35.63/10.01	YES
36.58/10.21	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
36.58/10.21	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
36.58/10.21	
36.58/10.21	
36.58/10.21	Termination w.r.t. Q of the given QTRS could be proven:
36.58/10.21	
36.58/10.21	(0) QTRS
36.58/10.21	(1) DependencyPairsProof [EQUIVALENT, 22 ms]
36.58/10.21	(2) QDP
36.58/10.21	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
36.58/10.21	(4) QDP
36.58/10.21	(5) QDPOrderProof [EQUIVALENT, 56 ms]
36.58/10.21	(6) QDP
36.58/10.21	(7) QDPOrderProof [EQUIVALENT, 85 ms]
36.58/10.21	(8) QDP
36.58/10.21	(9) PisEmptyProof [EQUIVALENT, 0 ms]
36.58/10.21	(10) YES
36.58/10.21	
36.58/10.21	
36.58/10.21	----------------------------------------
36.58/10.21	
36.58/10.21	(0)
36.58/10.21	Obligation:
36.58/10.21	Q restricted rewrite system:
36.58/10.21	The TRS R consists of the following rules:
36.58/10.21	
36.58/10.21	   a(b(x1)) -> x1
36.58/10.21	   a(c(x1)) -> b(b(c(c(a(a(x1))))))
36.58/10.21	   b(c(x1)) -> x1
36.58/10.21	
36.58/10.21	Q is empty.
36.58/10.21	
36.58/10.21	----------------------------------------
36.58/10.21	
36.58/10.21	(1) DependencyPairsProof (EQUIVALENT)
36.58/10.21	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
36.58/10.21	----------------------------------------
36.58/10.21	
36.58/10.21	(2)
36.58/10.21	Obligation:
36.58/10.21	Q DP problem:
36.58/10.21	The TRS P consists of the following rules:
36.58/10.21	
36.58/10.21	   A(c(x1)) -> B(b(c(c(a(a(x1))))))
36.58/10.21	   A(c(x1)) -> B(c(c(a(a(x1)))))
36.58/10.21	   A(c(x1)) -> A(a(x1))
36.58/10.21	   A(c(x1)) -> A(x1)
36.58/10.21	
36.58/10.21	The TRS R consists of the following rules:
36.58/10.21	
36.58/10.21	   a(b(x1)) -> x1
36.58/10.21	   a(c(x1)) -> b(b(c(c(a(a(x1))))))
36.58/10.21	   b(c(x1)) -> x1
36.58/10.21	
36.58/10.21	Q is empty.
36.58/10.21	We have to consider all minimal (P,Q,R)-chains.
36.58/10.21	----------------------------------------
36.58/10.21	
36.58/10.21	(3) DependencyGraphProof (EQUIVALENT)
36.58/10.21	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
36.58/10.21	----------------------------------------
36.58/10.21	
36.58/10.21	(4)
36.58/10.21	Obligation:
36.58/10.21	Q DP problem:
36.58/10.21	The TRS P consists of the following rules:
36.58/10.21	
36.58/10.21	   A(c(x1)) -> A(x1)
36.58/10.21	   A(c(x1)) -> A(a(x1))
36.58/10.21	
36.58/10.21	The TRS R consists of the following rules:
36.58/10.21	
36.58/10.21	   a(b(x1)) -> x1
36.58/10.21	   a(c(x1)) -> b(b(c(c(a(a(x1))))))
36.58/10.21	   b(c(x1)) -> x1
36.58/10.21	
36.58/10.21	Q is empty.
36.58/10.21	We have to consider all minimal (P,Q,R)-chains.
36.58/10.21	----------------------------------------
36.58/10.21	
36.58/10.21	(5) QDPOrderProof (EQUIVALENT)
36.58/10.21	We use the reduction pair processor [LPAR04,JAR06].
36.58/10.21	
36.58/10.21	
36.58/10.21	The following pairs can be oriented strictly and are deleted.
36.58/10.21	
36.58/10.21	   A(c(x1)) -> A(x1)
36.58/10.21	The remaining pairs can at least be oriented weakly.
36.58/10.21	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
36.58/10.21	
36.58/10.21	POL( A_1(x_1) ) = 2x_1
36.58/10.21	POL( a_1(x_1) ) = x_1 + 1
36.58/10.21	POL( b_1(x_1) ) = max{0, x_1 - 1}
36.58/10.21	POL( c_1(x_1) ) = x_1 + 1
36.58/10.21	
36.58/10.21	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
36.58/10.21	
36.58/10.21	   a(b(x1)) -> x1
36.58/10.21	   a(c(x1)) -> b(b(c(c(a(a(x1))))))
36.58/10.21	   b(c(x1)) -> x1
36.58/10.21	
36.58/10.21	
36.58/10.21	----------------------------------------
36.58/10.21	
36.58/10.21	(6)
36.58/10.21	Obligation:
36.58/10.21	Q DP problem:
36.58/10.21	The TRS P consists of the following rules:
36.58/10.21	
36.58/10.21	   A(c(x1)) -> A(a(x1))
36.58/10.21	
36.58/10.21	The TRS R consists of the following rules:
36.58/10.21	
36.58/10.21	   a(b(x1)) -> x1
36.58/10.21	   a(c(x1)) -> b(b(c(c(a(a(x1))))))
36.58/10.21	   b(c(x1)) -> x1
36.58/10.21	
36.58/10.21	Q is empty.
36.58/10.21	We have to consider all minimal (P,Q,R)-chains.
36.58/10.21	----------------------------------------
36.58/10.21	
36.58/10.21	(7) QDPOrderProof (EQUIVALENT)
36.58/10.21	We use the reduction pair processor [LPAR04,JAR06].
36.58/10.21	
36.58/10.21	
36.58/10.21	The following pairs can be oriented strictly and are deleted.
36.58/10.21	
36.58/10.21	   A(c(x1)) -> A(a(x1))
36.58/10.21	The remaining pairs can at least be oriented weakly.
36.58/10.21	Used ordering:  Polynomial interpretation [POLO,RATPOLO]:
36.58/10.21	
36.58/10.21	   POL(A(x_1)) = [1/4]x_1
36.58/10.21	   POL(a(x_1)) = [2]x_1
36.58/10.21	   POL(b(x_1)) = [1/2]x_1
36.58/10.21	   POL(c(x_1)) = [1/4] + [2]x_1
36.58/10.21	The value of delta used in the strict ordering is 1/16.
36.58/10.21	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
36.58/10.21	
36.58/10.21	   a(b(x1)) -> x1
36.58/10.21	   a(c(x1)) -> b(b(c(c(a(a(x1))))))
36.58/10.21	   b(c(x1)) -> x1
36.58/10.21	
36.58/10.21	
36.58/10.21	----------------------------------------
36.58/10.21	
36.58/10.21	(8)
36.58/10.21	Obligation:
36.58/10.21	Q DP problem:
36.58/10.21	P is empty.
36.58/10.21	The TRS R consists of the following rules:
36.58/10.21	
36.58/10.21	   a(b(x1)) -> x1
36.58/10.21	   a(c(x1)) -> b(b(c(c(a(a(x1))))))
36.58/10.21	   b(c(x1)) -> x1
36.58/10.21	
36.58/10.21	Q is empty.
36.58/10.21	We have to consider all minimal (P,Q,R)-chains.
36.58/10.21	----------------------------------------
36.58/10.21	
36.58/10.21	(9) PisEmptyProof (EQUIVALENT)
36.58/10.21	The TRS P is empty. Hence, there is no (P,Q,R) chain.
36.58/10.21	----------------------------------------
36.58/10.21	
36.58/10.21	(10)
36.58/10.21	YES
37.02/10.36	EOF