42.33/11.72	YES
42.33/11.75	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
42.33/11.75	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
42.33/11.75	
42.33/11.75	
42.33/11.75	Termination w.r.t. Q of the given QTRS could be proven:
42.33/11.75	
42.33/11.75	(0) QTRS
42.33/11.75	(1) DependencyPairsProof [EQUIVALENT, 29 ms]
42.33/11.75	(2) QDP
42.33/11.75	(3) DependencyGraphProof [EQUIVALENT, 4 ms]
42.33/11.75	(4) AND
42.33/11.75	    (5) QDP
42.33/11.75	        (6) UsableRulesProof [EQUIVALENT, 4 ms]
42.33/11.75	        (7) QDP
42.33/11.75	        (8) MNOCProof [EQUIVALENT, 7 ms]
42.33/11.75	        (9) QDP
42.33/11.75	        (10) MRRProof [EQUIVALENT, 19 ms]
42.33/11.75	        (11) QDP
42.33/11.75	        (12) QDPOrderProof [EQUIVALENT, 10 ms]
42.33/11.75	        (13) QDP
42.33/11.75	        (14) PisEmptyProof [EQUIVALENT, 0 ms]
42.33/11.75	        (15) YES
42.33/11.75	    (16) QDP
42.33/11.75	        (17) MRRProof [EQUIVALENT, 43 ms]
42.33/11.75	        (18) QDP
42.33/11.75	        (19) QDPOrderProof [EQUIVALENT, 0 ms]
42.33/11.75	        (20) QDP
42.33/11.75	        (21) PisEmptyProof [EQUIVALENT, 0 ms]
42.33/11.75	        (22) YES
42.33/11.75	
42.33/11.75	
42.33/11.75	----------------------------------------
42.33/11.75	
42.33/11.75	(0)
42.33/11.75	Obligation:
42.33/11.75	Q restricted rewrite system:
42.33/11.75	The TRS R consists of the following rules:
42.33/11.75	
42.33/11.75	   a(a(b(b(x1)))) -> b(b(c(c(a(a(x1))))))
42.33/11.75	   b(b(c(c(x1)))) -> c(c(b(b(b(b(x1))))))
42.33/11.75	   b(b(a(a(x1)))) -> a(a(c(c(b(b(x1))))))
42.33/11.75	
42.33/11.75	Q is empty.
42.33/11.75	
42.33/11.75	----------------------------------------
42.33/11.75	
42.33/11.75	(1) DependencyPairsProof (EQUIVALENT)
42.33/11.75	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
42.33/11.75	----------------------------------------
42.33/11.75	
42.33/11.75	(2)
42.33/11.75	Obligation:
42.33/11.75	Q DP problem:
42.33/11.75	The TRS P consists of the following rules:
42.33/11.75	
42.33/11.75	   A(a(b(b(x1)))) -> B(b(c(c(a(a(x1))))))
42.33/11.75	   A(a(b(b(x1)))) -> B(c(c(a(a(x1)))))
42.33/11.75	   A(a(b(b(x1)))) -> A(a(x1))
42.33/11.75	   A(a(b(b(x1)))) -> A(x1)
42.33/11.75	   B(b(c(c(x1)))) -> B(b(b(b(x1))))
42.33/11.75	   B(b(c(c(x1)))) -> B(b(b(x1)))
42.33/11.75	   B(b(c(c(x1)))) -> B(b(x1))
42.33/11.75	   B(b(c(c(x1)))) -> B(x1)
42.33/11.75	   B(b(a(a(x1)))) -> A(a(c(c(b(b(x1))))))
42.33/11.75	   B(b(a(a(x1)))) -> A(c(c(b(b(x1)))))
42.33/11.75	   B(b(a(a(x1)))) -> B(b(x1))
42.33/11.75	   B(b(a(a(x1)))) -> B(x1)
42.33/11.75	
42.33/11.75	The TRS R consists of the following rules:
42.33/11.75	
42.33/11.75	   a(a(b(b(x1)))) -> b(b(c(c(a(a(x1))))))
42.33/11.75	   b(b(c(c(x1)))) -> c(c(b(b(b(b(x1))))))
42.33/11.75	   b(b(a(a(x1)))) -> a(a(c(c(b(b(x1))))))
42.33/11.75	
42.33/11.75	Q is empty.
42.33/11.75	We have to consider all minimal (P,Q,R)-chains.
42.33/11.75	----------------------------------------
42.33/11.75	
42.33/11.75	(3) DependencyGraphProof (EQUIVALENT)
42.33/11.75	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 4 less nodes.
42.33/11.75	----------------------------------------
42.33/11.75	
42.33/11.75	(4)
42.33/11.75	Complex Obligation (AND)
42.33/11.75	
42.33/11.75	----------------------------------------
42.33/11.75	
42.33/11.75	(5)
42.33/11.75	Obligation:
42.33/11.75	Q DP problem:
42.33/11.75	The TRS P consists of the following rules:
42.33/11.75	
42.33/11.75	   B(b(c(c(x1)))) -> B(b(b(x1)))
42.33/11.75	   B(b(c(c(x1)))) -> B(b(b(b(x1))))
42.33/11.75	   B(b(c(c(x1)))) -> B(b(x1))
42.33/11.75	   B(b(c(c(x1)))) -> B(x1)
42.33/11.75	   B(b(a(a(x1)))) -> B(b(x1))
42.33/11.75	   B(b(a(a(x1)))) -> B(x1)
42.33/11.75	
42.33/11.75	The TRS R consists of the following rules:
42.33/11.75	
42.33/11.75	   a(a(b(b(x1)))) -> b(b(c(c(a(a(x1))))))
42.33/11.75	   b(b(c(c(x1)))) -> c(c(b(b(b(b(x1))))))
42.33/11.75	   b(b(a(a(x1)))) -> a(a(c(c(b(b(x1))))))
42.33/11.75	
42.33/11.75	Q is empty.
42.33/11.75	We have to consider all minimal (P,Q,R)-chains.
42.33/11.75	----------------------------------------
42.33/11.75	
42.33/11.75	(6) UsableRulesProof (EQUIVALENT)
42.33/11.75	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.
42.33/11.75	----------------------------------------
42.33/11.75	
42.33/11.75	(7)
42.33/11.75	Obligation:
42.33/11.75	Q DP problem:
42.33/11.75	The TRS P consists of the following rules:
42.33/11.75	
42.33/11.75	   B(b(c(c(x1)))) -> B(b(b(x1)))
42.33/11.75	   B(b(c(c(x1)))) -> B(b(b(b(x1))))
42.33/11.75	   B(b(c(c(x1)))) -> B(b(x1))
42.33/11.75	   B(b(c(c(x1)))) -> B(x1)
42.33/11.75	   B(b(a(a(x1)))) -> B(b(x1))
42.33/11.75	   B(b(a(a(x1)))) -> B(x1)
42.33/11.75	
42.33/11.75	The TRS R consists of the following rules:
42.33/11.75	
42.33/11.75	   b(b(c(c(x1)))) -> c(c(b(b(b(b(x1))))))
42.33/11.75	   b(b(a(a(x1)))) -> a(a(c(c(b(b(x1))))))
42.33/11.75	
42.33/11.75	Q is empty.
42.33/11.75	We have to consider all minimal (P,Q,R)-chains.
42.33/11.75	----------------------------------------
42.33/11.75	
42.33/11.75	(8) MNOCProof (EQUIVALENT)
42.33/11.75	We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.
42.33/11.75	----------------------------------------
42.33/11.75	
42.33/11.75	(9)
42.33/11.75	Obligation:
42.33/11.75	Q DP problem:
42.33/11.75	The TRS P consists of the following rules:
42.33/11.75	
42.33/11.75	   B(b(c(c(x1)))) -> B(b(b(x1)))
42.33/11.75	   B(b(c(c(x1)))) -> B(b(b(b(x1))))
42.33/11.75	   B(b(c(c(x1)))) -> B(b(x1))
42.33/11.75	   B(b(c(c(x1)))) -> B(x1)
42.33/11.75	   B(b(a(a(x1)))) -> B(b(x1))
42.33/11.75	   B(b(a(a(x1)))) -> B(x1)
42.33/11.75	
42.33/11.75	The TRS R consists of the following rules:
42.33/11.75	
42.33/11.75	   b(b(c(c(x1)))) -> c(c(b(b(b(b(x1))))))
42.33/11.75	   b(b(a(a(x1)))) -> a(a(c(c(b(b(x1))))))
42.33/11.75	
42.33/11.75	The set Q consists of the following terms:
42.33/11.75	
42.33/11.75	   b(b(c(c(x0))))
42.33/11.75	   b(b(a(a(x0))))
42.33/11.75	
42.33/11.75	We have to consider all minimal (P,Q,R)-chains.
42.33/11.75	----------------------------------------
42.33/11.75	
42.33/11.75	(10) MRRProof (EQUIVALENT)
42.33/11.75	By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
42.33/11.75	
42.33/11.75	Strictly oriented dependency pairs:
42.33/11.75	
42.33/11.75	   B(b(a(a(x1)))) -> B(b(x1))
42.33/11.75	   B(b(a(a(x1)))) -> B(x1)
42.33/11.75	
42.33/11.75	
42.33/11.75	Used ordering: Polynomial interpretation [POLO]:
42.33/11.75	
42.33/11.75	   POL(B(x_1)) = x_1
42.33/11.75	   POL(a(x_1)) = 1 + 2*x_1
42.33/11.75	   POL(b(x_1)) = x_1
42.33/11.75	   POL(c(x_1)) = x_1
42.33/11.75	
42.33/11.75	
42.33/11.75	----------------------------------------
42.33/11.75	
42.33/11.75	(11)
42.33/11.75	Obligation:
42.33/11.75	Q DP problem:
42.33/11.75	The TRS P consists of the following rules:
42.33/11.75	
42.33/11.75	   B(b(c(c(x1)))) -> B(b(b(x1)))
42.33/11.75	   B(b(c(c(x1)))) -> B(b(b(b(x1))))
42.33/11.75	   B(b(c(c(x1)))) -> B(b(x1))
42.33/11.75	   B(b(c(c(x1)))) -> B(x1)
42.33/11.75	
42.33/11.75	The TRS R consists of the following rules:
42.33/11.75	
42.33/11.75	   b(b(c(c(x1)))) -> c(c(b(b(b(b(x1))))))
42.33/11.75	   b(b(a(a(x1)))) -> a(a(c(c(b(b(x1))))))
42.33/11.75	
42.33/11.75	The set Q consists of the following terms:
42.33/11.75	
42.33/11.75	   b(b(c(c(x0))))
42.33/11.75	   b(b(a(a(x0))))
42.33/11.75	
42.33/11.75	We have to consider all minimal (P,Q,R)-chains.
42.33/11.75	----------------------------------------
42.33/11.75	
42.33/11.75	(12) QDPOrderProof (EQUIVALENT)
42.33/11.75	We use the reduction pair processor [LPAR04,JAR06].
42.33/11.75	
42.33/11.75	
42.33/11.75	The following pairs can be oriented strictly and are deleted.
42.33/11.75	
42.33/11.75	   B(b(c(c(x1)))) -> B(b(b(x1)))
42.33/11.75	   B(b(c(c(x1)))) -> B(b(b(b(x1))))
42.33/11.75	   B(b(c(c(x1)))) -> B(b(x1))
42.33/11.75	   B(b(c(c(x1)))) -> B(x1)
42.33/11.75	The remaining pairs can at least be oriented weakly.
42.33/11.75	Used ordering:  Polynomial interpretation [POLO]:
42.33/11.75	
42.33/11.75	   POL(B(x_1)) = x_1
42.33/11.75	   POL(a(x_1)) = 1
42.33/11.75	   POL(b(x_1)) = x_1
42.33/11.75	   POL(c(x_1)) = 1 + x_1
42.33/11.75	
42.33/11.75	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
42.33/11.75	
42.33/11.75	   b(b(c(c(x1)))) -> c(c(b(b(b(b(x1))))))
42.33/11.75	   b(b(a(a(x1)))) -> a(a(c(c(b(b(x1))))))
42.33/11.75	
42.33/11.75	
42.33/11.75	----------------------------------------
42.33/11.75	
42.33/11.75	(13)
42.33/11.75	Obligation:
42.33/11.75	Q DP problem:
42.33/11.75	P is empty.
42.33/11.75	The TRS R consists of the following rules:
42.33/11.75	
42.33/11.75	   b(b(c(c(x1)))) -> c(c(b(b(b(b(x1))))))
42.33/11.75	   b(b(a(a(x1)))) -> a(a(c(c(b(b(x1))))))
42.33/11.75	
42.33/11.75	The set Q consists of the following terms:
42.33/11.75	
42.33/11.75	   b(b(c(c(x0))))
42.33/11.75	   b(b(a(a(x0))))
42.33/11.75	
42.33/11.75	We have to consider all minimal (P,Q,R)-chains.
42.33/11.75	----------------------------------------
42.33/11.75	
42.33/11.75	(14) PisEmptyProof (EQUIVALENT)
42.33/11.75	The TRS P is empty. Hence, there is no (P,Q,R) chain.
42.33/11.75	----------------------------------------
42.33/11.75	
42.33/11.75	(15)
42.33/11.75	YES
42.33/11.75	
42.33/11.75	----------------------------------------
42.33/11.75	
42.33/11.75	(16)
42.33/11.75	Obligation:
42.33/11.75	Q DP problem:
42.33/11.75	The TRS P consists of the following rules:
42.33/11.75	
42.33/11.75	   A(a(b(b(x1)))) -> A(x1)
42.33/11.75	   A(a(b(b(x1)))) -> A(a(x1))
42.33/11.75	
42.33/11.75	The TRS R consists of the following rules:
42.33/11.75	
42.33/11.75	   a(a(b(b(x1)))) -> b(b(c(c(a(a(x1))))))
42.33/11.75	   b(b(c(c(x1)))) -> c(c(b(b(b(b(x1))))))
42.33/11.75	   b(b(a(a(x1)))) -> a(a(c(c(b(b(x1))))))
42.33/11.75	
42.33/11.75	Q is empty.
42.33/11.75	We have to consider all minimal (P,Q,R)-chains.
42.33/11.75	----------------------------------------
42.33/11.75	
42.33/11.75	(17) MRRProof (EQUIVALENT)
42.33/11.75	By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
42.33/11.75	
42.33/11.75	Strictly oriented dependency pairs:
42.33/11.75	
42.33/11.75	   A(a(b(b(x1)))) -> A(x1)
42.33/11.75	
42.33/11.75	
42.33/11.75	Used ordering: Polynomial interpretation [POLO]:
42.33/11.75	
42.33/11.75	   POL(A(x_1)) = 2*x_1
42.33/11.75	   POL(a(x_1)) = 2 + x_1
42.33/11.75	   POL(b(x_1)) = x_1
42.33/11.75	   POL(c(x_1)) = x_1
42.33/11.75	
42.33/11.75	
42.33/11.75	----------------------------------------
42.33/11.75	
42.33/11.75	(18)
42.33/11.75	Obligation:
42.33/11.75	Q DP problem:
42.33/11.75	The TRS P consists of the following rules:
42.33/11.75	
42.33/11.75	   A(a(b(b(x1)))) -> A(a(x1))
42.33/11.75	
42.33/11.75	The TRS R consists of the following rules:
42.33/11.75	
42.33/11.75	   a(a(b(b(x1)))) -> b(b(c(c(a(a(x1))))))
42.33/11.75	   b(b(c(c(x1)))) -> c(c(b(b(b(b(x1))))))
42.33/11.75	   b(b(a(a(x1)))) -> a(a(c(c(b(b(x1))))))
42.33/11.75	
42.33/11.75	Q is empty.
42.33/11.75	We have to consider all minimal (P,Q,R)-chains.
42.33/11.75	----------------------------------------
42.33/11.75	
42.33/11.75	(19) QDPOrderProof (EQUIVALENT)
42.33/11.75	We use the reduction pair processor [LPAR04,JAR06].
42.33/11.75	
42.33/11.75	
42.33/11.75	The following pairs can be oriented strictly and are deleted.
42.33/11.75	
42.33/11.75	   A(a(b(b(x1)))) -> A(a(x1))
42.33/11.75	The remaining pairs can at least be oriented weakly.
42.33/11.75	Used ordering:  Polynomial interpretation [POLO]:
42.33/11.75	
42.33/11.75	   POL(A(x_1)) = x_1
42.33/11.75	   POL(a(x_1)) = 1 + x_1
42.33/11.75	   POL(b(x_1)) = 1 + x_1
42.33/11.76	   POL(c(x_1)) = 1
42.33/11.76	
42.33/11.76	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
42.33/11.76	
42.33/11.76	   a(a(b(b(x1)))) -> b(b(c(c(a(a(x1))))))
42.33/11.76	   b(b(c(c(x1)))) -> c(c(b(b(b(b(x1))))))
42.33/11.76	
42.33/11.76	
42.33/11.76	----------------------------------------
42.33/11.76	
42.33/11.76	(20)
42.33/11.76	Obligation:
42.33/11.76	Q DP problem:
42.33/11.76	P is empty.
42.33/11.76	The TRS R consists of the following rules:
42.33/11.76	
42.33/11.76	   a(a(b(b(x1)))) -> b(b(c(c(a(a(x1))))))
42.33/11.76	   b(b(c(c(x1)))) -> c(c(b(b(b(b(x1))))))
42.33/11.76	   b(b(a(a(x1)))) -> a(a(c(c(b(b(x1))))))
42.33/11.76	
42.33/11.76	Q is empty.
42.33/11.76	We have to consider all minimal (P,Q,R)-chains.
42.33/11.76	----------------------------------------
42.33/11.76	
42.33/11.76	(21) PisEmptyProof (EQUIVALENT)
42.33/11.76	The TRS P is empty. Hence, there is no (P,Q,R) chain.
42.33/11.76	----------------------------------------
42.33/11.76	
42.33/11.76	(22)
42.33/11.76	YES
42.77/11.90	EOF