25.63/7.41	YES
26.70/7.67	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
26.70/7.67	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
26.70/7.67	
26.70/7.67	
26.70/7.67	Termination w.r.t. Q of the given QTRS could be proven:
26.70/7.67	
26.70/7.67	(0) QTRS
26.70/7.67	(1) DependencyPairsProof [EQUIVALENT, 27 ms]
26.70/7.67	(2) QDP
26.70/7.67	(3) QDPOrderProof [EQUIVALENT, 107 ms]
26.70/7.67	(4) QDP
26.70/7.67	(5) PisEmptyProof [EQUIVALENT, 0 ms]
26.70/7.67	(6) YES
26.70/7.67	
26.70/7.67	
26.70/7.67	----------------------------------------
26.70/7.67	
26.70/7.67	(0)
26.70/7.67	Obligation:
26.70/7.67	Q restricted rewrite system:
26.70/7.67	The TRS R consists of the following rules:
26.70/7.67	
26.70/7.67	   a(b(x1)) -> x1
26.70/7.67	   a(c(x1)) -> b(c(c(a(x1))))
26.70/7.67	   b(c(x1)) -> a(b(x1))
26.70/7.67	
26.70/7.67	Q is empty.
26.70/7.67	
26.70/7.67	----------------------------------------
26.70/7.67	
26.70/7.67	(1) DependencyPairsProof (EQUIVALENT)
26.70/7.67	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
26.70/7.67	----------------------------------------
26.70/7.67	
26.70/7.67	(2)
26.70/7.67	Obligation:
26.70/7.67	Q DP problem:
26.70/7.67	The TRS P consists of the following rules:
26.70/7.67	
26.70/7.67	   A(c(x1)) -> B(c(c(a(x1))))
26.70/7.67	   A(c(x1)) -> A(x1)
26.70/7.67	   B(c(x1)) -> A(b(x1))
26.70/7.67	   B(c(x1)) -> B(x1)
26.70/7.67	
26.70/7.67	The TRS R consists of the following rules:
26.70/7.67	
26.70/7.67	   a(b(x1)) -> x1
26.70/7.67	   a(c(x1)) -> b(c(c(a(x1))))
26.70/7.67	   b(c(x1)) -> a(b(x1))
26.70/7.67	
26.70/7.67	Q is empty.
26.70/7.67	We have to consider all minimal (P,Q,R)-chains.
26.70/7.67	----------------------------------------
26.70/7.67	
26.70/7.67	(3) QDPOrderProof (EQUIVALENT)
26.70/7.67	We use the reduction pair processor [LPAR04,JAR06].
26.70/7.67	
26.70/7.67	
26.70/7.67	The following pairs can be oriented strictly and are deleted.
26.70/7.67	
26.70/7.67	   A(c(x1)) -> B(c(c(a(x1))))
26.70/7.67	   A(c(x1)) -> A(x1)
26.70/7.67	   B(c(x1)) -> A(b(x1))
26.70/7.67	   B(c(x1)) -> B(x1)
26.70/7.67	The remaining pairs can at least be oriented weakly.
26.70/7.67	Used ordering:  Polynomial interpretation [POLO,RATPOLO]:
26.70/7.67	
26.70/7.67	   POL(A(x_1)) = [7/4] + [2]x_1
26.70/7.67	   POL(B(x_1)) = [1/2]x_1
26.70/7.67	   POL(a(x_1)) = [3/2] + [2]x_1
26.70/7.67	   POL(b(x_1)) = [1/2]x_1
26.70/7.67	   POL(c(x_1)) = [4] + [2]x_1
26.70/7.67	The value of delta used in the strict ordering is 1/4.
26.70/7.67	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
26.70/7.67	
26.70/7.67	   a(b(x1)) -> x1
26.70/7.67	   a(c(x1)) -> b(c(c(a(x1))))
26.70/7.67	   b(c(x1)) -> a(b(x1))
26.70/7.67	
26.70/7.67	
26.70/7.67	----------------------------------------
26.70/7.67	
26.70/7.67	(4)
26.70/7.67	Obligation:
26.70/7.67	Q DP problem:
26.70/7.67	P is empty.
26.70/7.67	The TRS R consists of the following rules:
26.70/7.67	
26.70/7.67	   a(b(x1)) -> x1
26.70/7.67	   a(c(x1)) -> b(c(c(a(x1))))
26.70/7.67	   b(c(x1)) -> a(b(x1))
26.70/7.67	
26.70/7.67	Q is empty.
26.70/7.67	We have to consider all minimal (P,Q,R)-chains.
26.70/7.67	----------------------------------------
26.70/7.67	
26.70/7.67	(5) PisEmptyProof (EQUIVALENT)
26.70/7.67	The TRS P is empty. Hence, there is no (P,Q,R) chain.
26.70/7.67	----------------------------------------
26.70/7.67	
26.70/7.67	(6)
26.70/7.67	YES
26.96/7.75	EOF