30.63/8.96	YES
31.15/8.99	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
31.15/8.99	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
31.15/8.99	
31.15/8.99	
31.15/8.99	Termination w.r.t. Q of the given QTRS could be proven:
31.15/8.99	
31.15/8.99	(0) QTRS
31.15/8.99	(1) DependencyPairsProof [EQUIVALENT, 18 ms]
31.15/8.99	(2) QDP
31.15/8.99	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
31.15/8.99	(4) QDP
31.15/8.99	(5) QDPOrderProof [EQUIVALENT, 120 ms]
31.15/8.99	(6) QDP
31.15/8.99	(7) MRRProof [EQUIVALENT, 31 ms]
31.15/8.99	(8) QDP
31.15/8.99	(9) QDPOrderProof [EQUIVALENT, 48 ms]
31.15/8.99	(10) QDP
31.15/8.99	(11) QDPOrderProof [EQUIVALENT, 0 ms]
31.15/8.99	(12) QDP
31.15/8.99	(13) DependencyGraphProof [EQUIVALENT, 0 ms]
31.15/8.99	(14) TRUE
31.15/8.99	
31.15/8.99	
31.15/8.99	----------------------------------------
31.15/8.99	
31.15/8.99	(0)
31.15/8.99	Obligation:
31.15/8.99	Q restricted rewrite system:
31.15/8.99	The TRS R consists of the following rules:
31.15/8.99	
31.15/8.99	   a(a(d(d(x1)))) -> d(d(b(b(x1))))
31.15/8.99	   a(a(x1)) -> b(b(b(b(b(b(x1))))))
31.15/8.99	   b(b(d(d(b(b(x1)))))) -> a(a(c(c(x1))))
31.15/8.99	   c(c(x1)) -> d(d(x1))
31.15/8.99	
31.15/8.99	Q is empty.
31.15/8.99	
31.15/8.99	----------------------------------------
31.15/8.99	
31.15/8.99	(1) DependencyPairsProof (EQUIVALENT)
31.15/8.99	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
31.15/8.99	----------------------------------------
31.15/8.99	
31.15/8.99	(2)
31.15/8.99	Obligation:
31.15/8.99	Q DP problem:
31.15/8.99	The TRS P consists of the following rules:
31.15/8.99	
31.15/8.99	   A(a(d(d(x1)))) -> B(b(x1))
31.15/8.99	   A(a(d(d(x1)))) -> B(x1)
31.15/8.99	   A(a(x1)) -> B(b(b(b(b(b(x1))))))
31.15/8.99	   A(a(x1)) -> B(b(b(b(b(x1)))))
31.15/8.99	   A(a(x1)) -> B(b(b(b(x1))))
31.15/8.99	   A(a(x1)) -> B(b(b(x1)))
31.15/8.99	   A(a(x1)) -> B(b(x1))
31.15/8.99	   A(a(x1)) -> B(x1)
31.15/8.99	   B(b(d(d(b(b(x1)))))) -> A(a(c(c(x1))))
31.15/8.99	   B(b(d(d(b(b(x1)))))) -> A(c(c(x1)))
31.15/8.99	   B(b(d(d(b(b(x1)))))) -> C(c(x1))
31.15/8.99	   B(b(d(d(b(b(x1)))))) -> C(x1)
31.15/8.99	
31.15/8.99	The TRS R consists of the following rules:
31.15/8.99	
31.15/8.99	   a(a(d(d(x1)))) -> d(d(b(b(x1))))
31.15/8.99	   a(a(x1)) -> b(b(b(b(b(b(x1))))))
31.15/8.99	   b(b(d(d(b(b(x1)))))) -> a(a(c(c(x1))))
31.15/8.99	   c(c(x1)) -> d(d(x1))
31.15/8.99	
31.15/8.99	Q is empty.
31.15/8.99	We have to consider all minimal (P,Q,R)-chains.
31.15/8.99	----------------------------------------
31.15/8.99	
31.15/8.99	(3) DependencyGraphProof (EQUIVALENT)
31.15/8.99	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
31.15/8.99	----------------------------------------
31.15/8.99	
31.15/8.99	(4)
31.15/8.99	Obligation:
31.15/8.99	Q DP problem:
31.15/8.99	The TRS P consists of the following rules:
31.15/8.99	
31.15/8.99	   B(b(d(d(b(b(x1)))))) -> A(a(c(c(x1))))
31.15/8.99	   A(a(d(d(x1)))) -> B(b(x1))
31.15/8.99	   B(b(d(d(b(b(x1)))))) -> A(c(c(x1)))
31.15/8.99	   A(a(d(d(x1)))) -> B(x1)
31.15/8.99	   A(a(x1)) -> B(b(b(b(b(b(x1))))))
31.15/8.99	   A(a(x1)) -> B(b(b(b(b(x1)))))
31.15/8.99	   A(a(x1)) -> B(b(b(b(x1))))
31.15/8.99	   A(a(x1)) -> B(b(b(x1)))
31.15/8.99	   A(a(x1)) -> B(b(x1))
31.15/8.99	   A(a(x1)) -> B(x1)
31.15/8.99	
31.15/8.99	The TRS R consists of the following rules:
31.15/8.99	
31.15/8.99	   a(a(d(d(x1)))) -> d(d(b(b(x1))))
31.15/8.99	   a(a(x1)) -> b(b(b(b(b(b(x1))))))
31.15/8.99	   b(b(d(d(b(b(x1)))))) -> a(a(c(c(x1))))
31.15/8.99	   c(c(x1)) -> d(d(x1))
31.15/8.99	
31.15/8.99	Q is empty.
31.15/8.99	We have to consider all minimal (P,Q,R)-chains.
31.15/8.99	----------------------------------------
31.15/8.99	
31.15/8.99	(5) QDPOrderProof (EQUIVALENT)
31.15/8.99	We use the reduction pair processor [LPAR04,JAR06].
31.15/8.99	
31.15/8.99	
31.15/8.99	The following pairs can be oriented strictly and are deleted.
31.15/8.99	
31.15/8.99	   B(b(d(d(b(b(x1)))))) -> A(c(c(x1)))
31.15/8.99	The remaining pairs can at least be oriented weakly.
31.15/8.99	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
31.15/8.99	
31.15/8.99	POL( A_1(x_1) ) = max{0, 2x_1 - 1}
31.15/8.99	POL( B_1(x_1) ) = 1
31.15/8.99	POL( a_1(x_1) ) = 1
31.15/8.99	POL( c_1(x_1) ) = 0
31.15/8.99	POL( d_1(x_1) ) = max{0, -2}
31.15/8.99	POL( b_1(x_1) ) = 1
31.15/8.99	
31.15/8.99	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
31.15/8.99	
31.15/8.99	   c(c(x1)) -> d(d(x1))
31.15/8.99	   a(a(d(d(x1)))) -> d(d(b(b(x1))))
31.15/8.99	   a(a(x1)) -> b(b(b(b(b(b(x1))))))
31.15/8.99	   b(b(d(d(b(b(x1)))))) -> a(a(c(c(x1))))
31.15/8.99	
31.15/8.99	
31.15/8.99	----------------------------------------
31.15/8.99	
31.15/8.99	(6)
31.15/8.99	Obligation:
31.15/8.99	Q DP problem:
31.15/8.99	The TRS P consists of the following rules:
31.15/8.99	
31.15/8.99	   B(b(d(d(b(b(x1)))))) -> A(a(c(c(x1))))
31.15/8.99	   A(a(d(d(x1)))) -> B(b(x1))
31.15/8.99	   A(a(d(d(x1)))) -> B(x1)
31.15/8.99	   A(a(x1)) -> B(b(b(b(b(b(x1))))))
31.15/8.99	   A(a(x1)) -> B(b(b(b(b(x1)))))
31.15/8.99	   A(a(x1)) -> B(b(b(b(x1))))
31.15/8.99	   A(a(x1)) -> B(b(b(x1)))
31.15/8.99	   A(a(x1)) -> B(b(x1))
31.15/8.99	   A(a(x1)) -> B(x1)
31.15/8.99	
31.15/8.99	The TRS R consists of the following rules:
31.15/8.99	
31.15/8.99	   a(a(d(d(x1)))) -> d(d(b(b(x1))))
31.15/8.99	   a(a(x1)) -> b(b(b(b(b(b(x1))))))
31.15/8.99	   b(b(d(d(b(b(x1)))))) -> a(a(c(c(x1))))
31.15/8.99	   c(c(x1)) -> d(d(x1))
31.15/8.99	
31.15/8.99	Q is empty.
31.15/8.99	We have to consider all minimal (P,Q,R)-chains.
31.15/8.99	----------------------------------------
31.15/8.99	
31.15/8.99	(7) MRRProof (EQUIVALENT)
31.15/8.99	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.
31.15/8.99	
31.15/8.99	Strictly oriented dependency pairs:
31.15/8.99	
31.15/8.99	   A(a(d(d(x1)))) -> B(b(x1))
31.15/8.99	   A(a(d(d(x1)))) -> B(x1)
31.15/8.99	
31.15/8.99	
31.15/8.99	Used ordering: Polynomial interpretation [POLO]:
31.15/8.99	
31.15/8.99	   POL(A(x_1)) = 2*x_1
31.15/8.99	   POL(B(x_1)) = 2*x_1
31.15/8.99	   POL(a(x_1)) = x_1
31.15/8.99	   POL(b(x_1)) = x_1
31.15/8.99	   POL(c(x_1)) = 2 + 2*x_1
31.15/8.99	   POL(d(x_1)) = 2 + 2*x_1
31.15/8.99	
31.15/8.99	
31.15/8.99	----------------------------------------
31.15/8.99	
31.15/8.99	(8)
31.15/8.99	Obligation:
31.15/8.99	Q DP problem:
31.15/8.99	The TRS P consists of the following rules:
31.15/8.99	
31.15/8.99	   B(b(d(d(b(b(x1)))))) -> A(a(c(c(x1))))
31.15/8.99	   A(a(x1)) -> B(b(b(b(b(b(x1))))))
31.15/8.99	   A(a(x1)) -> B(b(b(b(b(x1)))))
31.15/8.99	   A(a(x1)) -> B(b(b(b(x1))))
31.15/8.99	   A(a(x1)) -> B(b(b(x1)))
31.15/8.99	   A(a(x1)) -> B(b(x1))
31.15/8.99	   A(a(x1)) -> B(x1)
31.15/8.99	
31.15/8.99	The TRS R consists of the following rules:
31.15/8.99	
31.15/8.99	   a(a(d(d(x1)))) -> d(d(b(b(x1))))
31.15/8.99	   a(a(x1)) -> b(b(b(b(b(b(x1))))))
31.15/8.99	   b(b(d(d(b(b(x1)))))) -> a(a(c(c(x1))))
31.15/8.99	   c(c(x1)) -> d(d(x1))
31.15/8.99	
31.15/8.99	Q is empty.
31.15/8.99	We have to consider all minimal (P,Q,R)-chains.
31.15/8.99	----------------------------------------
31.15/8.99	
31.15/8.99	(9) QDPOrderProof (EQUIVALENT)
31.15/8.99	We use the reduction pair processor [LPAR04,JAR06].
31.15/8.99	
31.15/8.99	
31.15/8.99	The following pairs can be oriented strictly and are deleted.
31.15/8.99	
31.15/8.99	   A(a(x1)) -> B(x1)
31.15/8.99	The remaining pairs can at least be oriented weakly.
31.15/8.99	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
31.15/8.99	
31.15/8.99	POL( A_1(x_1) ) = max{0, 2x_1 - 1}
31.15/8.99	POL( a_1(x_1) ) = x_1 + 1
31.15/8.99	POL( B_1(x_1) ) = max{0, x_1 - 1}
31.15/8.99	POL( c_1(x_1) ) = 0
31.15/8.99	POL( d_1(x_1) ) = max{0, -2}
31.15/8.99	POL( b_1(x_1) ) = 2
31.15/8.99	
31.15/8.99	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
31.15/8.99	
31.15/8.99	   c(c(x1)) -> d(d(x1))
31.15/8.99	   a(a(d(d(x1)))) -> d(d(b(b(x1))))
31.15/8.99	   a(a(x1)) -> b(b(b(b(b(b(x1))))))
31.15/8.99	   b(b(d(d(b(b(x1)))))) -> a(a(c(c(x1))))
31.15/8.99	
31.15/8.99	
31.15/8.99	----------------------------------------
31.15/8.99	
31.15/8.99	(10)
31.15/8.99	Obligation:
31.15/8.99	Q DP problem:
31.15/8.99	The TRS P consists of the following rules:
31.15/8.99	
31.15/8.99	   B(b(d(d(b(b(x1)))))) -> A(a(c(c(x1))))
31.15/8.99	   A(a(x1)) -> B(b(b(b(b(b(x1))))))
31.15/8.99	   A(a(x1)) -> B(b(b(b(b(x1)))))
31.15/8.99	   A(a(x1)) -> B(b(b(b(x1))))
31.15/8.99	   A(a(x1)) -> B(b(b(x1)))
31.15/8.99	   A(a(x1)) -> B(b(x1))
31.15/8.99	
31.15/8.99	The TRS R consists of the following rules:
31.15/8.99	
31.15/8.99	   a(a(d(d(x1)))) -> d(d(b(b(x1))))
31.15/8.99	   a(a(x1)) -> b(b(b(b(b(b(x1))))))
31.15/8.99	   b(b(d(d(b(b(x1)))))) -> a(a(c(c(x1))))
31.15/8.99	   c(c(x1)) -> d(d(x1))
31.15/8.99	
31.15/8.99	Q is empty.
31.15/8.99	We have to consider all minimal (P,Q,R)-chains.
31.15/8.99	----------------------------------------
31.15/8.99	
31.15/8.99	(11) QDPOrderProof (EQUIVALENT)
31.15/8.99	We use the reduction pair processor [LPAR04,JAR06].
31.15/8.99	
31.15/8.99	
31.15/8.99	The following pairs can be oriented strictly and are deleted.
31.15/8.99	
31.15/8.99	   B(b(d(d(b(b(x1)))))) -> A(a(c(c(x1))))
31.15/8.99	   A(a(x1)) -> B(b(b(b(b(x1)))))
31.15/8.99	   A(a(x1)) -> B(b(b(b(x1))))
31.15/8.99	   A(a(x1)) -> B(b(b(x1)))
31.15/8.99	   A(a(x1)) -> B(b(x1))
31.15/8.99	The remaining pairs can at least be oriented weakly.
31.15/8.99	Used ordering:  Polynomial interpretation [POLO]:
31.15/8.99	
31.15/8.99	   POL(A(x_1)) = 2*x_1
31.15/8.99	   POL(B(x_1)) = 2*x_1
31.15/8.99	   POL(a(x_1)) = 5 + x_1
31.15/8.99	   POL(b(x_1)) = 1 + x_1
31.15/8.99	   POL(c(x_1)) = 2*x_1
31.15/8.99	   POL(d(x_1)) = 2*x_1
31.15/8.99	
31.15/8.99	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
31.15/8.99	
31.15/8.99	   c(c(x1)) -> d(d(x1))
31.15/8.99	   a(a(d(d(x1)))) -> d(d(b(b(x1))))
31.15/8.99	   a(a(x1)) -> b(b(b(b(b(b(x1))))))
31.15/8.99	   b(b(d(d(b(b(x1)))))) -> a(a(c(c(x1))))
31.15/8.99	
31.15/8.99	
31.15/8.99	----------------------------------------
31.15/8.99	
31.15/8.99	(12)
31.15/8.99	Obligation:
31.15/8.99	Q DP problem:
31.15/8.99	The TRS P consists of the following rules:
31.15/8.99	
31.15/8.99	   A(a(x1)) -> B(b(b(b(b(b(x1))))))
31.15/8.99	
31.15/8.99	The TRS R consists of the following rules:
31.15/8.99	
31.15/8.99	   a(a(d(d(x1)))) -> d(d(b(b(x1))))
31.15/8.99	   a(a(x1)) -> b(b(b(b(b(b(x1))))))
31.15/8.99	   b(b(d(d(b(b(x1)))))) -> a(a(c(c(x1))))
31.15/8.99	   c(c(x1)) -> d(d(x1))
31.15/8.99	
31.15/8.99	Q is empty.
31.15/8.99	We have to consider all minimal (P,Q,R)-chains.
31.15/8.99	----------------------------------------
31.15/8.99	
31.15/8.99	(13) DependencyGraphProof (EQUIVALENT)
31.15/8.99	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
31.15/8.99	----------------------------------------
31.15/8.99	
31.15/8.99	(14)
31.15/8.99	TRUE
31.26/9.07	EOF