13.12/4.31	YES
13.12/4.32	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
13.12/4.32	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
13.12/4.32	
13.12/4.32	
13.12/4.32	Termination w.r.t. Q of the given QTRS could be proven:
13.12/4.32	
13.12/4.32	(0) QTRS
13.12/4.32	(1) DependencyPairsProof [EQUIVALENT, 25 ms]
13.12/4.32	(2) QDP
13.12/4.32	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
13.12/4.32	(4) QDP
13.12/4.32	(5) QDPOrderProof [EQUIVALENT, 131 ms]
13.12/4.32	(6) QDP
13.12/4.32	(7) UsableRulesProof [EQUIVALENT, 0 ms]
13.12/4.32	(8) QDP
13.12/4.32	(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
13.12/4.32	(10) YES
13.12/4.32	
13.12/4.32	
13.12/4.32	----------------------------------------
13.12/4.32	
13.12/4.32	(0)
13.12/4.32	Obligation:
13.12/4.32	Q restricted rewrite system:
13.12/4.32	The TRS R consists of the following rules:
13.12/4.32	
13.12/4.32	   a(x1) -> b(c(b(x1)))
13.12/4.32	   a(b(x1)) -> x1
13.12/4.32	   c(c(b(x1))) -> a(c(c(x1)))
13.12/4.32	
13.12/4.32	Q is empty.
13.12/4.32	
13.12/4.32	----------------------------------------
13.12/4.32	
13.12/4.32	(1) DependencyPairsProof (EQUIVALENT)
13.12/4.32	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
13.12/4.32	----------------------------------------
13.12/4.32	
13.12/4.32	(2)
13.12/4.32	Obligation:
13.12/4.32	Q DP problem:
13.12/4.32	The TRS P consists of the following rules:
13.12/4.32	
13.12/4.32	   A(x1) -> C(b(x1))
13.12/4.32	   C(c(b(x1))) -> A(c(c(x1)))
13.12/4.32	   C(c(b(x1))) -> C(c(x1))
13.12/4.32	   C(c(b(x1))) -> C(x1)
13.12/4.32	
13.12/4.32	The TRS R consists of the following rules:
13.12/4.32	
13.12/4.32	   a(x1) -> b(c(b(x1)))
13.12/4.32	   a(b(x1)) -> x1
13.12/4.32	   c(c(b(x1))) -> a(c(c(x1)))
13.12/4.32	
13.12/4.32	Q is empty.
13.12/4.32	We have to consider all minimal (P,Q,R)-chains.
13.12/4.32	----------------------------------------
13.12/4.32	
13.12/4.32	(3) DependencyGraphProof (EQUIVALENT)
13.12/4.32	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
13.12/4.32	----------------------------------------
13.12/4.32	
13.12/4.32	(4)
13.12/4.32	Obligation:
13.12/4.32	Q DP problem:
13.12/4.32	The TRS P consists of the following rules:
13.12/4.32	
13.12/4.32	   C(c(b(x1))) -> C(x1)
13.12/4.32	   C(c(b(x1))) -> C(c(x1))
13.12/4.32	
13.12/4.32	The TRS R consists of the following rules:
13.12/4.32	
13.12/4.32	   a(x1) -> b(c(b(x1)))
13.12/4.32	   a(b(x1)) -> x1
13.12/4.32	   c(c(b(x1))) -> a(c(c(x1)))
13.12/4.32	
13.12/4.32	Q is empty.
13.12/4.32	We have to consider all minimal (P,Q,R)-chains.
13.12/4.32	----------------------------------------
13.12/4.32	
13.12/4.32	(5) QDPOrderProof (EQUIVALENT)
13.12/4.32	We use the reduction pair processor [LPAR04,JAR06].
13.12/4.32	
13.12/4.32	
13.12/4.32	The following pairs can be oriented strictly and are deleted.
13.12/4.32	
13.12/4.32	   C(c(b(x1))) -> C(c(x1))
13.12/4.32	The remaining pairs can at least be oriented weakly.
13.12/4.32	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
13.12/4.32	
13.12/4.32	   <<<
13.12/4.32	 POL(C(x_1)) =  	[[0A]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
13.12/4.32	>>>
13.12/4.32	
13.12/4.32	   <<<
13.12/4.32	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, -I], [0A, -I, 0A], [-I, 0A, -I]] 	* 	x_1
13.12/4.32	>>>
13.12/4.32	
13.12/4.32	   <<<
13.12/4.32	 POL(b(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[-I, 0A, -I], [0A, 1A, 0A], [-I, 0A, -I]] 	* 	x_1
13.12/4.32	>>>
13.12/4.32	
13.12/4.32	   <<<
13.12/4.32	 POL(a(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[-I, 0A, -I], [0A, 1A, 0A], [-I, 0A, -I]] 	* 	x_1
13.12/4.32	>>>
13.12/4.32	
13.12/4.32	
13.12/4.32	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
13.12/4.32	
13.12/4.32	   c(c(b(x1))) -> a(c(c(x1)))
13.12/4.32	   a(x1) -> b(c(b(x1)))
13.12/4.32	   a(b(x1)) -> x1
13.12/4.32	
13.12/4.32	
13.12/4.32	----------------------------------------
13.12/4.32	
13.12/4.32	(6)
13.12/4.32	Obligation:
13.12/4.32	Q DP problem:
13.12/4.32	The TRS P consists of the following rules:
13.12/4.32	
13.12/4.32	   C(c(b(x1))) -> C(x1)
13.12/4.32	
13.12/4.32	The TRS R consists of the following rules:
13.12/4.32	
13.12/4.32	   a(x1) -> b(c(b(x1)))
13.12/4.32	   a(b(x1)) -> x1
13.12/4.32	   c(c(b(x1))) -> a(c(c(x1)))
13.12/4.32	
13.12/4.32	Q is empty.
13.12/4.32	We have to consider all minimal (P,Q,R)-chains.
13.12/4.32	----------------------------------------
13.12/4.32	
13.12/4.32	(7) UsableRulesProof (EQUIVALENT)
13.12/4.32	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.
13.12/4.32	----------------------------------------
13.12/4.32	
13.12/4.32	(8)
13.12/4.32	Obligation:
13.12/4.32	Q DP problem:
13.12/4.32	The TRS P consists of the following rules:
13.12/4.32	
13.12/4.32	   C(c(b(x1))) -> C(x1)
13.12/4.32	
13.12/4.32	R is empty.
13.12/4.32	Q is empty.
13.12/4.32	We have to consider all minimal (P,Q,R)-chains.
13.12/4.32	----------------------------------------
13.12/4.32	
13.12/4.32	(9) QDPSizeChangeProof (EQUIVALENT)
13.12/4.32	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. 
13.12/4.32	
13.12/4.32	From the DPs we obtained the following set of size-change graphs:
13.12/4.32	*C(c(b(x1))) -> C(x1)
13.12/4.32	The graph contains the following edges 1 > 1
13.12/4.32	
13.12/4.32	
13.12/4.32	----------------------------------------
13.12/4.32	
13.12/4.32	(10)
13.12/4.32	YES
13.66/4.39	EOF