(b(b(x1)))))
24.80/7.28	   b(b(c(x1))) -> a(x1)
24.80/7.28	   c(c(x1)) -> x1
24.80/7.28	
24.80/7.28	Q is empty.
24.80/7.28	We have to consider all minimal (P,Q,R)-chains.
24.80/7.28	----------------------------------------
24.80/7.28	
24.80/7.28	(7) QDPOrderProof (EQUIVALENT)
24.80/7.28	We use the reduction pair processor [LPAR04,JAR06].
24.80/7.28	
24.80/7.28	
24.80/7.28	The following pairs can be oriented strictly and are deleted.
24.80/7.28	
24.80/7.28	   A(x1) -> B(b(x1))
24.80/7.28	The remaining pairs can at least be oriented weakly.
24.80/7.28	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
24.80/7.28	
24.80/7.28	   <<<
24.80/7.28	 POL(A(x_1)) =  	[[1A]] 	 +  	[[0A, 1A, 0A]] 	* 	x_1
24.80/7.28	>>>
24.80/7.28	
24.80/7.28	   <<<
24.80/7.28	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, -I, -I]] 	* 	x_1
24.80/7.28	>>>
24.80/7.28	
24.80/7.28	   <<<
24.80/7.28	 POL(c(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[-I, 0A, 0A], [0A, 1A, 1A], [-I, 0A, 0A]] 	* 	x_1
24.80/7.28	>>>
24.80/7.28	
24.80/7.28	   <<<
24.80/7.28	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, -I], [0A, -I, 0A], [-I, -I, 0A]] 	* 	x_1
24.80/7.28	>>>
24.80/7.28	
24.80/7.28	   <<<
24.80/7.28	 POL(a(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[-I, 0A, 0A], [0A, 1A, 1A], [-I, 0A, 0A]] 	* 	x_1
24.80/7.28	>>>
24.80/7.28	
24.80/7.28	
24.80/7.28	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
24.80/7.28	
24.80/7.28	   b(b(c(x1))) -> a(x1)
24.80/7.28	   c(c(x1)) -> x1
24.80/7.28	   a(x1) -> c(b(c(b(b(x1)))))
24.80/7.28	
24.80/7.28	
24.80/7.28	----------------------------------------
24.80/7.28	
24.80/7.28	(8)
24.80/7.28	Obligation:
24.80/7.28	Q DP problem:
24.80/7.28	The TRS P consists of the following rules:
24.80/7.28	
24.80/7.28	   A(x1) -> B(c(b(b(x1))))
24.80/7.28	   B(b(c(x1))) -> A(x1)
24.80/7.28	   A(x1) -> B(x1)
24.80/7.28	
24.80/7.28	The TRS R consists of the following rules:
24.80/7.28	
24.80/7.28	   a(x1) -> c(b(c(b(b(x1)))))
24.80/7.28	   b(b(c(x1))) -> a(x1)
24.80/7.28	   c(c(x1)) -> x1
24.80/7.28	
24.80/7.28	Q is empty.
24.80/7.28	We have to consider all minimal (P,Q,R)-chains.
24.80/7.28	----------------------------------------
24.80/7.28	
24.80/7.28	(9) QDPOrderProof (EQUIVALENT)
24.80/7.28	We use the reduction pair processor [LPAR04,JAR06].
24.80/7.28	
24.80/7.28	
24.80/7.28	The following pairs can be oriented strictly and are deleted.
24.80/7.28	
24.80/7.28	   A(x1) -> B(c(b(b(x1))))
24.80/7.28	The remaining pairs can at least be oriented weakly.
24.80/7.28	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
24.80/7.28	
24.80/7.28	   <<<
24.80/7.28	 POL(A(x_1)) =  	[[1A]] 	 +  	[[0A, 0A, 1A]] 	* 	x_1
24.80/7.28	>>>
24.80/7.28	
24.80/7.28	   <<<
24.80/7.28	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, -I, -I]] 	* 	x_1
24.80/7.28	>>>
24.80/7.28	
24.80/7.28	   <<<
24.80/7.28	 POL(c(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[-I, -I, 0A], [0A, 0A, 0A], [0A, 0A, 1A]] 	* 	x_1
24.80/7.28	>>>
24.80/7.28	
24.80/7.28	   <<<
24.80/7.28	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, 0A], [0A, 0A, -I], [0A, -I, -I]] 	* 	x_1
24.80/7.28	>>>
24.80/7.28	
24.80/7.28	   <<<
24.80/7.28	 POL(a(x_1)) =  	[[0A], [1A], [1A]] 	 +  	[[-I, -I, 0A], [0A, 0A, 1A], [0A, 0A, 1A]] 	* 	x_1
24.80/7.28	>>>
24.80/7.28	
24.80/7.28	
24.80/7.28	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
24.80/7.28	
24.80/7.28	   b(b(c(x1))) -> a(x1)
24.80/7.28	   c(c(x1)) -> x1
24.80/7.28	   a(x1) -> c(b(c(b(b(x1)))))
24.80/7.28	
24.80/7.28	
24.80/7.28	----------------------------------------
24.80/7.28	
24.80/7.28	(10)
24.80/7.28	Obligation:
24.80/7.28	Q DP problem:
24.80/7.28	The TRS P consists of the following rules:
24.80/7.28	
24.80/7.28	   B(b(c(x1))) -> A(x1)
24.80/7.28	   A(x1) -> B(x1)
24.80/7.28	
24.80/7.28	The TRS R consists of the following rules:
24.80/7.28	
24.80/7.28	   a(x1) -> c(b(c(b(b(x1)))))
24.80/7.28	   b(b(c(x1))) -> a(x1)
24.80/7.28	   c(c(x1)) -> x1
24.80/7.28	
24.80/7.28	Q is empty.
24.80/7.28	We have to consider all minimal (P,Q,R)-chains.
24.80/7.28	----------------------------------------
24.80/7.28	
24.80/7.28	(11) UsableRulesProof (EQUIVALENT)
24.80/7.28	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.
24.80/7.28	----------------------------------------
24.80/7.28	
24.80/7.28	(12)
24.80/7.28	Obligation:
24.80/7.28	Q DP problem:
24.80/7.28	The TRS P consists of the following rules:
24.80/7.28	
24.80/7.28	   B(b(c(x1))) -> A(x1)
24.80/7.28	   A(x1) -> B(x1)
24.80/7.28	
24.80/7.28	R is empty.
24.80/7.28	Q is empty.
24.80/7.28	We have to consider all minimal (P,Q,R)-chains.
24.80/7.28	----------------------------------------
24.80/7.28	
24.80/7.28	(13) QDPSizeChangeProof (EQUIVALENT)
24.80/7.28	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. 
24.80/7.28	
24.80/7.28	From the DPs we obtained the following set of size-change graphs:
24.80/7.28	*A(x1) -> B(x1)
24.80/7.28	The graph contains the following edges 1 >= 1
24.80/7.28	
24.80/7.28	
24.80/7.28	*B(b(c(x1))) -> A(x1)
24.80/7.28	The graph contains the following edges 1 > 1
24.80/7.28	
24.80/7.28	
24.80/7.28	----------------------------------------
24.80/7.28	
24.80/7.28	(14)
24.80/7.28	YES
25.17/7.35	EOF