30.02/8.63	YES
30.37/8.78	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
30.37/8.78	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
30.37/8.78	
30.37/8.78	
30.37/8.78	Termination w.r.t. Q of the given QTRS could be proven:
30.37/8.78	
30.37/8.78	(0) QTRS
30.37/8.78	(1) DependencyPairsProof [EQUIVALENT, 31 ms]
30.37/8.78	(2) QDP
30.37/8.78	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
30.37/8.78	(4) QDP
30.37/8.78	(5) MRRProof [EQUIVALENT, 25 ms]
30.37/8.78	(6) QDP
30.37/8.78	(7) MRRProof [EQUIVALENT, 16 ms]
30.37/8.78	(8) QDP
30.37/8.78	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
30.37/8.78	(10) TRUE
30.37/8.78	
30.37/8.78	
30.37/8.78	----------------------------------------
30.37/8.78	
30.37/8.78	(0)
30.37/8.78	Obligation:
30.37/8.78	Q restricted rewrite system:
30.37/8.78	The TRS R consists of the following rules:
30.37/8.78	
30.37/8.78	   a(d(x1)) -> d(b(x1))
30.37/8.78	   a(x1) -> b(b(b(x1)))
30.37/8.78	   b(d(b(x1))) -> a(c(x1))
30.37/8.78	   c(x1) -> d(x1)
30.37/8.78	
30.37/8.78	Q is empty.
30.37/8.78	
30.37/8.78	----------------------------------------
30.37/8.78	
30.37/8.78	(1) DependencyPairsProof (EQUIVALENT)
30.37/8.78	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
30.37/8.78	----------------------------------------
30.37/8.78	
30.37/8.78	(2)
30.37/8.78	Obligation:
30.37/8.78	Q DP problem:
30.37/8.78	The TRS P consists of the following rules:
30.37/8.78	
30.37/8.78	   A(d(x1)) -> B(x1)
30.37/8.78	   A(x1) -> B(b(b(x1)))
30.37/8.78	   A(x1) -> B(b(x1))
30.37/8.78	   A(x1) -> B(x1)
30.37/8.78	   B(d(b(x1))) -> A(c(x1))
30.37/8.78	   B(d(b(x1))) -> C(x1)
30.37/8.78	
30.37/8.78	The TRS R consists of the following rules:
30.37/8.78	
30.37/8.78	   a(d(x1)) -> d(b(x1))
30.37/8.78	   a(x1) -> b(b(b(x1)))
30.37/8.78	   b(d(b(x1))) -> a(c(x1))
30.37/8.78	   c(x1) -> d(x1)
30.37/8.78	
30.37/8.78	Q is empty.
30.37/8.78	We have to consider all minimal (P,Q,R)-chains.
30.37/8.78	----------------------------------------
30.37/8.78	
30.37/8.78	(3) DependencyGraphProof (EQUIVALENT)
30.37/8.78	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
30.37/8.78	----------------------------------------
30.37/8.78	
30.37/8.78	(4)
30.37/8.78	Obligation:
30.37/8.78	Q DP problem:
30.37/8.78	The TRS P consists of the following rules:
30.37/8.78	
30.37/8.78	   B(d(b(x1))) -> A(c(x1))
30.37/8.78	   A(d(x1)) -> B(x1)
30.37/8.78	   A(x1) -> B(b(b(x1)))
30.37/8.78	   A(x1) -> B(b(x1))
30.37/8.78	   A(x1) -> B(x1)
30.37/8.78	
30.37/8.78	The TRS R consists of the following rules:
30.37/8.78	
30.37/8.78	   a(d(x1)) -> d(b(x1))
30.37/8.78	   a(x1) -> b(b(b(x1)))
30.37/8.78	   b(d(b(x1))) -> a(c(x1))
30.37/8.78	   c(x1) -> d(x1)
30.37/8.78	
30.37/8.78	Q is empty.
30.37/8.78	We have to consider all minimal (P,Q,R)-chains.
30.37/8.78	----------------------------------------
30.37/8.78	
30.37/8.78	(5) MRRProof (EQUIVALENT)
30.37/8.78	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.
30.37/8.78	
30.37/8.78	Strictly oriented dependency pairs:
30.37/8.78	
30.37/8.78	   A(d(x1)) -> B(x1)
30.37/8.78	
30.37/8.78	
30.37/8.78	Used ordering: Polynomial interpretation [POLO]:
30.37/8.78	
30.37/8.78	   POL(A(x_1)) = x_1
30.37/8.78	   POL(B(x_1)) = x_1
30.37/8.78	   POL(a(x_1)) = x_1
30.37/8.78	   POL(b(x_1)) = x_1
30.37/8.78	   POL(c(x_1)) = 1 + x_1
30.37/8.78	   POL(d(x_1)) = 1 + x_1
30.37/8.78	
30.37/8.78	
30.37/8.78	----------------------------------------
30.37/8.78	
30.37/8.78	(6)
30.37/8.78	Obligation:
30.37/8.78	Q DP problem:
30.37/8.78	The TRS P consists of the following rules:
30.37/8.78	
30.37/8.78	   B(d(b(x1))) -> A(c(x1))
30.37/8.78	   A(x1) -> B(b(b(x1)))
30.37/8.78	   A(x1) -> B(b(x1))
30.37/8.78	   A(x1) -> B(x1)
30.37/8.78	
30.37/8.78	The TRS R consists of the following rules:
30.37/8.78	
30.37/8.78	   a(d(x1)) -> d(b(x1))
30.37/8.78	   a(x1) -> b(b(b(x1)))
30.37/8.78	   b(d(b(x1))) -> a(c(x1))
30.37/8.78	   c(x1) -> d(x1)
30.37/8.78	
30.37/8.78	Q is empty.
30.37/8.78	We have to consider all minimal (P,Q,R)-chains.
30.37/8.78	----------------------------------------
30.37/8.78	
30.37/8.78	(7) MRRProof (EQUIVALENT)
30.37/8.78	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.
30.37/8.78	
30.37/8.78	Strictly oriented dependency pairs:
30.37/8.78	
30.37/8.78	   A(x1) -> B(b(b(x1)))
30.37/8.78	   A(x1) -> B(b(x1))
30.37/8.78	   A(x1) -> B(x1)
30.37/8.78	
30.37/8.78	Strictly oriented rules of the TRS R:
30.37/8.78	
30.37/8.78	   b(d(b(x1))) -> a(c(x1))
30.37/8.78	
30.37/8.78	Used ordering: Polynomial interpretation [POLO]:
30.37/8.78	
30.37/8.78	   POL(A(x_1)) = 3 + x_1
30.37/8.78	   POL(B(x_1)) = x_1
30.37/8.78	   POL(a(x_1)) = 3 + x_1
30.37/8.78	   POL(b(x_1)) = 1 + x_1
30.37/8.78	   POL(c(x_1)) = 3*x_1
30.37/8.78	   POL(d(x_1)) = 3*x_1
30.37/8.78	
30.37/8.78	
30.37/8.78	----------------------------------------
30.37/8.78	
30.37/8.78	(8)
30.37/8.78	Obligation:
30.37/8.78	Q DP problem:
30.37/8.78	The TRS P consists of the following rules:
30.37/8.78	
30.37/8.78	   B(d(b(x1))) -> A(c(x1))
30.37/8.78	
30.37/8.78	The TRS R consists of the following rules:
30.37/8.78	
30.37/8.78	   a(d(x1)) -> d(b(x1))
30.37/8.78	   a(x1) -> b(b(b(x1)))
30.37/8.78	   c(x1) -> d(x1)
30.37/8.78	
30.37/8.78	Q is empty.
30.37/8.78	We have to consider all minimal (P,Q,R)-chains.
30.37/8.78	----------------------------------------
30.37/8.78	
30.37/8.78	(9) DependencyGraphProof (EQUIVALENT)
30.37/8.78	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
30.37/8.78	----------------------------------------
30.37/8.78	
30.37/8.78	(10)
30.37/8.78	TRUE
30.79/8.83	EOF