13.29/4.30	YES
13.90/4.45	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
13.90/4.45	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
13.90/4.45	
13.90/4.45	
13.90/4.45	Termination w.r.t. Q of the given QTRS could be proven:
13.90/4.45	
13.90/4.45	(0) QTRS
13.90/4.45	(1) QTRS Reverse [EQUIVALENT, 0 ms]
13.90/4.45	(2) QTRS
13.90/4.45	(3) DependencyPairsProof [EQUIVALENT, 4 ms]
13.90/4.45	(4) QDP
13.90/4.45	(5) MRRProof [EQUIVALENT, 27 ms]
13.90/4.45	(6) QDP
13.90/4.45	(7) DependencyGraphProof [EQUIVALENT, 5 ms]
13.90/4.45	(8) TRUE
13.90/4.45	
13.90/4.45	
13.90/4.45	----------------------------------------
13.90/4.45	
13.90/4.45	(0)
13.90/4.45	Obligation:
13.90/4.45	Q restricted rewrite system:
13.90/4.45	The TRS R consists of the following rules:
13.90/4.45	
13.90/4.45	   a(p(x1)) -> p(a(A(x1)))
13.90/4.45	   a(A(x1)) -> A(a(x1))
13.90/4.45	   p(A(A(x1))) -> a(p(x1))
13.90/4.45	
13.90/4.45	Q is empty.
13.90/4.45	
13.90/4.45	----------------------------------------
13.90/4.45	
13.90/4.45	(1) QTRS Reverse (EQUIVALENT)
13.90/4.45	We applied the QTRS Reverse Processor [REVERSE].
13.90/4.45	----------------------------------------
13.90/4.45	
13.90/4.45	(2)
13.90/4.45	Obligation:
13.90/4.45	Q restricted rewrite system:
13.90/4.45	The TRS R consists of the following rules:
13.90/4.45	
13.90/4.45	   p(a(x1)) -> A(a(p(x1)))
13.90/4.45	   A(a(x1)) -> a(A(x1))
13.90/4.45	   A(A(p(x1))) -> p(a(x1))
13.90/4.45	
13.90/4.45	Q is empty.
13.90/4.45	
13.90/4.45	----------------------------------------
13.90/4.45	
13.90/4.45	(3) DependencyPairsProof (EQUIVALENT)
13.90/4.45	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
13.90/4.45	----------------------------------------
13.90/4.45	
13.90/4.45	(4)
13.90/4.45	Obligation:
13.90/4.45	Q DP problem:
13.90/4.45	The TRS P consists of the following rules:
13.90/4.45	
13.90/4.45	   P(a(x1)) -> A^1(a(p(x1)))
13.90/4.45	   P(a(x1)) -> P(x1)
13.90/4.45	   A^1(a(x1)) -> A^1(x1)
13.90/4.45	   A^1(A(p(x1))) -> P(a(x1))
13.90/4.45	
13.90/4.45	The TRS R consists of the following rules:
13.90/4.45	
13.90/4.45	   p(a(x1)) -> A(a(p(x1)))
13.90/4.45	   A(a(x1)) -> a(A(x1))
13.90/4.45	   A(A(p(x1))) -> p(a(x1))
13.90/4.45	
13.90/4.45	Q is empty.
13.90/4.45	We have to consider all minimal (P,Q,R)-chains.
13.90/4.45	----------------------------------------
13.90/4.45	
13.90/4.45	(5) MRRProof (EQUIVALENT)
13.90/4.45	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.
13.90/4.45	
13.90/4.45	Strictly oriented dependency pairs:
13.90/4.45	
13.90/4.45	   P(a(x1)) -> P(x1)
13.90/4.45	   A^1(a(x1)) -> A^1(x1)
13.90/4.45	   A^1(A(p(x1))) -> P(a(x1))
13.90/4.45	
13.90/4.45	Strictly oriented rules of the TRS R:
13.90/4.45	
13.90/4.45	   A(A(p(x1))) -> p(a(x1))
13.90/4.45	
13.90/4.45	Used ordering: Polynomial interpretation [POLO]:
13.90/4.45	
13.90/4.45	   POL(A(x_1)) = 2 + x_1
13.90/4.45	   POL(A^1(x_1)) = 2 + x_1
13.90/4.45	   POL(P(x_1)) = 3*x_1
13.90/4.45	   POL(a(x_1)) = 1 + x_1
13.90/4.45	   POL(p(x_1)) = 3*x_1
13.90/4.45	
13.90/4.45	
13.90/4.45	----------------------------------------
13.90/4.45	
13.90/4.45	(6)
13.90/4.45	Obligation:
13.90/4.45	Q DP problem:
13.90/4.45	The TRS P consists of the following rules:
13.90/4.45	
13.90/4.45	   P(a(x1)) -> A^1(a(p(x1)))
13.90/4.45	
13.90/4.45	The TRS R consists of the following rules:
13.90/4.45	
13.90/4.45	   p(a(x1)) -> A(a(p(x1)))
13.90/4.45	   A(a(x1)) -> a(A(x1))
13.90/4.45	
13.90/4.45	Q is empty.
13.90/4.45	We have to consider all minimal (P,Q,R)-chains.
13.90/4.45	----------------------------------------
13.90/4.45	
13.90/4.45	(7) DependencyGraphProof (EQUIVALENT)
13.90/4.45	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
13.90/4.45	----------------------------------------
13.90/4.45	
13.90/4.45	(8)
13.90/4.45	TRUE
14.19/4.49	EOF