37.20/10.37	YES
37.42/10.39	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
37.42/10.39	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
37.42/10.39	
37.42/10.39	
37.42/10.39	Termination w.r.t. Q of the given QTRS could be proven:
37.42/10.39	
37.42/10.39	(0) QTRS
37.42/10.39	(1) DependencyPairsProof [EQUIVALENT, 0 ms]
37.42/10.39	(2) QDP
37.42/10.39	(3) QDPOrderProof [EQUIVALENT, 227 ms]
37.42/10.39	(4) QDP
37.42/10.39	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
37.42/10.39	(6) QDP
37.42/10.39	(7) UsableRulesProof [EQUIVALENT, 0 ms]
37.42/10.39	(8) QDP
37.42/10.39	(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
37.42/10.39	(10) YES
37.42/10.39	
37.42/10.39	
37.42/10.39	----------------------------------------
37.42/10.39	
37.42/10.39	(0)
37.42/10.39	Obligation:
37.42/10.39	Q restricted rewrite system:
37.42/10.39	The TRS R consists of the following rules:
37.42/10.39	
37.42/10.39	   a(b(x1)) -> x1
37.42/10.39	   a(c(x1)) -> c(b(c(c(x1))))
37.42/10.39	   b(c(x1)) -> a(b(x1))
37.42/10.39	
37.42/10.39	Q is empty.
37.42/10.39	
37.42/10.39	----------------------------------------
37.42/10.39	
37.42/10.39	(1) DependencyPairsProof (EQUIVALENT)
37.42/10.39	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
37.42/10.39	----------------------------------------
37.42/10.39	
37.42/10.39	(2)
37.42/10.39	Obligation:
37.42/10.39	Q DP problem:
37.42/10.39	The TRS P consists of the following rules:
37.42/10.39	
37.42/10.39	   A(c(x1)) -> B(c(c(x1)))
37.42/10.39	   B(c(x1)) -> A(b(x1))
37.42/10.39	   B(c(x1)) -> B(x1)
37.42/10.39	
37.42/10.39	The TRS R consists of the following rules:
37.42/10.39	
37.42/10.39	   a(b(x1)) -> x1
37.42/10.39	   a(c(x1)) -> c(b(c(c(x1))))
37.42/10.39	   b(c(x1)) -> a(b(x1))
37.42/10.39	
37.42/10.39	Q is empty.
37.42/10.39	We have to consider all minimal (P,Q,R)-chains.
37.42/10.39	----------------------------------------
37.42/10.39	
37.42/10.39	(3) QDPOrderProof (EQUIVALENT)
37.42/10.39	We use the reduction pair processor [LPAR04,JAR06].
37.42/10.39	
37.42/10.39	
37.42/10.39	The following pairs can be oriented strictly and are deleted.
37.42/10.39	
37.42/10.39	   B(c(x1)) -> A(b(x1))
37.42/10.39	The remaining pairs can at least be oriented weakly.
37.42/10.39	Used ordering:  Matrix interpretation [MATRO] with arctic integers [ARCTIC,STERNAGEL_THIEMANN_RTA14]:
37.42/10.39	
37.42/10.39	   <<<
37.42/10.39	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
37.42/10.39	>>>
37.42/10.39	
37.42/10.39	   <<<
37.42/10.39	 POL(c(x_1)) =  	[[0A], [1A], [-I]] 	 +  	[[-I, -1A, -I], [-I, 1A, -I], [-1A, 0A, 1A]] 	* 	x_1
37.42/10.39	>>>
37.42/10.39	
37.42/10.39	   <<<
37.42/10.39	 POL(B(x_1)) =  	[[1A]] 	 +  	[[-I, -1A, -I]] 	* 	x_1
37.42/10.39	>>>
37.42/10.39	
37.42/10.39	   <<<
37.42/10.39	 POL(b(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[-1A, 0A, 1A], [-I, -1A, -I], [-I, -I, -1A]] 	* 	x_1
37.42/10.39	>>>
37.42/10.39	
37.42/10.39	   <<<
37.42/10.39	 POL(a(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[1A, -1A, -I], [-I, 1A, -I], [-I, 0A, 1A]] 	* 	x_1
37.42/10.39	>>>
37.42/10.39	
37.42/10.39	
37.42/10.39	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
37.42/10.39	
37.42/10.39	   b(c(x1)) -> a(b(x1))
37.42/10.39	   a(c(x1)) -> c(b(c(c(x1))))
37.42/10.39	   a(b(x1)) -> x1
37.42/10.39	
37.42/10.39	
37.42/10.39	----------------------------------------
37.42/10.39	
37.42/10.39	(4)
37.42/10.39	Obligation:
37.42/10.39	Q DP problem:
37.42/10.39	The TRS P consists of the following rules:
37.42/10.39	
37.42/10.39	   A(c(x1)) -> B(c(c(x1)))
37.42/10.39	   B(c(x1)) -> B(x1)
37.42/10.39	
37.42/10.39	The TRS R consists of the following rules:
37.42/10.39	
37.42/10.39	   a(b(x1)) -> x1
37.42/10.39	   a(c(x1)) -> c(b(c(c(x1))))
37.42/10.39	   b(c(x1)) -> a(b(x1))
37.42/10.39	
37.42/10.39	Q is empty.
37.42/10.39	We have to consider all minimal (P,Q,R)-chains.
37.42/10.39	----------------------------------------
37.42/10.39	
37.42/10.39	(5) DependencyGraphProof (EQUIVALENT)
37.42/10.39	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
37.42/10.39	----------------------------------------
37.42/10.39	
37.42/10.39	(6)
37.42/10.39	Obligation:
37.42/10.39	Q DP problem:
37.42/10.39	The TRS P consists of the following rules:
37.42/10.39	
37.42/10.39	   B(c(x1)) -> B(x1)
37.42/10.39	
37.42/10.39	The TRS R consists of the following rules:
37.42/10.39	
37.42/10.39	   a(b(x1)) -> x1
37.42/10.39	   a(c(x1)) -> c(b(c(c(x1))))
37.42/10.39	   b(c(x1)) -> a(b(x1))
37.42/10.39	
37.42/10.39	Q is empty.
37.42/10.39	We have to consider all minimal (P,Q,R)-chains.
37.42/10.39	----------------------------------------
37.42/10.39	
37.42/10.39	(7) UsableRulesProof (EQUIVALENT)
37.42/10.39	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.
37.42/10.39	----------------------------------------
37.42/10.39	
37.42/10.39	(8)
37.42/10.39	Obligation:
37.42/10.39	Q DP problem:
37.42/10.39	The TRS P consists of the following rules:
37.42/10.39	
37.42/10.39	   B(c(x1)) -> B(x1)
37.42/10.39	
37.42/10.39	R is empty.
37.42/10.39	Q is empty.
37.42/10.39	We have to consider all minimal (P,Q,R)-chains.
37.42/10.39	----------------------------------------
37.42/10.39	
37.42/10.39	(9) QDPSizeChangeProof (EQUIVALENT)
37.42/10.39	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. 
37.42/10.39	
37.42/10.39	From the DPs we obtained the following set of size-change graphs:
37.42/10.39	*B(c(x1)) -> B(x1)
37.42/10.39	The graph contains the following edges 1 > 1
37.42/10.39	
37.42/10.39	
37.42/10.39	----------------------------------------
37.42/10.39	
37.42/10.39	(10)
37.42/10.39	YES
37.53/10.45	EOF