33.15/9.30	YES
33.27/9.33	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
33.27/9.33	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
33.27/9.33	
33.27/9.33	
33.27/9.33	Termination w.r.t. Q of the given QTRS could be proven:
33.27/9.33	
33.27/9.33	(0) QTRS
33.27/9.33	(1) DependencyPairsProof [EQUIVALENT, 1 ms]
33.27/9.33	(2) QDP
33.27/9.33	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
33.27/9.33	(4) QDP
33.27/9.33	(5) QDPOrderProof [EQUIVALENT, 55 ms]
33.27/9.33	(6) QDP
33.27/9.33	(7) QDPOrderProof [EQUIVALENT, 309 ms]
33.27/9.33	(8) QDP
33.27/9.33	(9) PisEmptyProof [EQUIVALENT, 0 ms]
33.27/9.33	(10) YES
33.27/9.33	
33.27/9.33	
33.27/9.33	----------------------------------------
33.27/9.33	
33.27/9.33	(0)
33.27/9.33	Obligation:
33.27/9.33	Q restricted rewrite system:
33.27/9.33	The TRS R consists of the following rules:
33.27/9.33	
33.27/9.33	   a(b(x1)) -> x1
33.27/9.33	   a(c(x1)) -> a(b(b(c(c(a(x1))))))
33.27/9.33	   b(c(x1)) -> x1
33.27/9.33	
33.27/9.33	Q is empty.
33.27/9.33	
33.27/9.33	----------------------------------------
33.27/9.33	
33.27/9.33	(1) DependencyPairsProof (EQUIVALENT)
33.27/9.33	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
33.27/9.33	----------------------------------------
33.27/9.33	
33.27/9.33	(2)
33.27/9.33	Obligation:
33.27/9.33	Q DP problem:
33.27/9.33	The TRS P consists of the following rules:
33.27/9.33	
33.27/9.33	   A(c(x1)) -> A(b(b(c(c(a(x1))))))
33.27/9.33	   A(c(x1)) -> B(b(c(c(a(x1)))))
33.27/9.33	   A(c(x1)) -> B(c(c(a(x1))))
33.27/9.33	   A(c(x1)) -> A(x1)
33.27/9.33	
33.27/9.33	The TRS R consists of the following rules:
33.27/9.33	
33.27/9.33	   a(b(x1)) -> x1
33.27/9.33	   a(c(x1)) -> a(b(b(c(c(a(x1))))))
33.27/9.33	   b(c(x1)) -> x1
33.27/9.33	
33.27/9.33	Q is empty.
33.27/9.33	We have to consider all minimal (P,Q,R)-chains.
33.27/9.33	----------------------------------------
33.27/9.33	
33.27/9.33	(3) DependencyGraphProof (EQUIVALENT)
33.27/9.33	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
33.27/9.33	----------------------------------------
33.27/9.33	
33.27/9.33	(4)
33.27/9.33	Obligation:
33.27/9.33	Q DP problem:
33.27/9.33	The TRS P consists of the following rules:
33.27/9.33	
33.27/9.33	   A(c(x1)) -> A(x1)
33.27/9.33	   A(c(x1)) -> A(b(b(c(c(a(x1))))))
33.27/9.33	
33.27/9.33	The TRS R consists of the following rules:
33.27/9.33	
33.27/9.33	   a(b(x1)) -> x1
33.27/9.33	   a(c(x1)) -> a(b(b(c(c(a(x1))))))
33.27/9.33	   b(c(x1)) -> x1
33.27/9.33	
33.27/9.33	Q is empty.
33.27/9.33	We have to consider all minimal (P,Q,R)-chains.
33.27/9.33	----------------------------------------
33.27/9.33	
33.27/9.33	(5) QDPOrderProof (EQUIVALENT)
33.27/9.33	We use the reduction pair processor [LPAR04,JAR06].
33.27/9.33	
33.27/9.33	
33.27/9.33	The following pairs can be oriented strictly and are deleted.
33.27/9.33	
33.27/9.33	   A(c(x1)) -> A(x1)
33.27/9.33	The remaining pairs can at least be oriented weakly.
33.27/9.33	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
33.27/9.33	
33.27/9.33	POL( A_1(x_1) ) = x_1
33.27/9.33	POL( b_1(x_1) ) = max{0, x_1 - 1}
33.27/9.33	POL( c_1(x_1) ) = x_1 + 1
33.27/9.33	POL( a_1(x_1) ) = x_1 + 1
33.27/9.33	
33.27/9.33	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
33.27/9.33	
33.27/9.33	   a(b(x1)) -> x1
33.27/9.33	   a(c(x1)) -> a(b(b(c(c(a(x1))))))
33.27/9.33	   b(c(x1)) -> x1
33.27/9.33	
33.27/9.33	
33.27/9.33	----------------------------------------
33.27/9.33	
33.27/9.33	(6)
33.27/9.33	Obligation:
33.27/9.33	Q DP problem:
33.27/9.33	The TRS P consists of the following rules:
33.27/9.33	
33.27/9.33	   A(c(x1)) -> A(b(b(c(c(a(x1))))))
33.27/9.33	
33.27/9.33	The TRS R consists of the following rules:
33.27/9.33	
33.27/9.33	   a(b(x1)) -> x1
33.27/9.33	   a(c(x1)) -> a(b(b(c(c(a(x1))))))
33.27/9.33	   b(c(x1)) -> x1
33.27/9.33	
33.27/9.33	Q is empty.
33.27/9.33	We have to consider all minimal (P,Q,R)-chains.
33.27/9.33	----------------------------------------
33.27/9.33	
33.27/9.33	(7) QDPOrderProof (EQUIVALENT)
33.27/9.33	We use the reduction pair processor [LPAR04,JAR06].
33.27/9.33	
33.27/9.33	
33.27/9.33	The following pairs can be oriented strictly and are deleted.
33.27/9.33	
33.27/9.33	   A(c(x1)) -> A(b(b(c(c(a(x1))))))
33.27/9.33	The remaining pairs can at least be oriented weakly.
33.27/9.33	Used ordering:  Polynomial interpretation [POLO,RATPOLO]:
33.27/9.33	
33.27/9.33	   POL(A(x_1)) = [1/4]x_1
33.27/9.33	   POL(a(x_1)) = [4]x_1
33.27/9.33	   POL(b(x_1)) = [1/4]x_1
33.27/9.33	   POL(c(x_1)) = [1/2] + [4]x_1
33.27/9.33	The value of delta used in the strict ordering is 11/128.
33.27/9.33	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
33.27/9.33	
33.27/9.33	   a(b(x1)) -> x1
33.27/9.33	   a(c(x1)) -> a(b(b(c(c(a(x1))))))
33.27/9.33	   b(c(x1)) -> x1
33.27/9.33	
33.27/9.33	
33.27/9.33	----------------------------------------
33.27/9.33	
33.27/9.33	(8)
33.27/9.33	Obligation:
33.27/9.33	Q DP problem:
33.27/9.33	P is empty.
33.27/9.33	The TRS R consists of the following rules:
33.27/9.33	
33.27/9.33	   a(b(x1)) -> x1
33.27/9.33	   a(c(x1)) -> a(b(b(c(c(a(x1))))))
33.27/9.33	   b(c(x1)) -> x1
33.27/9.33	
33.27/9.33	Q is empty.
33.27/9.33	We have to consider all minimal (P,Q,R)-chains.
33.27/9.33	----------------------------------------
33.27/9.33	
33.27/9.33	(9) PisEmptyProof (EQUIVALENT)
33.27/9.33	The TRS P is empty. Hence, there is no (P,Q,R) chain.
33.27/9.33	----------------------------------------
33.27/9.33	
33.27/9.33	(10)
33.27/9.33	YES
33.27/9.40	EOF