11.38/3.75	YES
11.38/3.78	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
11.38/3.78	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.38/3.78	
11.38/3.78	
11.38/3.78	Termination w.r.t. Q of the given QTRS could be proven:
11.38/3.78	
11.38/3.78	(0) QTRS
11.38/3.78	(1) QTRSRRRProof [EQUIVALENT, 35 ms]
11.38/3.78	(2) QTRS
11.38/3.78	(3) DependencyPairsProof [EQUIVALENT, 26 ms]
11.38/3.78	(4) QDP
11.38/3.78	(5) QDPOrderProof [EQUIVALENT, 26 ms]
11.38/3.78	(6) QDP
11.38/3.78	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
11.38/3.78	(8) TRUE
11.38/3.78	
11.38/3.78	
11.38/3.78	----------------------------------------
11.38/3.78	
11.38/3.78	(0)
11.38/3.78	Obligation:
11.38/3.78	Q restricted rewrite system:
11.38/3.78	The TRS R consists of the following rules:
11.38/3.78	
11.38/3.78	   a(a(x1)) -> d(c(x1))
11.38/3.78	   a(b(x1)) -> c(c(c(x1)))
11.38/3.78	   b(b(x1)) -> a(c(c(x1)))
11.38/3.78	   c(c(x1)) -> b(x1)
11.38/3.78	   c(d(x1)) -> a(a(x1))
11.38/3.78	   d(d(x1)) -> b(a(c(x1)))
11.38/3.78	
11.38/3.78	Q is empty.
11.38/3.78	
11.38/3.78	----------------------------------------
11.38/3.78	
11.38/3.78	(1) QTRSRRRProof (EQUIVALENT)
11.38/3.78	Used ordering:
11.38/3.78	Polynomial interpretation [POLO]:
11.38/3.78	
11.38/3.78	   POL(a(x_1)) = 15 + x_1
11.38/3.78	   POL(b(x_1)) = 17 + x_1
11.38/3.78	   POL(c(x_1)) = 9 + x_1
11.38/3.78	   POL(d(x_1)) = 21 + x_1
11.38/3.78	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
11.38/3.78	
11.38/3.78	   a(b(x1)) -> c(c(c(x1)))
11.38/3.78	   b(b(x1)) -> a(c(c(x1)))
11.38/3.78	   c(c(x1)) -> b(x1)
11.38/3.78	   d(d(x1)) -> b(a(c(x1)))
11.38/3.78	
11.38/3.78	
11.38/3.78	
11.38/3.78	
11.38/3.78	----------------------------------------
11.38/3.78	
11.38/3.78	(2)
11.38/3.78	Obligation:
11.38/3.78	Q restricted rewrite system:
11.38/3.78	The TRS R consists of the following rules:
11.38/3.78	
11.38/3.78	   a(a(x1)) -> d(c(x1))
11.38/3.78	   c(d(x1)) -> a(a(x1))
11.38/3.78	
11.38/3.78	Q is empty.
11.38/3.78	
11.38/3.78	----------------------------------------
11.38/3.78	
11.38/3.78	(3) DependencyPairsProof (EQUIVALENT)
11.38/3.78	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
11.38/3.78	----------------------------------------
11.38/3.78	
11.38/3.78	(4)
11.38/3.78	Obligation:
11.38/3.78	Q DP problem:
11.38/3.78	The TRS P consists of the following rules:
11.38/3.78	
11.38/3.78	   A(a(x1)) -> C(x1)
11.38/3.78	   C(d(x1)) -> A(a(x1))
11.38/3.78	   C(d(x1)) -> A(x1)
11.38/3.78	
11.38/3.78	The TRS R consists of the following rules:
11.38/3.78	
11.38/3.78	   a(a(x1)) -> d(c(x1))
11.38/3.78	   c(d(x1)) -> a(a(x1))
11.38/3.78	
11.38/3.78	Q is empty.
11.38/3.78	We have to consider all minimal (P,Q,R)-chains.
11.38/3.78	----------------------------------------
11.38/3.78	
11.38/3.78	(5) QDPOrderProof (EQUIVALENT)
11.38/3.78	We use the reduction pair processor [LPAR04,JAR06].
11.38/3.78	
11.38/3.78	
11.38/3.78	The following pairs can be oriented strictly and are deleted.
11.38/3.78	
11.38/3.78	   C(d(x1)) -> A(a(x1))
11.38/3.78	   C(d(x1)) -> A(x1)
11.38/3.78	The remaining pairs can at least be oriented weakly.
11.38/3.78	Used ordering:  Polynomial interpretation [POLO]:
11.38/3.78	
11.38/3.78	   POL(A(x_1)) = x_1
11.38/3.78	   POL(C(x_1)) = 1 + x_1
11.38/3.78	   POL(a(x_1)) = 1 + x_1
11.38/3.78	   POL(c(x_1)) = 1 + x_1
11.38/3.78	   POL(d(x_1)) = 1 + x_1
11.38/3.78	
11.38/3.78	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
11.38/3.78	
11.38/3.78	   a(a(x1)) -> d(c(x1))
11.38/3.78	   c(d(x1)) -> a(a(x1))
11.38/3.78	
11.38/3.78	
11.38/3.78	----------------------------------------
11.38/3.78	
11.38/3.78	(6)
11.38/3.78	Obligation:
11.38/3.78	Q DP problem:
11.38/3.78	The TRS P consists of the following rules:
11.38/3.78	
11.38/3.78	   A(a(x1)) -> C(x1)
11.38/3.78	
11.38/3.78	The TRS R consists of the following rules:
11.38/3.78	
11.38/3.78	   a(a(x1)) -> d(c(x1))
11.38/3.78	   c(d(x1)) -> a(a(x1))
11.38/3.78	
11.38/3.78	Q is empty.
11.38/3.78	We have to consider all minimal (P,Q,R)-chains.
11.38/3.78	----------------------------------------
11.38/3.78	
11.38/3.78	(7) DependencyGraphProof (EQUIVALENT)
11.38/3.78	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
11.38/3.78	----------------------------------------
11.38/3.78	
11.38/3.78	(8)
11.38/3.78	TRUE
11.70/3.89	EOF