22.39/6.71	YES
22.77/6.73	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
22.77/6.73	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
22.77/6.73	
22.77/6.73	
22.77/6.73	Termination w.r.t. Q of the given QTRS could be proven:
22.77/6.73	
22.77/6.73	(0) QTRS
22.77/6.73	(1) DependencyPairsProof [EQUIVALENT, 25 ms]
22.77/6.73	(2) QDP
22.77/6.73	(3) DependencyGraphProof [EQUIVALENT, 6 ms]
22.77/6.73	(4) QDP
22.77/6.73	(5) QDPOrderProof [EQUIVALENT, 110 ms]
22.77/6.73	(6) QDP
22.77/6.73	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
22.77/6.73	(8) QDP
22.77/6.73	(9) UsableRulesProof [EQUIVALENT, 0 ms]
22.77/6.73	(10) QDP
22.77/6.73	(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
22.77/6.73	(12) YES
22.77/6.73	
22.77/6.73	
22.77/6.73	----------------------------------------
22.77/6.73	
22.77/6.73	(0)
22.77/6.73	Obligation:
22.77/6.73	Q restricted rewrite system:
22.77/6.73	The TRS R consists of the following rules:
22.77/6.73	
22.77/6.73	   a(x1) -> b(c(b(c(x1))))
22.77/6.73	   b(b(x1)) -> x1
22.77/6.73	   c(c(b(x1))) -> a(c(x1))
22.77/6.73	
22.77/6.73	Q is empty.
22.77/6.73	
22.77/6.73	----------------------------------------
22.77/6.73	
22.77/6.73	(1) DependencyPairsProof (EQUIVALENT)
22.77/6.73	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
22.77/6.73	----------------------------------------
22.77/6.73	
22.77/6.73	(2)
22.77/6.73	Obligation:
22.77/6.73	Q DP problem:
22.77/6.73	The TRS P consists of the following rules:
22.77/6.73	
22.77/6.73	   A(x1) -> B(c(b(c(x1))))
22.77/6.73	   A(x1) -> C(b(c(x1)))
22.77/6.73	   A(x1) -> B(c(x1))
22.77/6.73	   A(x1) -> C(x1)
22.77/6.73	   C(c(b(x1))) -> A(c(x1))
22.77/6.73	   C(c(b(x1))) -> C(x1)
22.77/6.73	
22.77/6.73	The TRS R consists of the following rules:
22.77/6.73	
22.77/6.73	   a(x1) -> b(c(b(c(x1))))
22.77/6.73	   b(b(x1)) -> x1
22.77/6.73	   c(c(b(x1))) -> a(c(x1))
22.77/6.73	
22.77/6.73	Q is empty.
22.77/6.73	We have to consider all minimal (P,Q,R)-chains.
22.77/6.73	----------------------------------------
22.77/6.73	
22.77/6.73	(3) DependencyGraphProof (EQUIVALENT)
22.77/6.73	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
22.77/6.73	----------------------------------------
22.77/6.73	
22.77/6.73	(4)
22.77/6.73	Obligation:
22.77/6.73	Q DP problem:
22.77/6.73	The TRS P consists of the following rules:
22.77/6.73	
22.77/6.73	   A(x1) -> C(b(c(x1)))
22.77/6.73	   C(c(b(x1))) -> A(c(x1))
22.77/6.73	   A(x1) -> C(x1)
22.77/6.73	   C(c(b(x1))) -> C(x1)
22.77/6.73	
22.77/6.73	The TRS R consists of the following rules:
22.77/6.73	
22.77/6.73	   a(x1) -> b(c(b(c(x1))))
22.77/6.73	   b(b(x1)) -> x1
22.77/6.73	   c(c(b(x1))) -> a(c(x1))
22.77/6.73	
22.77/6.73	Q is empty.
22.77/6.73	We have to consider all minimal (P,Q,R)-chains.
22.77/6.73	----------------------------------------
22.77/6.73	
22.77/6.73	(5) QDPOrderProof (EQUIVALENT)
22.77/6.73	We use the reduction pair processor [LPAR04,JAR06].
22.77/6.73	
22.77/6.73	
22.77/6.73	The following pairs can be oriented strictly and are deleted.
22.77/6.73	
22.77/6.73	   C(c(b(x1))) -> A(c(x1))
22.77/6.73	The remaining pairs can at least be oriented weakly.
22.77/6.73	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
22.77/6.73	
22.77/6.73	   <<<
22.77/6.73	 POL(A(x_1)) =  	[[1A]] 	 +  	[[1A, 0A, 0A]] 	* 	x_1
22.77/6.73	>>>
22.77/6.73	
22.77/6.73	   <<<
22.77/6.73	 POL(C(x_1)) =  	[[0A]] 	 +  	[[1A, 0A, 0A]] 	* 	x_1
22.77/6.73	>>>
22.77/6.73	
22.77/6.73	   <<<
22.77/6.73	 POL(b(x_1)) =  	[[0A], [1A], [-I]] 	 +  	[[-I, 0A, -I], [0A, 1A, 0A], [0A, 0A, 0A]] 	* 	x_1
22.77/6.73	>>>
22.77/6.73	
22.77/6.73	   <<<
22.77/6.73	 POL(c(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[-I, 0A, -I], [0A, -I, -I], [0A, 0A, 0A]] 	* 	x_1
22.77/6.73	>>>
22.77/6.73	
22.77/6.73	   <<<
22.77/6.73	 POL(a(x_1)) =  	[[0A], [1A], [1A]] 	 +  	[[0A, -I, -I], [1A, 0A, 0A], [1A, 0A, 0A]] 	* 	x_1
22.77/6.73	>>>
22.77/6.73	
22.77/6.73	
22.77/6.73	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
22.77/6.73	
22.77/6.73	   c(c(b(x1))) -> a(c(x1))
22.77/6.73	   b(b(x1)) -> x1
22.77/6.73	   a(x1) -> b(c(b(c(x1))))
22.77/6.73	
22.77/6.73	
22.77/6.73	----------------------------------------
22.77/6.73	
22.77/6.73	(6)
22.77/6.73	Obligation:
22.77/6.73	Q DP problem:
22.77/6.73	The TRS P consists of the following rules:
22.77/6.73	
22.77/6.73	   A(x1) -> C(b(c(x1)))
22.77/6.73	   A(x1) -> C(x1)
22.77/6.73	   C(c(b(x1))) -> C(x1)
22.77/6.73	
22.77/6.73	The TRS R consists of the following rules:
22.77/6.73	
22.77/6.73	   a(x1) -> b(c(b(c(x1))))
22.77/6.73	   b(b(x1)) -> x1
22.77/6.73	   c(c(b(x1))) -> a(c(x1))
22.77/6.73	
22.77/6.73	Q is empty.
22.77/6.73	We have to consider all minimal (P,Q,R)-chains.
22.77/6.73	----------------------------------------
22.77/6.73	
22.77/6.73	(7) DependencyGraphProof (EQUIVALENT)
22.77/6.73	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
22.77/6.73	----------------------------------------
22.77/6.73	
22.77/6.73	(8)
22.77/6.73	Obligation:
22.77/6.73	Q DP problem:
22.77/6.73	The TRS P consists of the following rules:
22.77/6.73	
22.77/6.73	   C(c(b(x1))) -> C(x1)
22.77/6.73	
22.77/6.73	The TRS R consists of the following rules:
22.77/6.73	
22.77/6.73	   a(x1) -> b(c(b(c(x1))))
22.77/6.73	   b(b(x1)) -> x1
22.77/6.73	   c(c(b(x1))) -> a(c(x1))
22.77/6.73	
22.77/6.73	Q is empty.
22.77/6.73	We have to consider all minimal (P,Q,R)-chains.
22.77/6.73	----------------------------------------
22.77/6.73	
22.77/6.73	(9) UsableRulesProof (EQUIVALENT)
22.77/6.73	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.
22.77/6.73	----------------------------------------
22.77/6.73	
22.77/6.73	(10)
22.77/6.73	Obligation:
22.77/6.73	Q DP problem:
22.77/6.73	The TRS P consists of the following rules:
22.77/6.73	
22.77/6.73	   C(c(b(x1))) -> C(x1)
22.77/6.73	
22.77/6.73	R is empty.
22.77/6.73	Q is empty.
22.77/6.73	We have to consider all minimal (P,Q,R)-chains.
22.77/6.73	----------------------------------------
22.77/6.73	
22.77/6.73	(11) QDPSizeChangeProof (EQUIVALENT)
22.77/6.73	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. 
22.77/6.73	
22.77/6.73	From the DPs we obtained the following set of size-change graphs:
22.77/6.73	*C(c(b(x1))) -> C(x1)
22.77/6.73	The graph contains the following edges 1 > 1
22.77/6.73	
22.77/6.73	
22.77/6.73	----------------------------------------
22.77/6.73	
22.77/6.73	(12)
22.77/6.73	YES
22.77/6.79	EOF