25.77/7.54	YES
27.62/8.00	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
27.62/8.00	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
27.62/8.00	
27.62/8.00	
27.62/8.00	Termination w.r.t. Q of the given QTRS could be proven:
27.62/8.00	
27.62/8.00	(0) QTRS
27.62/8.00	(1) DependencyPairsProof [EQUIVALENT, 7 ms]
27.62/8.00	(2) QDP
27.62/8.00	(3) MRRProof [EQUIVALENT, 68 ms]
27.62/8.00	(4) QDP
27.62/8.00	(5) DependencyGraphProof [EQUIVALENT, 1 ms]
27.62/8.00	(6) AND
27.62/8.00	    (7) QDP
27.62/8.00	        (8) UsableRulesProof [EQUIVALENT, 0 ms]
27.62/8.00	        (9) QDP
27.62/8.00	        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
27.62/8.00	        (11) YES
27.62/8.00	    (12) QDP
27.62/8.00	        (13) UsableRulesProof [EQUIVALENT, 0 ms]
27.62/8.00	        (14) QDP
27.62/8.00	        (15) QDPSizeChangeProof [EQUIVALENT, 0 ms]
27.62/8.00	        (16) YES
27.62/8.00	
27.62/8.00	
27.62/8.00	----------------------------------------
27.62/8.00	
27.62/8.00	(0)
27.62/8.00	Obligation:
27.62/8.00	Q restricted rewrite system:
27.62/8.00	The TRS R consists of the following rules:
27.62/8.00	
27.62/8.00	   a(b(x1)) -> b(r(x1))
27.62/8.00	   r(a(x1)) -> d(r(x1))
27.62/8.00	   r(x1) -> d(x1)
27.62/8.00	   d(a(x1)) -> a(a(d(x1)))
27.62/8.00	   d(x1) -> a(x1)
27.62/8.00	
27.62/8.00	Q is empty.
27.62/8.00	
27.62/8.00	----------------------------------------
27.62/8.00	
27.62/8.00	(1) DependencyPairsProof (EQUIVALENT)
27.62/8.00	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
27.62/8.00	----------------------------------------
27.62/8.00	
27.62/8.00	(2)
27.62/8.00	Obligation:
27.62/8.00	Q DP problem:
27.62/8.00	The TRS P consists of the following rules:
27.62/8.00	
27.62/8.00	   A(b(x1)) -> R(x1)
27.62/8.00	   R(a(x1)) -> D(r(x1))
27.62/8.00	   R(a(x1)) -> R(x1)
27.62/8.00	   R(x1) -> D(x1)
27.62/8.00	   D(a(x1)) -> A(a(d(x1)))
27.62/8.00	   D(a(x1)) -> A(d(x1))
27.62/8.00	   D(a(x1)) -> D(x1)
27.62/8.00	   D(x1) -> A(x1)
27.62/8.00	
27.62/8.00	The TRS R consists of the following rules:
27.62/8.00	
27.62/8.00	   a(b(x1)) -> b(r(x1))
27.62/8.00	   r(a(x1)) -> d(r(x1))
27.62/8.00	   r(x1) -> d(x1)
27.62/8.00	   d(a(x1)) -> a(a(d(x1)))
27.62/8.00	   d(x1) -> a(x1)
27.62/8.00	
27.62/8.00	Q is empty.
27.62/8.00	We have to consider all minimal (P,Q,R)-chains.
27.62/8.00	----------------------------------------
27.62/8.00	
27.62/8.00	(3) MRRProof (EQUIVALENT)
27.62/8.00	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.
27.62/8.00	
27.62/8.00	Strictly oriented dependency pairs:
27.62/8.00	
27.62/8.00	   A(b(x1)) -> R(x1)
27.62/8.00	
27.62/8.00	
27.62/8.00	Used ordering: Polynomial interpretation [POLO]:
27.62/8.00	
27.62/8.00	   POL(A(x_1)) = x_1
27.62/8.00	   POL(D(x_1)) = 2*x_1
27.62/8.00	   POL(R(x_1)) = 3*x_1
27.62/8.00	   POL(a(x_1)) = x_1
27.62/8.00	   POL(b(x_1)) = 1 + 3*x_1
27.62/8.00	   POL(d(x_1)) = x_1
27.62/8.00	   POL(r(x_1)) = x_1
27.62/8.00	
27.62/8.00	
27.62/8.00	----------------------------------------
27.62/8.00	
27.62/8.00	(4)
27.62/8.00	Obligation:
27.62/8.00	Q DP problem:
27.62/8.00	The TRS P consists of the following rules:
27.62/8.00	
27.62/8.00	   R(a(x1)) -> D(r(x1))
27.62/8.00	   R(a(x1)) -> R(x1)
27.62/8.00	   R(x1) -> D(x1)
27.62/8.00	   D(a(x1)) -> A(a(d(x1)))
27.62/8.00	   D(a(x1)) -> A(d(x1))
27.62/8.00	   D(a(x1)) -> D(x1)
27.62/8.00	   D(x1) -> A(x1)
27.62/8.00	
27.62/8.00	The TRS R consists of the following rules:
27.62/8.00	
27.62/8.00	   a(b(x1)) -> b(r(x1))
27.62/8.00	   r(a(x1)) -> d(r(x1))
27.62/8.00	   r(x1) -> d(x1)
27.62/8.00	   d(a(x1)) -> a(a(d(x1)))
27.62/8.00	   d(x1) -> a(x1)
27.62/8.00	
27.62/8.00	Q is empty.
27.62/8.00	We have to consider all minimal (P,Q,R)-chains.
27.62/8.00	----------------------------------------
27.62/8.00	
27.62/8.00	(5) DependencyGraphProof (EQUIVALENT)
27.62/8.00	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 5 less nodes.
27.62/8.00	----------------------------------------
27.62/8.00	
27.62/8.00	(6)
27.62/8.00	Complex Obligation (AND)
27.62/8.00	
27.62/8.00	----------------------------------------
27.62/8.00	
27.62/8.00	(7)
27.62/8.00	Obligation:
27.62/8.00	Q DP problem:
27.62/8.00	The TRS P consists of the following rules:
27.62/8.00	
27.62/8.00	   D(a(x1)) -> D(x1)
27.62/8.00	
27.62/8.00	The TRS R consists of the following rules:
27.62/8.00	
27.62/8.00	   a(b(x1)) -> b(r(x1))
27.62/8.00	   r(a(x1)) -> d(r(x1))
27.62/8.00	   r(x1) -> d(x1)
27.62/8.00	   d(a(x1)) -> a(a(d(x1)))
27.62/8.00	   d(x1) -> a(x1)
27.62/8.00	
27.62/8.00	Q is empty.
27.62/8.00	We have to consider all minimal (P,Q,R)-chains.
27.62/8.00	----------------------------------------
27.62/8.00	
27.62/8.00	(8) UsableRulesProof (EQUIVALENT)
27.62/8.00	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.
27.62/8.00	----------------------------------------
27.62/8.00	
27.62/8.00	(9)
27.62/8.00	Obligation:
27.62/8.00	Q DP problem:
27.62/8.00	The TRS P consists of the following rules:
27.62/8.00	
27.62/8.00	   D(a(x1)) -> D(x1)
27.62/8.00	
27.62/8.00	R is empty.
27.62/8.00	Q is empty.
27.62/8.00	We have to consider all minimal (P,Q,R)-chains.
27.62/8.00	----------------------------------------
27.62/8.00	
27.62/8.00	(10) QDPSizeChangeProof (EQUIVALENT)
27.62/8.00	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. 
27.62/8.00	
27.62/8.00	From the DPs we obtained the following set of size-change graphs:
27.62/8.00	*D(a(x1)) -> D(x1)
27.62/8.00	The graph contains the following edges 1 > 1
27.62/8.00	
27.62/8.00	
27.62/8.00	----------------------------------------
27.62/8.00	
27.62/8.00	(11)
27.62/8.00	YES
27.62/8.00	
27.62/8.00	----------------------------------------
27.62/8.00	
27.62/8.00	(12)
27.62/8.00	Obligation:
27.62/8.00	Q DP problem:
27.62/8.00	The TRS P consists of the following rules:
27.62/8.00	
27.62/8.00	   R(a(x1)) -> R(x1)
27.62/8.00	
27.62/8.00	The TRS R consists of the following rules:
27.62/8.00	
27.62/8.00	   a(b(x1)) -> b(r(x1))
27.62/8.00	   r(a(x1)) -> d(r(x1))
27.62/8.00	   r(x1) -> d(x1)
27.62/8.00	   d(a(x1)) -> a(a(d(x1)))
27.62/8.00	   d(x1) -> a(x1)
27.62/8.00	
27.62/8.00	Q is empty.
27.62/8.00	We have to consider all minimal (P,Q,R)-chains.
27.62/8.00	----------------------------------------
27.62/8.00	
27.62/8.00	(13) UsableRulesProof (EQUIVALENT)
27.62/8.00	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.
27.62/8.00	----------------------------------------
27.62/8.00	
27.62/8.00	(14)
27.62/8.00	Obligation:
27.62/8.00	Q DP problem:
27.62/8.00	The TRS P consists of the following rules:
27.62/8.00	
27.62/8.00	   R(a(x1)) -> R(x1)
27.62/8.00	
27.62/8.00	R is empty.
27.62/8.00	Q is empty.
27.62/8.00	We have to consider all minimal (P,Q,R)-chains.
27.62/8.00	----------------------------------------
27.62/8.00	
27.62/8.00	(15) QDPSizeChangeProof (EQUIVALENT)
27.62/8.00	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. 
27.62/8.00	
27.62/8.00	From the DPs we obtained the following set of size-change graphs:
27.62/8.00	*R(a(x1)) -> R(x1)
27.62/8.00	The graph contains the following edges 1 > 1
27.62/8.00	
27.62/8.00	
27.62/8.00	----------------------------------------
27.62/8.00	
27.62/8.00	(16)
27.62/8.00	YES
28.09/11.83	EOF