42.52/12.26	YES
43.10/12.33	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
43.10/12.33	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
43.10/12.33	
43.10/12.33	
43.10/12.33	Termination w.r.t. Q of the given QTRS could be proven:
43.10/12.33	
43.10/12.33	(0) QTRS
43.10/12.33	(1) QTRS Reverse [EQUIVALENT, 0 ms]
43.10/12.33	(2) QTRS
43.10/12.33	(3) DependencyPairsProof [EQUIVALENT, 8 ms]
43.10/12.33	(4) QDP
43.10/12.33	(5) DependencyGraphProof [EQUIVALENT, 3 ms]
43.10/12.33	(6) AND
43.10/12.33	    (7) QDP
43.10/12.33	        (8) UsableRulesProof [EQUIVALENT, 0 ms]
43.10/12.33	        (9) QDP
43.10/12.33	        (10) DependencyGraphProof [EQUIVALENT, 0 ms]
43.10/12.33	        (11) QDP
43.10/12.33	        (12) MNOCProof [EQUIVALENT, 0 ms]
43.10/12.33	        (13) QDP
43.10/12.33	        (14) UsableRulesProof [EQUIVALENT, 0 ms]
43.10/12.33	        (15) QDP
43.10/12.33	        (16) QReductionProof [EQUIVALENT, 1 ms]
43.10/12.33	        (17) QDP
43.10/12.33	        (18) QDPSizeChangeProof [EQUIVALENT, 0 ms]
43.10/12.33	        (19) YES
43.10/12.33	    (20) QDP
43.10/12.33	        (21) QDPOrderProof [EQUIVALENT, 22 ms]
43.10/12.33	        (22) QDP
43.10/12.33	        (23) QDPOrderProof [EQUIVALENT, 679 ms]
43.10/12.33	        (24) QDP
43.10/12.33	        (25) PisEmptyProof [EQUIVALENT, 0 ms]
43.10/12.33	        (26) YES
43.10/12.33	
43.10/12.33	
43.10/12.33	----------------------------------------
43.10/12.33	
43.10/12.33	(0)
43.10/12.33	Obligation:
43.10/12.33	Q restricted rewrite system:
43.10/12.33	The TRS R consists of the following rules:
43.10/12.33	
43.10/12.33	   b(b(b(a(x1)))) -> a(b(a(a(x1))))
43.10/12.33	   a(a(a(a(x1)))) -> b(a(a(b(x1))))
43.10/12.33	   a(a(a(b(x1)))) -> b(b(b(b(x1))))
43.10/12.33	
43.10/12.33	Q is empty.
43.10/12.33	
43.10/12.33	----------------------------------------
43.10/12.33	
43.10/12.33	(1) QTRS Reverse (EQUIVALENT)
43.10/12.33	We applied the QTRS Reverse Processor [REVERSE].
43.10/12.33	----------------------------------------
43.10/12.33	
43.10/12.33	(2)
43.10/12.33	Obligation:
43.10/12.33	Q restricted rewrite system:
43.10/12.33	The TRS R consists of the following rules:
43.10/12.33	
43.10/12.33	   a(b(b(b(x1)))) -> a(a(b(a(x1))))
43.10/12.33	   a(a(a(a(x1)))) -> b(a(a(b(x1))))
43.10/12.33	   b(a(a(a(x1)))) -> b(b(b(b(x1))))
43.10/12.33	
43.10/12.33	Q is empty.
43.10/12.33	
43.10/12.33	----------------------------------------
43.10/12.33	
43.10/12.33	(3) DependencyPairsProof (EQUIVALENT)
43.10/12.33	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
43.10/12.33	----------------------------------------
43.10/12.33	
43.10/12.33	(4)
43.10/12.33	Obligation:
43.10/12.33	Q DP problem:
43.10/12.33	The TRS P consists of the following rules:
43.10/12.33	
43.10/12.33	   A(b(b(b(x1)))) -> A(a(b(a(x1))))
43.10/12.33	   A(b(b(b(x1)))) -> A(b(a(x1)))
43.10/12.33	   A(b(b(b(x1)))) -> B(a(x1))
43.10/12.33	   A(b(b(b(x1)))) -> A(x1)
43.10/12.33	   A(a(a(a(x1)))) -> B(a(a(b(x1))))
43.10/12.33	   A(a(a(a(x1)))) -> A(a(b(x1)))
43.10/12.33	   A(a(a(a(x1)))) -> A(b(x1))
43.10/12.33	   A(a(a(a(x1)))) -> B(x1)
43.10/12.33	   B(a(a(a(x1)))) -> B(b(b(b(x1))))
43.10/12.33	   B(a(a(a(x1)))) -> B(b(b(x1)))
43.10/12.33	   B(a(a(a(x1)))) -> B(b(x1))
43.10/12.33	   B(a(a(a(x1)))) -> B(x1)
43.10/12.33	
43.10/12.33	The TRS R consists of the following rules:
43.10/12.33	
43.10/12.33	   a(b(b(b(x1)))) -> a(a(b(a(x1))))
43.10/12.33	   a(a(a(a(x1)))) -> b(a(a(b(x1))))
43.10/12.33	   b(a(a(a(x1)))) -> b(b(b(b(x1))))
43.10/12.33	
43.10/12.33	Q is empty.
43.10/12.33	We have to consider all minimal (P,Q,R)-chains.
43.10/12.33	----------------------------------------
43.10/12.33	
43.10/12.33	(5) DependencyGraphProof (EQUIVALENT)
43.10/12.33	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 3 less nodes.
43.10/12.33	----------------------------------------
43.10/12.33	
43.10/12.33	(6)
43.10/12.33	Complex Obligation (AND)
43.10/12.33	
43.10/12.33	----------------------------------------
43.10/12.33	
43.10/12.33	(7)
43.10/12.33	Obligation:
43.10/12.33	Q DP problem:
43.10/12.33	The TRS P consists of the following rules:
43.10/12.33	
43.10/12.33	   B(a(a(a(x1)))) -> B(b(b(x1)))
43.10/12.33	   B(a(a(a(x1)))) -> B(b(b(b(x1))))
43.10/12.33	   B(a(a(a(x1)))) -> B(b(x1))
43.10/12.33	   B(a(a(a(x1)))) -> B(x1)
43.10/12.33	
43.10/12.33	The TRS R consists of the following rules:
43.10/12.33	
43.10/12.33	   a(b(b(b(x1)))) -> a(a(b(a(x1))))
43.10/12.33	   a(a(a(a(x1)))) -> b(a(a(b(x1))))
43.10/12.33	   b(a(a(a(x1)))) -> b(b(b(b(x1))))
43.10/12.33	
43.10/12.33	Q is empty.
43.10/12.33	We have to consider all minimal (P,Q,R)-chains.
43.10/12.33	----------------------------------------
43.10/12.33	
43.10/12.33	(8) UsableRulesProof (EQUIVALENT)
43.10/12.33	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.
43.10/12.33	----------------------------------------
43.10/12.33	
43.10/12.33	(9)
43.10/12.33	Obligation:
43.10/12.33	Q DP problem:
43.10/12.33	The TRS P consists of the following rules:
43.10/12.33	
43.10/12.33	   B(a(a(a(x1)))) -> B(b(b(x1)))
43.10/12.33	   B(a(a(a(x1)))) -> B(b(b(b(x1))))
43.10/12.33	   B(a(a(a(x1)))) -> B(b(x1))
43.10/12.33	   B(a(a(a(x1)))) -> B(x1)
43.10/12.33	
43.10/12.33	The TRS R consists of the following rules:
43.10/12.33	
43.10/12.33	   b(a(a(a(x1)))) -> b(b(b(b(x1))))
43.10/12.33	
43.10/12.33	Q is empty.
43.10/12.33	We have to consider all minimal (P,Q,R)-chains.
43.10/12.33	----------------------------------------
43.10/12.33	
43.10/12.33	(10) DependencyGraphProof (EQUIVALENT)
43.10/12.33	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
43.10/12.33	----------------------------------------
43.10/12.33	
43.10/12.33	(11)
43.10/12.33	Obligation:
43.10/12.33	Q DP problem:
43.10/12.33	The TRS P consists of the following rules:
43.10/12.33	
43.10/12.33	   B(a(a(a(x1)))) -> B(x1)
43.10/12.33	
43.10/12.33	The TRS R consists of the following rules:
43.10/12.33	
43.10/12.33	   b(a(a(a(x1)))) -> b(b(b(b(x1))))
43.10/12.33	
43.10/12.33	Q is empty.
43.10/12.33	We have to consider all minimal (P,Q,R)-chains.
43.10/12.33	----------------------------------------
43.10/12.33	
43.10/12.33	(12) MNOCProof (EQUIVALENT)
43.10/12.33	We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.
43.10/12.33	----------------------------------------
43.10/12.33	
43.10/12.33	(13)
43.10/12.33	Obligation:
43.10/12.33	Q DP problem:
43.10/12.33	The TRS P consists of the following rules:
43.10/12.33	
43.10/12.33	   B(a(a(a(x1)))) -> B(x1)
43.10/12.33	
43.10/12.33	The TRS R consists of the following rules:
43.10/12.33	
43.10/12.33	   b(a(a(a(x1)))) -> b(b(b(b(x1))))
43.10/12.33	
43.10/12.33	The set Q consists of the following terms:
43.10/12.33	
43.10/12.33	   b(a(a(a(x0))))
43.10/12.33	
43.10/12.33	We have to consider all minimal (P,Q,R)-chains.
43.10/12.33	----------------------------------------
43.10/12.33	
43.10/12.33	(14) UsableRulesProof (EQUIVALENT)
43.10/12.33	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
43.10/12.33	----------------------------------------
43.10/12.33	
43.10/12.33	(15)
43.10/12.33	Obligation:
43.10/12.33	Q DP problem:
43.10/12.33	The TRS P consists of the following rules:
43.10/12.33	
43.10/12.33	   B(a(a(a(x1)))) -> B(x1)
43.10/12.33	
43.10/12.33	R is empty.
43.10/12.33	The set Q consists of the following terms:
43.10/12.33	
43.10/12.33	   b(a(a(a(x0))))
43.10/12.33	
43.10/12.33	We have to consider all minimal (P,Q,R)-chains.
43.10/12.33	----------------------------------------
43.10/12.33	
43.10/12.33	(16) QReductionProof (EQUIVALENT)
43.10/12.33	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
43.10/12.33	
43.10/12.33	   b(a(a(a(x0))))
43.10/12.33	
43.10/12.33	
43.10/12.33	----------------------------------------
43.10/12.33	
43.10/12.33	(17)
43.10/12.33	Obligation:
43.10/12.33	Q DP problem:
43.10/12.33	The TRS P consists of the following rules:
43.10/12.33	
43.10/12.33	   B(a(a(a(x1)))) -> B(x1)
43.10/12.33	
43.10/12.33	R is empty.
43.10/12.33	Q is empty.
43.10/12.33	We have to consider all minimal (P,Q,R)-chains.
43.10/12.33	----------------------------------------
43.10/12.33	
43.10/12.33	(18) QDPSizeChangeProof (EQUIVALENT)
43.10/12.33	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. 
43.10/12.33	
43.10/12.33	From the DPs we obtained the following set of size-change graphs:
43.10/12.33	*B(a(a(a(x1)))) -> B(x1)
43.10/12.33	The graph contains the following edges 1 > 1
43.10/12.33	
43.10/12.33	
43.10/12.33	----------------------------------------
43.10/12.33	
43.10/12.33	(19)
43.10/12.33	YES
43.10/12.33	
43.10/12.33	----------------------------------------
43.10/12.33	
43.10/12.33	(20)
43.10/12.33	Obligation:
43.10/12.33	Q DP problem:
43.10/12.33	The TRS P consists of the following rules:
43.10/12.33	
43.10/12.33	   A(b(b(b(x1)))) -> A(b(a(x1)))
43.10/12.33	   A(b(b(b(x1)))) -> A(a(b(a(x1))))
43.10/12.33	   A(b(b(b(x1)))) -> A(x1)
43.10/12.33	   A(a(a(a(x1)))) -> A(a(b(x1)))
43.10/12.33	   A(a(a(a(x1)))) -> A(b(x1))
43.10/12.33	
43.10/12.33	The TRS R consists of the following rules:
43.10/12.33	
43.10/12.33	   a(b(b(b(x1)))) -> a(a(b(a(x1))))
43.10/12.33	   a(a(a(a(x1)))) -> b(a(a(b(x1))))
43.10/12.33	   b(a(a(a(x1)))) -> b(b(b(b(x1))))
43.10/12.33	
43.10/12.33	Q is empty.
43.10/12.33	We have to consider all minimal (P,Q,R)-chains.
43.10/12.33	----------------------------------------
43.10/12.33	
43.10/12.33	(21) QDPOrderProof (EQUIVALENT)
43.10/12.33	We use the reduction pair processor [LPAR04,JAR06].
43.10/12.33	
43.10/12.33	
43.10/12.33	The following pairs can be oriented strictly and are deleted.
43.10/12.33	
43.10/12.33	   A(b(b(b(x1)))) -> A(b(a(x1)))
43.10/12.33	   A(b(b(b(x1)))) -> A(x1)
43.10/12.33	   A(a(a(a(x1)))) -> A(a(b(x1)))
43.10/12.33	   A(a(a(a(x1)))) -> A(b(x1))
43.10/12.33	The remaining pairs can at least be oriented weakly.
43.10/12.33	Used ordering:  Polynomial interpretation [POLO]:
43.10/12.33	
43.10/12.33	   POL(A(x_1)) = x_1
43.10/12.33	   POL(a(x_1)) = 1 + x_1
43.10/12.33	   POL(b(x_1)) = 1 + x_1
43.10/12.33	
43.10/12.33	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
43.10/12.33	
43.10/12.33	   a(b(b(b(x1)))) -> a(a(b(a(x1))))
43.10/12.33	   a(a(a(a(x1)))) -> b(a(a(b(x1))))
43.10/12.33	   b(a(a(a(x1)))) -> b(b(b(b(x1))))
43.10/12.33	
43.10/12.33	
43.10/12.33	----------------------------------------
43.10/12.33	
43.10/12.33	(22)
43.10/12.33	Obligation:
43.10/12.33	Q DP problem:
43.10/12.33	The TRS P consists of the following rules:
43.10/12.33	
43.10/12.33	   A(b(b(b(x1)))) -> A(a(b(a(x1))))
43.10/12.33	
43.10/12.33	The TRS R consists of the following rules:
43.10/12.33	
43.10/12.33	   a(b(b(b(x1)))) -> a(a(b(a(x1))))
43.10/12.33	   a(a(a(a(x1)))) -> b(a(a(b(x1))))
43.10/12.33	   b(a(a(a(x1)))) -> b(b(b(b(x1))))
43.10/12.33	
43.10/12.33	Q is empty.
43.10/12.33	We have to consider all minimal (P,Q,R)-chains.
43.10/12.33	----------------------------------------
43.10/12.33	
43.10/12.33	(23) QDPOrderProof (EQUIVALENT)
43.10/12.33	We use the reduction pair processor [LPAR04,JAR06].
43.10/12.33	
43.10/12.33	
43.10/12.33	The following pairs can be oriented strictly and are deleted.
43.10/12.33	
43.10/12.33	   A(b(b(b(x1)))) -> A(a(b(a(x1))))
43.10/12.33	The remaining pairs can at least be oriented weakly.
43.10/12.33	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
43.10/12.33	
43.10/12.33	   <<<
43.10/12.33	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, 1A, 0A]] 	* 	x_1
43.10/12.33	>>>
43.10/12.33	
43.10/12.33	   <<<
43.10/12.33	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, 0A], [0A, 0A, 1A], [0A, -I, 0A]] 	* 	x_1
43.10/12.33	>>>
43.10/12.33	
43.10/12.33	   <<<
43.10/12.33	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[1A, 0A, 0A], [0A, -I, -I], [0A, -I, 0A]] 	* 	x_1
43.10/12.33	>>>
43.10/12.33	
43.10/12.33	
43.10/12.33	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
43.10/12.33	
43.10/12.33	   a(b(b(b(x1)))) -> a(a(b(a(x1))))
43.10/12.33	   a(a(a(a(x1)))) -> b(a(a(b(x1))))
43.10/12.33	   b(a(a(a(x1)))) -> b(b(b(b(x1))))
43.10/12.33	
43.10/12.33	
43.10/12.33	----------------------------------------
43.10/12.33	
43.10/12.33	(24)
43.10/12.33	Obligation:
43.10/12.33	Q DP problem:
43.10/12.33	P is empty.
43.10/12.33	The TRS R consists of the following rules:
43.10/12.33	
43.10/12.33	   a(b(b(b(x1)))) -> a(a(b(a(x1))))
43.10/12.33	   a(a(a(a(x1)))) -> b(a(a(b(x1))))
43.10/12.33	   b(a(a(a(x1)))) -> b(b(b(b(x1))))
43.10/12.33	
43.10/12.33	Q is empty.
43.10/12.33	We have to consider all minimal (P,Q,R)-chains.
43.10/12.33	----------------------------------------
43.10/12.33	
43.10/12.33	(25) PisEmptyProof (EQUIVALENT)
43.10/12.33	The TRS P is empty. Hence, there is no (P,Q,R) chain.
43.10/12.33	----------------------------------------
43.10/12.33	
43.10/12.33	(26)
43.10/12.33	YES
43.36/14.09	EOF