17.92/5.46	YES
17.98/5.49	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
17.98/5.49	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
17.98/5.49	
17.98/5.49	
17.98/5.49	Termination w.r.t. Q of the given QTRS could be proven:
17.98/5.49	
17.98/5.49	(0) QTRS
17.98/5.49	(1) DependencyPairsProof [EQUIVALENT, 12 ms]
17.98/5.49	(2) QDP
17.98/5.49	(3) QDPOrderProof [EQUIVALENT, 120 ms]
17.98/5.49	(4) QDP
17.98/5.49	(5) UsableRulesProof [EQUIVALENT, 0 ms]
17.98/5.49	(6) QDP
17.98/5.49	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
17.98/5.49	(8) YES
17.98/5.49	
17.98/5.49	
17.98/5.49	----------------------------------------
17.98/5.49	
17.98/5.49	(0)
17.98/5.49	Obligation:
17.98/5.49	Q restricted rewrite system:
17.98/5.49	The TRS R consists of the following rules:
17.98/5.49	
17.98/5.49	   a(b(b(x1))) -> b(a(a(a(x1))))
17.98/5.49	   a(a(b(x1))) -> b(a(b(x1)))
17.98/5.49	
17.98/5.49	Q is empty.
17.98/5.49	
17.98/5.49	----------------------------------------
17.98/5.49	
17.98/5.49	(1) DependencyPairsProof (EQUIVALENT)
17.98/5.49	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
17.98/5.49	----------------------------------------
17.98/5.49	
17.98/5.49	(2)
17.98/5.49	Obligation:
17.98/5.49	Q DP problem:
17.98/5.49	The TRS P consists of the following rules:
17.98/5.49	
17.98/5.49	   A(b(b(x1))) -> A(a(a(x1)))
17.98/5.49	   A(b(b(x1))) -> A(a(x1))
17.98/5.49	   A(b(b(x1))) -> A(x1)
17.98/5.49	
17.98/5.49	The TRS R consists of the following rules:
17.98/5.49	
17.98/5.49	   a(b(b(x1))) -> b(a(a(a(x1))))
17.98/5.49	   a(a(b(x1))) -> b(a(b(x1)))
17.98/5.49	
17.98/5.49	Q is empty.
17.98/5.49	We have to consider all minimal (P,Q,R)-chains.
17.98/5.49	----------------------------------------
17.98/5.49	
17.98/5.49	(3) QDPOrderProof (EQUIVALENT)
17.98/5.49	We use the reduction pair processor [LPAR04,JAR06].
17.98/5.49	
17.98/5.49	
17.98/5.49	The following pairs can be oriented strictly and are deleted.
17.98/5.49	
17.98/5.49	   A(b(b(x1))) -> A(a(a(x1)))
17.98/5.49	   A(b(b(x1))) -> A(a(x1))
17.98/5.49	The remaining pairs can at least be oriented weakly.
17.98/5.49	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
17.98/5.49	
17.98/5.49	   <<<
17.98/5.49	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
17.98/5.49	>>>
17.98/5.49	
17.98/5.49	   <<<
17.98/5.49	 POL(b(x_1)) =  	[[0A], [1A], [-I]] 	 +  	[[-I, -I, -I], [-I, -I, 0A], [0A, 1A, -I]] 	* 	x_1
17.98/5.49	>>>
17.98/5.49	
17.98/5.49	   <<<
17.98/5.49	 POL(a(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[-I, 0A, -I], [-I, 0A, -I], [1A, 0A, -I]] 	* 	x_1
17.98/5.49	>>>
17.98/5.49	
17.98/5.49	
17.98/5.49	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
17.98/5.49	
17.98/5.49	   a(b(b(x1))) -> b(a(a(a(x1))))
17.98/5.49	   a(a(b(x1))) -> b(a(b(x1)))
17.98/5.49	
17.98/5.49	
17.98/5.49	----------------------------------------
17.98/5.49	
17.98/5.49	(4)
17.98/5.49	Obligation:
17.98/5.49	Q DP problem:
17.98/5.49	The TRS P consists of the following rules:
17.98/5.49	
17.98/5.49	   A(b(b(x1))) -> A(x1)
17.98/5.49	
17.98/5.49	The TRS R consists of the following rules:
17.98/5.49	
17.98/5.49	   a(b(b(x1))) -> b(a(a(a(x1))))
17.98/5.49	   a(a(b(x1))) -> b(a(b(x1)))
17.98/5.49	
17.98/5.49	Q is empty.
17.98/5.49	We have to consider all minimal (P,Q,R)-chains.
17.98/5.49	----------------------------------------
17.98/5.49	
17.98/5.49	(5) UsableRulesProof (EQUIVALENT)
17.98/5.49	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.
17.98/5.49	----------------------------------------
17.98/5.49	
17.98/5.49	(6)
17.98/5.49	Obligation:
17.98/5.49	Q DP problem:
17.98/5.49	The TRS P consists of the following rules:
17.98/5.49	
17.98/5.49	   A(b(b(x1))) -> A(x1)
17.98/5.49	
17.98/5.49	R is empty.
17.98/5.49	Q is empty.
17.98/5.49	We have to consider all minimal (P,Q,R)-chains.
17.98/5.49	----------------------------------------
17.98/5.49	
17.98/5.49	(7) QDPSizeChangeProof (EQUIVALENT)
17.98/5.49	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. 
17.98/5.49	
17.98/5.49	From the DPs we obtained the following set of size-change graphs:
17.98/5.49	*A(b(b(x1))) -> A(x1)
17.98/5.49	The graph contains the following edges 1 > 1
17.98/5.49	
17.98/5.49	
17.98/5.49	----------------------------------------
17.98/5.49	
17.98/5.49	(8)
17.98/5.49	YES
18.20/5.56	EOF