22.82/6.81	YES
25.85/7.85	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
25.85/7.85	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
25.85/7.85	
25.85/7.85	
25.85/7.85	Termination w.r.t. Q of the given QTRS could be proven:
25.85/7.85	
25.85/7.85	(0) QTRS
25.85/7.85	(1) DependencyPairsProof [EQUIVALENT, 21 ms]
25.85/7.85	(2) QDP
25.85/7.85	(3) QDPOrderProof [EQUIVALENT, 154 ms]
25.85/7.85	(4) QDP
25.85/7.85	(5) UsableRulesProof [EQUIVALENT, 0 ms]
25.85/7.85	(6) QDP
25.85/7.85	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
25.85/7.85	(8) YES
25.85/7.85	
25.85/7.85	
25.85/7.85	----------------------------------------
25.85/7.85	
25.85/7.85	(0)
25.85/7.85	Obligation:
25.85/7.85	Q restricted rewrite system:
25.85/7.85	The TRS R consists of the following rules:
25.85/7.85	
25.85/7.85	   a(x1) -> b(c(x1))
25.85/7.85	   a(a(x1)) -> x1
25.85/7.85	   a(b(b(x1))) -> b(b(a(a(x1))))
25.85/7.85	
25.85/7.85	Q is empty.
25.85/7.85	
25.85/7.85	----------------------------------------
25.85/7.85	
25.85/7.85	(1) DependencyPairsProof (EQUIVALENT)
25.85/7.85	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
25.85/7.85	----------------------------------------
25.85/7.85	
25.85/7.85	(2)
25.85/7.85	Obligation:
25.85/7.85	Q DP problem:
25.85/7.85	The TRS P consists of the following rules:
25.85/7.85	
25.85/7.85	   A(b(b(x1))) -> A(a(x1))
25.85/7.85	   A(b(b(x1))) -> A(x1)
25.85/7.85	
25.85/7.85	The TRS R consists of the following rules:
25.85/7.85	
25.85/7.85	   a(x1) -> b(c(x1))
25.85/7.85	   a(a(x1)) -> x1
25.85/7.85	   a(b(b(x1))) -> b(b(a(a(x1))))
25.85/7.85	
25.85/7.85	Q is empty.
25.85/7.85	We have to consider all minimal (P,Q,R)-chains.
25.85/7.85	----------------------------------------
25.85/7.85	
25.85/7.85	(3) QDPOrderProof (EQUIVALENT)
25.85/7.85	We use the reduction pair processor [LPAR04,JAR06].
25.85/7.85	
25.85/7.85	
25.85/7.85	The following pairs can be oriented strictly and are deleted.
25.85/7.85	
25.85/7.85	   A(b(b(x1))) -> A(a(x1))
25.85/7.85	The remaining pairs can at least be oriented weakly.
25.85/7.85	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
25.85/7.85	
25.85/7.85	   <<<
25.85/7.85	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, -I, -I]] 	* 	x_1
25.85/7.85	>>>
25.85/7.85	
25.85/7.85	   <<<
25.85/7.85	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 1A, -I], [-I, 0A, 0A], [0A, 1A, 0A]] 	* 	x_1
25.85/7.85	>>>
25.85/7.85	
25.85/7.85	   <<<
25.85/7.85	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, 0A], [0A, -I, -I], [0A, -I, -I]] 	* 	x_1
25.85/7.85	>>>
25.85/7.85	
25.85/7.85	   <<<
25.85/7.85	 POL(c(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[-I, -I, -I], [-I, -I, -I], [0A, -I, -I]] 	* 	x_1
25.85/7.85	>>>
25.85/7.85	
25.85/7.85	
25.85/7.85	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
25.85/7.85	
25.85/7.85	   a(x1) -> b(c(x1))
25.85/7.85	   a(a(x1)) -> x1
25.85/7.85	   a(b(b(x1))) -> b(b(a(a(x1))))
25.85/7.85	
25.85/7.85	
25.85/7.85	----------------------------------------
25.85/7.85	
25.85/7.85	(4)
25.85/7.85	Obligation:
25.85/7.85	Q DP problem:
25.85/7.85	The TRS P consists of the following rules:
25.85/7.85	
25.85/7.85	   A(b(b(x1))) -> A(x1)
25.85/7.85	
25.85/7.85	The TRS R consists of the following rules:
25.85/7.85	
25.85/7.85	   a(x1) -> b(c(x1))
25.85/7.85	   a(a(x1)) -> x1
25.85/7.85	   a(b(b(x1))) -> b(b(a(a(x1))))
25.85/7.85	
25.85/7.85	Q is empty.
25.85/7.85	We have to consider all minimal (P,Q,R)-chains.
25.85/7.85	----------------------------------------
25.85/7.85	
25.85/7.85	(5) UsableRulesProof (EQUIVALENT)
25.85/7.85	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.
25.85/7.85	----------------------------------------
25.85/7.85	
25.85/7.85	(6)
25.85/7.85	Obligation:
25.85/7.85	Q DP problem:
25.85/7.85	The TRS P consists of the following rules:
25.85/7.85	
25.85/7.85	   A(b(b(x1))) -> A(x1)
25.85/7.85	
25.85/7.85	R is empty.
25.85/7.85	Q is empty.
25.85/7.85	We have to consider all minimal (P,Q,R)-chains.
25.85/7.85	----------------------------------------
25.85/7.85	
25.85/7.85	(7) QDPSizeChangeProof (EQUIVALENT)
25.85/7.85	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. 
25.85/7.85	
25.85/7.85	From the DPs we obtained the following set of size-change graphs:
25.85/7.85	*A(b(b(x1))) -> A(x1)
25.85/7.85	The graph contains the following edges 1 > 1
25.85/7.85	
25.85/7.85	
25.85/7.85	----------------------------------------
25.85/7.85	
25.85/7.85	(8)
25.85/7.85	YES
26.78/9.11	EOF