21.01/6.18	YES
21.01/6.21	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
21.01/6.21	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
21.01/6.21	
21.01/6.21	
21.01/6.21	Termination w.r.t. Q of the given QTRS could be proven:
21.01/6.21	
21.01/6.21	(0) QTRS
21.01/6.21	(1) QTRS Reverse [EQUIVALENT, 0 ms]
21.01/6.21	(2) QTRS
21.01/6.21	(3) DependencyPairsProof [EQUIVALENT, 5 ms]
21.01/6.21	(4) QDP
21.01/6.21	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
21.01/6.21	(6) QDP
21.01/6.21	(7) QDPOrderProof [EQUIVALENT, 148 ms]
21.01/6.21	(8) QDP
21.01/6.21	(9) UsableRulesProof [EQUIVALENT, 1 ms]
21.01/6.21	(10) QDP
21.01/6.21	(11) QDPSizeChangeProof [EQUIVALENT, 2 ms]
21.01/6.21	(12) YES
21.01/6.21	
21.01/6.21	
21.01/6.21	----------------------------------------
21.01/6.21	
21.01/6.21	(0)
21.01/6.21	Obligation:
21.01/6.21	Q restricted rewrite system:
21.01/6.21	The TRS R consists of the following rules:
21.01/6.21	
21.01/6.21	   a(x1) -> b(x1)
21.01/6.21	   a(b(x1)) -> x1
21.01/6.21	   b(c(c(x1))) -> c(c(b(c(a(x1)))))
21.01/6.21	
21.01/6.21	Q is empty.
21.01/6.21	
21.01/6.21	----------------------------------------
21.01/6.21	
21.01/6.21	(1) QTRS Reverse (EQUIVALENT)
21.01/6.21	We applied the QTRS Reverse Processor [REVERSE].
21.01/6.21	----------------------------------------
21.01/6.21	
21.01/6.21	(2)
21.01/6.21	Obligation:
21.01/6.21	Q restricted rewrite system:
21.01/6.21	The TRS R consists of the following rules:
21.01/6.21	
21.01/6.21	   a(x1) -> b(x1)
21.01/6.21	   b(a(x1)) -> x1
21.01/6.21	   c(c(b(x1))) -> a(c(b(c(c(x1)))))
21.01/6.21	
21.01/6.21	Q is empty.
21.01/6.21	
21.01/6.21	----------------------------------------
21.01/6.21	
21.01/6.21	(3) DependencyPairsProof (EQUIVALENT)
21.01/6.21	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
21.01/6.21	----------------------------------------
21.01/6.21	
21.01/6.21	(4)
21.01/6.21	Obligation:
21.01/6.21	Q DP problem:
21.01/6.21	The TRS P consists of the following rules:
21.01/6.21	
21.01/6.21	   A(x1) -> B(x1)
21.01/6.21	   C(c(b(x1))) -> A(c(b(c(c(x1)))))
21.01/6.21	   C(c(b(x1))) -> C(b(c(c(x1))))
21.01/6.21	   C(c(b(x1))) -> B(c(c(x1)))
21.01/6.21	   C(c(b(x1))) -> C(c(x1))
21.01/6.21	   C(c(b(x1))) -> C(x1)
21.01/6.21	
21.01/6.21	The TRS R consists of the following rules:
21.01/6.21	
21.01/6.21	   a(x1) -> b(x1)
21.01/6.21	   b(a(x1)) -> x1
21.01/6.21	   c(c(b(x1))) -> a(c(b(c(c(x1)))))
21.01/6.21	
21.01/6.21	Q is empty.
21.01/6.21	We have to consider all minimal (P,Q,R)-chains.
21.01/6.21	----------------------------------------
21.01/6.21	
21.01/6.21	(5) DependencyGraphProof (EQUIVALENT)
21.01/6.21	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
21.01/6.21	----------------------------------------
21.01/6.21	
21.01/6.21	(6)
21.01/6.21	Obligation:
21.01/6.21	Q DP problem:
21.01/6.21	The TRS P consists of the following rules:
21.01/6.21	
21.01/6.21	   C(c(b(x1))) -> C(c(x1))
21.01/6.21	   C(c(b(x1))) -> C(b(c(c(x1))))
21.01/6.21	   C(c(b(x1))) -> C(x1)
21.01/6.21	
21.01/6.21	The TRS R consists of the following rules:
21.01/6.21	
21.01/6.21	   a(x1) -> b(x1)
21.01/6.21	   b(a(x1)) -> x1
21.01/6.21	   c(c(b(x1))) -> a(c(b(c(c(x1)))))
21.01/6.21	
21.01/6.21	Q is empty.
21.01/6.21	We have to consider all minimal (P,Q,R)-chains.
21.01/6.21	----------------------------------------
21.01/6.21	
21.01/6.21	(7) QDPOrderProof (EQUIVALENT)
21.01/6.21	We use the reduction pair processor [LPAR04,JAR06].
21.01/6.21	
21.01/6.21	
21.01/6.21	The following pairs can be oriented strictly and are deleted.
21.01/6.21	
21.01/6.21	   C(c(b(x1))) -> C(c(x1))
21.01/6.21	   C(c(b(x1))) -> C(b(c(c(x1))))
21.01/6.21	The remaining pairs can at least be oriented weakly.
21.01/6.21	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
21.01/6.21	
21.01/6.21	   <<<
21.01/6.21	 POL(C(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
21.01/6.21	>>>
21.01/6.21	
21.01/6.21	   <<<
21.01/6.21	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, 0A], [-I, -I, -I], [0A, 0A, -I]] 	* 	x_1
21.01/6.21	>>>
21.01/6.21	
21.01/6.21	   <<<
21.01/6.21	 POL(b(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[-I, 0A, 0A], [-I, 0A, -I], [0A, 0A, 1A]] 	* 	x_1
21.01/6.21	>>>
21.01/6.21	
21.01/6.21	   <<<
21.01/6.21	 POL(a(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[-I, 0A, 0A], [-I, 0A, -I], [0A, 0A, 1A]] 	* 	x_1
21.01/6.21	>>>
21.01/6.21	
21.01/6.21	
21.01/6.21	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
21.01/6.21	
21.01/6.21	   c(c(b(x1))) -> a(c(b(c(c(x1)))))
21.01/6.21	   b(a(x1)) -> x1
21.01/6.21	   a(x1) -> b(x1)
21.01/6.21	
21.01/6.21	
21.01/6.21	----------------------------------------
21.01/6.21	
21.01/6.21	(8)
21.01/6.21	Obligation:
21.01/6.21	Q DP problem:
21.01/6.21	The TRS P consists of the following rules:
21.01/6.21	
21.01/6.21	   C(c(b(x1))) -> C(x1)
21.01/6.21	
21.01/6.21	The TRS R consists of the following rules:
21.01/6.21	
21.01/6.21	   a(x1) -> b(x1)
21.01/6.21	   b(a(x1)) -> x1
21.01/6.21	   c(c(b(x1))) -> a(c(b(c(c(x1)))))
21.01/6.21	
21.01/6.21	Q is empty.
21.01/6.21	We have to consider all minimal (P,Q,R)-chains.
21.01/6.21	----------------------------------------
21.01/6.21	
21.01/6.21	(9) UsableRulesProof (EQUIVALENT)
21.01/6.21	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.
21.01/6.21	----------------------------------------
21.01/6.21	
21.01/6.21	(10)
21.01/6.21	Obligation:
21.01/6.21	Q DP problem:
21.01/6.21	The TRS P consists of the following rules:
21.01/6.21	
21.01/6.21	   C(c(b(x1))) -> C(x1)
21.01/6.21	
21.01/6.21	R is empty.
21.01/6.21	Q is empty.
21.01/6.21	We have to consider all minimal (P,Q,R)-chains.
21.01/6.21	----------------------------------------
21.01/6.21	
21.01/6.21	(11) QDPSizeChangeProof (EQUIVALENT)
21.01/6.21	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. 
21.01/6.21	
21.01/6.21	From the DPs we obtained the following set of size-change graphs:
21.01/6.21	*C(c(b(x1))) -> C(x1)
21.01/6.21	The graph contains the following edges 1 > 1
21.01/6.21	
21.01/6.21	
21.01/6.21	----------------------------------------
21.01/6.21	
21.01/6.21	(12)
21.01/6.21	YES
21.01/6.25	EOF