10.71/3.59	YES
10.97/3.64	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
10.97/3.64	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.97/3.64	
10.97/3.64	
10.97/3.64	Termination w.r.t. Q of the given QTRS could be proven:
10.97/3.64	
10.97/3.64	(0) QTRS
10.97/3.64	(1) QTRS Reverse [EQUIVALENT, 0 ms]
10.97/3.64	(2) QTRS
10.97/3.64	(3) QTRSRRRProof [EQUIVALENT, 2 ms]
10.97/3.64	(4) QTRS
10.97/3.64	(5) Overlay + Local Confluence [EQUIVALENT, 0 ms]
10.97/3.64	(6) QTRS
10.97/3.64	(7) DependencyPairsProof [EQUIVALENT, 1 ms]
10.97/3.64	(8) QDP
10.97/3.64	(9) DependencyGraphProof [EQUIVALENT, 1 ms]
10.97/3.64	(10) QDP
10.97/3.64	(11) UsableRulesProof [EQUIVALENT, 0 ms]
10.97/3.64	(12) QDP
10.97/3.64	(13) QReductionProof [EQUIVALENT, 0 ms]
10.97/3.64	(14) QDP
10.97/3.64	(15) QDPSizeChangeProof [EQUIVALENT, 1 ms]
10.97/3.64	(16) YES
10.97/3.64	
10.97/3.64	
10.97/3.64	----------------------------------------
10.97/3.64	
10.97/3.64	(0)
10.97/3.64	Obligation:
10.97/3.64	Q restricted rewrite system:
10.97/3.64	The TRS R consists of the following rules:
10.97/3.64	
10.97/3.64	   a(a(x1)) -> b(b(b(x1)))
10.97/3.64	   a(x1) -> d(c(d(x1)))
10.97/3.64	   b(b(x1)) -> c(c(c(x1)))
10.97/3.64	   c(c(x1)) -> d(d(d(x1)))
10.97/3.64	   c(d(d(x1))) -> a(x1)
10.97/3.64	
10.97/3.64	Q is empty.
10.97/3.64	
10.97/3.64	----------------------------------------
10.97/3.64	
10.97/3.64	(1) QTRS Reverse (EQUIVALENT)
10.97/3.64	We applied the QTRS Reverse Processor [REVERSE].
10.97/3.64	----------------------------------------
10.97/3.64	
10.97/3.64	(2)
10.97/3.64	Obligation:
10.97/3.64	Q restricted rewrite system:
10.97/3.64	The TRS R consists of the following rules:
10.97/3.64	
10.97/3.64	   a(a(x1)) -> b(b(b(x1)))
10.97/3.64	   a(x1) -> d(c(d(x1)))
10.97/3.64	   b(b(x1)) -> c(c(c(x1)))
10.97/3.64	   c(c(x1)) -> d(d(d(x1)))
10.97/3.64	   d(d(c(x1))) -> a(x1)
10.97/3.64	
10.97/3.64	Q is empty.
10.97/3.64	
10.97/3.64	----------------------------------------
10.97/3.64	
10.97/3.64	(3) QTRSRRRProof (EQUIVALENT)
10.97/3.64	Used ordering:
10.97/3.64	Polynomial interpretation [POLO]:
10.97/3.64	
10.97/3.64	   POL(a(x_1)) = 53 + x_1
10.97/3.64	   POL(b(x_1)) = 35 + x_1
10.97/3.64	   POL(c(x_1)) = 23 + x_1
10.97/3.64	   POL(d(x_1)) = 15 + x_1
10.97/3.64	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
10.97/3.64	
10.97/3.64	   a(a(x1)) -> b(b(b(x1)))
10.97/3.64	   b(b(x1)) -> c(c(c(x1)))
10.97/3.64	   c(c(x1)) -> d(d(d(x1)))
10.97/3.64	
10.97/3.64	
10.97/3.64	
10.97/3.64	
10.97/3.64	----------------------------------------
10.97/3.64	
10.97/3.64	(4)
10.97/3.64	Obligation:
10.97/3.64	Q restricted rewrite system:
10.97/3.64	The TRS R consists of the following rules:
10.97/3.64	
10.97/3.64	   a(x1) -> d(c(d(x1)))
10.97/3.64	   d(d(c(x1))) -> a(x1)
10.97/3.64	
10.97/3.64	Q is empty.
10.97/3.64	
10.97/3.64	----------------------------------------
10.97/3.64	
10.97/3.64	(5) Overlay + Local Confluence (EQUIVALENT)
10.97/3.64	The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
10.97/3.64	----------------------------------------
10.97/3.64	
10.97/3.64	(6)
10.97/3.64	Obligation:
10.97/3.64	Q restricted rewrite system:
10.97/3.64	The TRS R consists of the following rules:
10.97/3.64	
10.97/3.64	   a(x1) -> d(c(d(x1)))
10.97/3.64	   d(d(c(x1))) -> a(x1)
10.97/3.64	
10.97/3.64	The set Q consists of the following terms:
10.97/3.64	
10.97/3.64	   a(x0)
10.97/3.64	   d(d(c(x0)))
10.97/3.64	
10.97/3.64	
10.97/3.64	----------------------------------------
10.97/3.64	
10.97/3.64	(7) DependencyPairsProof (EQUIVALENT)
10.97/3.64	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
10.97/3.64	----------------------------------------
10.97/3.64	
10.97/3.64	(8)
10.97/3.64	Obligation:
10.97/3.64	Q DP problem:
10.97/3.64	The TRS P consists of the following rules:
10.97/3.64	
10.97/3.64	   A(x1) -> D(c(d(x1)))
10.97/3.64	   A(x1) -> D(x1)
10.97/3.64	   D(d(c(x1))) -> A(x1)
10.97/3.64	
10.97/3.64	The TRS R consists of the following rules:
10.97/3.64	
10.97/3.64	   a(x1) -> d(c(d(x1)))
10.97/3.64	   d(d(c(x1))) -> a(x1)
10.97/3.64	
10.97/3.64	The set Q consists of the following terms:
10.97/3.64	
10.97/3.64	   a(x0)
10.97/3.64	   d(d(c(x0)))
10.97/3.64	
10.97/3.64	We have to consider all minimal (P,Q,R)-chains.
10.97/3.64	----------------------------------------
10.97/3.64	
10.97/3.64	(9) DependencyGraphProof (EQUIVALENT)
10.97/3.64	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
10.97/3.64	----------------------------------------
10.97/3.64	
10.97/3.64	(10)
10.97/3.64	Obligation:
10.97/3.64	Q DP problem:
10.97/3.64	The TRS P consists of the following rules:
10.97/3.64	
10.97/3.64	   A(x1) -> D(x1)
10.97/3.64	   D(d(c(x1))) -> A(x1)
10.97/3.64	
10.97/3.64	The TRS R consists of the following rules:
10.97/3.64	
10.97/3.64	   a(x1) -> d(c(d(x1)))
10.97/3.64	   d(d(c(x1))) -> a(x1)
10.97/3.64	
10.97/3.64	The set Q consists of the following terms:
10.97/3.64	
10.97/3.64	   a(x0)
10.97/3.64	   d(d(c(x0)))
10.97/3.64	
10.97/3.64	We have to consider all minimal (P,Q,R)-chains.
10.97/3.64	----------------------------------------
10.97/3.64	
10.97/3.64	(11) UsableRulesProof (EQUIVALENT)
10.97/3.64	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
10.97/3.64	----------------------------------------
10.97/3.64	
10.97/3.64	(12)
10.97/3.64	Obligation:
10.97/3.64	Q DP problem:
10.97/3.64	The TRS P consists of the following rules:
10.97/3.64	
10.97/3.64	   A(x1) -> D(x1)
10.97/3.64	   D(d(c(x1))) -> A(x1)
10.97/3.64	
10.97/3.64	R is empty.
10.97/3.64	The set Q consists of the following terms:
10.97/3.64	
10.97/3.64	   a(x0)
10.97/3.64	   d(d(c(x0)))
10.97/3.64	
10.97/3.64	We have to consider all minimal (P,Q,R)-chains.
10.97/3.64	----------------------------------------
10.97/3.64	
10.97/3.64	(13) QReductionProof (EQUIVALENT)
10.97/3.64	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
10.97/3.64	
10.97/3.64	   a(x0)
10.97/3.64	
10.97/3.64	
10.97/3.64	----------------------------------------
10.97/3.64	
10.97/3.64	(14)
10.97/3.64	Obligation:
10.97/3.64	Q DP problem:
10.97/3.64	The TRS P consists of the following rules:
10.97/3.64	
10.97/3.64	   A(x1) -> D(x1)
10.97/3.64	   D(d(c(x1))) -> A(x1)
10.97/3.64	
10.97/3.64	R is empty.
10.97/3.64	The set Q consists of the following terms:
10.97/3.64	
10.97/3.64	   d(d(c(x0)))
10.97/3.64	
10.97/3.64	We have to consider all minimal (P,Q,R)-chains.
10.97/3.64	----------------------------------------
10.97/3.64	
10.97/3.64	(15) QDPSizeChangeProof (EQUIVALENT)
10.97/3.64	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. 
10.97/3.64	
10.97/3.64	From the DPs we obtained the following set of size-change graphs:
10.97/3.64	*D(d(c(x1))) -> A(x1)
10.97/3.64	The graph contains the following edges 1 > 1
10.97/3.64	
10.97/3.64	
10.97/3.64	*A(x1) -> D(x1)
10.97/3.64	The graph contains the following edges 1 >= 1
10.97/3.64	
10.97/3.64	
10.97/3.64	----------------------------------------
10.97/3.64	
10.97/3.64	(16)
10.97/3.64	YES
11.17/3.72	EOF