10.21/3.42	YES
10.21/3.45	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
10.21/3.45	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.21/3.45	
10.21/3.45	
10.21/3.45	Termination w.r.t. Q of the given QTRS could be proven:
10.21/3.45	
10.21/3.45	(0) QTRS
10.21/3.45	(1) QTRS Reverse [EQUIVALENT, 0 ms]
10.21/3.45	(2) QTRS
10.21/3.45	(3) QTRSRRRProof [EQUIVALENT, 18 ms]
10.21/3.45	(4) QTRS
10.21/3.45	(5) DependencyPairsProof [EQUIVALENT, 27 ms]
10.21/3.45	(6) QDP
10.21/3.45	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
10.21/3.45	(8) AND
10.21/3.45	    (9) QDP
10.21/3.45	        (10) UsableRulesProof [EQUIVALENT, 0 ms]
10.21/3.45	        (11) QDP
10.21/3.45	        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.21/3.45	        (13) YES
10.21/3.45	    (14) QDP
10.21/3.45	        (15) UsableRulesProof [EQUIVALENT, 0 ms]
10.21/3.45	        (16) QDP
10.21/3.45	        (17) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.21/3.45	        (18) YES
10.21/3.45	
10.21/3.45	
10.21/3.45	----------------------------------------
10.21/3.45	
10.21/3.45	(0)
10.21/3.45	Obligation:
10.21/3.45	Q restricted rewrite system:
10.21/3.45	The TRS R consists of the following rules:
10.21/3.45	
10.21/3.45	   f(f(x1)) -> b(b(b(x1)))
10.21/3.45	   a(f(x1)) -> f(a(a(x1)))
10.21/3.45	   b(b(x1)) -> c(c(a(c(x1))))
10.21/3.45	   d(b(x1)) -> d(a(b(x1)))
10.21/3.45	   c(c(x1)) -> d(d(d(x1)))
10.21/3.45	   b(d(x1)) -> d(b(x1))
10.21/3.45	   c(d(d(x1))) -> f(x1)
10.21/3.45	
10.21/3.45	Q is empty.
10.21/3.45	
10.21/3.45	----------------------------------------
10.21/3.45	
10.21/3.45	(1) QTRS Reverse (EQUIVALENT)
10.21/3.45	We applied the QTRS Reverse Processor [REVERSE].
10.21/3.45	----------------------------------------
10.21/3.45	
10.21/3.45	(2)
10.21/3.45	Obligation:
10.21/3.45	Q restricted rewrite system:
10.21/3.45	The TRS R consists of the following rules:
10.21/3.45	
10.21/3.45	   f(f(x1)) -> b(b(b(x1)))
10.21/3.45	   f(a(x1)) -> a(a(f(x1)))
10.21/3.45	   b(b(x1)) -> c(a(c(c(x1))))
10.21/3.45	   b(d(x1)) -> b(a(d(x1)))
10.21/3.45	   c(c(x1)) -> d(d(d(x1)))
10.21/3.45	   d(b(x1)) -> b(d(x1))
10.21/3.45	   d(d(c(x1))) -> f(x1)
10.21/3.45	
10.21/3.45	Q is empty.
10.21/3.45	
10.21/3.45	----------------------------------------
10.21/3.45	
10.21/3.45	(3) QTRSRRRProof (EQUIVALENT)
10.21/3.45	Used ordering:
10.21/3.45	Polynomial interpretation [POLO]:
10.21/3.45	
10.21/3.45	   POL(a(x_1)) = x_1
10.21/3.45	   POL(b(x_1)) = 53 + x_1
10.21/3.45	   POL(c(x_1)) = 35 + x_1
10.21/3.45	   POL(d(x_1)) = 23 + x_1
10.21/3.45	   POL(f(x_1)) = 80 + x_1
10.21/3.45	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
10.21/3.45	
10.21/3.45	   f(f(x1)) -> b(b(b(x1)))
10.21/3.45	   b(b(x1)) -> c(a(c(c(x1))))
10.21/3.45	   c(c(x1)) -> d(d(d(x1)))
10.21/3.45	   d(d(c(x1))) -> f(x1)
10.21/3.45	
10.21/3.45	
10.21/3.45	
10.21/3.45	
10.21/3.45	----------------------------------------
10.21/3.45	
10.21/3.45	(4)
10.21/3.45	Obligation:
10.21/3.45	Q restricted rewrite system:
10.21/3.45	The TRS R consists of the following rules:
10.21/3.45	
10.21/3.45	   f(a(x1)) -> a(a(f(x1)))
10.21/3.45	   b(d(x1)) -> b(a(d(x1)))
10.21/3.45	   d(b(x1)) -> b(d(x1))
10.21/3.45	
10.21/3.45	Q is empty.
10.21/3.45	
10.21/3.45	----------------------------------------
10.21/3.45	
10.21/3.45	(5) DependencyPairsProof (EQUIVALENT)
10.21/3.45	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
10.21/3.45	----------------------------------------
10.21/3.45	
10.21/3.45	(6)
10.21/3.45	Obligation:
10.21/3.45	Q DP problem:
10.21/3.45	The TRS P consists of the following rules:
10.21/3.45	
10.21/3.45	   F(a(x1)) -> F(x1)
10.21/3.45	   B(d(x1)) -> B(a(d(x1)))
10.21/3.45	   D(b(x1)) -> B(d(x1))
10.21/3.45	   D(b(x1)) -> D(x1)
10.21/3.45	
10.21/3.45	The TRS R consists of the following rules:
10.21/3.45	
10.21/3.45	   f(a(x1)) -> a(a(f(x1)))
10.21/3.45	   b(d(x1)) -> b(a(d(x1)))
10.21/3.45	   d(b(x1)) -> b(d(x1))
10.21/3.45	
10.21/3.45	Q is empty.
10.21/3.45	We have to consider all minimal (P,Q,R)-chains.
10.21/3.45	----------------------------------------
10.21/3.45	
10.21/3.45	(7) DependencyGraphProof (EQUIVALENT)
10.21/3.45	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 2 less nodes.
10.21/3.45	----------------------------------------
10.21/3.45	
10.21/3.45	(8)
10.21/3.45	Complex Obligation (AND)
10.21/3.45	
10.21/3.45	----------------------------------------
10.21/3.45	
10.21/3.45	(9)
10.21/3.45	Obligation:
10.21/3.45	Q DP problem:
10.21/3.45	The TRS P consists of the following rules:
10.21/3.45	
10.21/3.45	   D(b(x1)) -> D(x1)
10.21/3.45	
10.21/3.45	The TRS R consists of the following rules:
10.21/3.45	
10.21/3.45	   f(a(x1)) -> a(a(f(x1)))
10.21/3.45	   b(d(x1)) -> b(a(d(x1)))
10.21/3.45	   d(b(x1)) -> b(d(x1))
10.21/3.45	
10.21/3.45	Q is empty.
10.21/3.45	We have to consider all minimal (P,Q,R)-chains.
10.21/3.45	----------------------------------------
10.21/3.45	
10.21/3.45	(10) UsableRulesProof (EQUIVALENT)
10.21/3.45	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.
10.21/3.45	----------------------------------------
10.21/3.45	
10.21/3.45	(11)
10.21/3.45	Obligation:
10.21/3.45	Q DP problem:
10.21/3.45	The TRS P consists of the following rules:
10.21/3.45	
10.21/3.45	   D(b(x1)) -> D(x1)
10.21/3.45	
10.21/3.45	R is empty.
10.21/3.45	Q is empty.
10.21/3.45	We have to consider all minimal (P,Q,R)-chains.
10.21/3.45	----------------------------------------
10.21/3.45	
10.21/3.45	(12) QDPSizeChangeProof (EQUIVALENT)
10.21/3.45	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. 
10.21/3.45	
10.21/3.45	From the DPs we obtained the following set of size-change graphs:
10.21/3.45	*D(b(x1)) -> D(x1)
10.21/3.45	The graph contains the following edges 1 > 1
10.21/3.45	
10.21/3.45	
10.21/3.45	----------------------------------------
10.21/3.45	
10.21/3.45	(13)
10.21/3.45	YES
10.21/3.45	
10.21/3.45	----------------------------------------
10.21/3.45	
10.21/3.45	(14)
10.21/3.45	Obligation:
10.21/3.45	Q DP problem:
10.21/3.45	The TRS P consists of the following rules:
10.21/3.45	
10.21/3.45	   F(a(x1)) -> F(x1)
10.21/3.45	
10.21/3.45	The TRS R consists of the following rules:
10.21/3.45	
10.21/3.45	   f(a(x1)) -> a(a(f(x1)))
10.21/3.45	   b(d(x1)) -> b(a(d(x1)))
10.21/3.45	   d(b(x1)) -> b(d(x1))
10.21/3.45	
10.21/3.45	Q is empty.
10.21/3.45	We have to consider all minimal (P,Q,R)-chains.
10.21/3.45	----------------------------------------
10.21/3.45	
10.21/3.45	(15) UsableRulesProof (EQUIVALENT)
10.21/3.45	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.
10.21/3.45	----------------------------------------
10.21/3.45	
10.21/3.45	(16)
10.21/3.45	Obligation:
10.21/3.45	Q DP problem:
10.21/3.45	The TRS P consists of the following rules:
10.21/3.45	
10.21/3.45	   F(a(x1)) -> F(x1)
10.21/3.45	
10.21/3.45	R is empty.
10.21/3.45	Q is empty.
10.21/3.45	We have to consider all minimal (P,Q,R)-chains.
10.21/3.45	----------------------------------------
10.21/3.45	
10.21/3.45	(17) QDPSizeChangeProof (EQUIVALENT)
10.21/3.45	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. 
10.21/3.45	
10.21/3.45	From the DPs we obtained the following set of size-change graphs:
10.21/3.45	*F(a(x1)) -> F(x1)
10.21/3.45	The graph contains the following edges 1 > 1
10.21/3.45	
10.21/3.45	
10.21/3.45	----------------------------------------
10.21/3.45	
10.21/3.45	(18)
10.21/3.45	YES
10.61/3.59	EOF