33.15/9.40	YES
33.57/9.47	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
33.57/9.47	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
33.57/9.47	
33.57/9.47	
33.57/9.47	Termination w.r.t. Q of the given QTRS could be proven:
33.57/9.47	
33.57/9.47	(0) QTRS
33.57/9.47	(1) DependencyPairsProof [EQUIVALENT, 23 ms]
33.57/9.47	(2) QDP
33.57/9.47	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
33.57/9.47	(4) QDP
33.57/9.47	(5) QDPOrderProof [EQUIVALENT, 73 ms]
33.57/9.47	(6) QDP
33.57/9.47	(7) PisEmptyProof [EQUIVALENT, 0 ms]
33.57/9.47	(8) YES
33.57/9.47	
33.57/9.47	
33.57/9.47	----------------------------------------
33.57/9.47	
33.57/9.47	(0)
33.57/9.47	Obligation:
33.57/9.47	Q restricted rewrite system:
33.57/9.47	The TRS R consists of the following rules:
33.57/9.47	
33.57/9.47	   a(b(x1)) -> x1
33.57/9.47	   a(c(x1)) -> b(c(c(a(a(b(x1))))))
33.57/9.47	   b(c(x1)) -> x1
33.57/9.47	
33.57/9.47	Q is empty.
33.57/9.47	
33.57/9.47	----------------------------------------
33.57/9.47	
33.57/9.47	(1) DependencyPairsProof (EQUIVALENT)
33.57/9.47	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
33.57/9.47	----------------------------------------
33.57/9.47	
33.57/9.47	(2)
33.57/9.47	Obligation:
33.57/9.47	Q DP problem:
33.57/9.47	The TRS P consists of the following rules:
33.57/9.47	
33.57/9.47	   A(c(x1)) -> B(c(c(a(a(b(x1))))))
33.57/9.47	   A(c(x1)) -> A(a(b(x1)))
33.57/9.47	   A(c(x1)) -> A(b(x1))
33.57/9.47	   A(c(x1)) -> B(x1)
33.57/9.47	
33.57/9.47	The TRS R consists of the following rules:
33.57/9.47	
33.57/9.47	   a(b(x1)) -> x1
33.57/9.47	   a(c(x1)) -> b(c(c(a(a(b(x1))))))
33.57/9.47	   b(c(x1)) -> x1
33.57/9.47	
33.57/9.47	Q is empty.
33.57/9.47	We have to consider all minimal (P,Q,R)-chains.
33.57/9.47	----------------------------------------
33.57/9.47	
33.57/9.47	(3) DependencyGraphProof (EQUIVALENT)
33.57/9.47	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
33.57/9.47	----------------------------------------
33.57/9.47	
33.57/9.47	(4)
33.57/9.47	Obligation:
33.57/9.47	Q DP problem:
33.57/9.47	The TRS P consists of the following rules:
33.57/9.47	
33.57/9.47	   A(c(x1)) -> A(b(x1))
33.57/9.47	   A(c(x1)) -> A(a(b(x1)))
33.57/9.47	
33.57/9.47	The TRS R consists of the following rules:
33.57/9.47	
33.57/9.47	   a(b(x1)) -> x1
33.57/9.47	   a(c(x1)) -> b(c(c(a(a(b(x1))))))
33.57/9.47	   b(c(x1)) -> x1
33.57/9.47	
33.57/9.47	Q is empty.
33.57/9.47	We have to consider all minimal (P,Q,R)-chains.
33.57/9.47	----------------------------------------
33.57/9.47	
33.57/9.47	(5) QDPOrderProof (EQUIVALENT)
33.57/9.47	We use the reduction pair processor [LPAR04,JAR06].
33.57/9.47	
33.57/9.47	
33.57/9.47	The following pairs can be oriented strictly and are deleted.
33.57/9.47	
33.57/9.47	   A(c(x1)) -> A(b(x1))
33.57/9.47	   A(c(x1)) -> A(a(b(x1)))
33.57/9.47	The remaining pairs can at least be oriented weakly.
33.57/9.47	Used ordering:  Polynomial interpretation [POLO,RATPOLO]:
33.57/9.47	
33.57/9.47	   POL(A(x_1)) = [1/2]x_1
33.57/9.47	   POL(a(x_1)) = [4]x_1
33.57/9.47	   POL(b(x_1)) = [1/4]x_1
33.57/9.47	   POL(c(x_1)) = [1/2] + [4]x_1
33.57/9.47	The value of delta used in the strict ordering is 1/4.
33.57/9.47	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
33.57/9.47	
33.57/9.47	   b(c(x1)) -> x1
33.57/9.47	   a(b(x1)) -> x1
33.57/9.47	   a(c(x1)) -> b(c(c(a(a(b(x1))))))
33.57/9.47	
33.57/9.47	
33.57/9.47	----------------------------------------
33.57/9.47	
33.57/9.47	(6)
33.57/9.47	Obligation:
33.57/9.47	Q DP problem:
33.57/9.47	P is empty.
33.57/9.47	The TRS R consists of the following rules:
33.57/9.47	
33.57/9.47	   a(b(x1)) -> x1
33.57/9.47	   a(c(x1)) -> b(c(c(a(a(b(x1))))))
33.57/9.47	   b(c(x1)) -> x1
33.57/9.47	
33.57/9.47	Q is empty.
33.57/9.47	We have to consider all minimal (P,Q,R)-chains.
33.57/9.47	----------------------------------------
33.57/9.47	
33.57/9.47	(7) PisEmptyProof (EQUIVALENT)
33.57/9.47	The TRS P is empty. Hence, there is no (P,Q,R) chain.
33.57/9.47	----------------------------------------
33.57/9.47	
33.57/9.47	(8)
33.57/9.47	YES
33.67/9.53	EOF