40.20/11.23	YES
40.61/11.26	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
40.61/11.26	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
40.61/11.26	
40.61/11.26	
40.61/11.26	Termination w.r.t. Q of the given QTRS could be proven:
40.61/11.26	
40.61/11.26	(0) QTRS
40.61/11.26	(1) DependencyPairsProof [EQUIVALENT, 6 ms]
40.61/11.26	(2) QDP
40.61/11.26	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
40.61/11.26	(4) QDP
40.61/11.26	(5) QDPOrderProof [EQUIVALENT, 47 ms]
40.61/11.26	(6) QDP
40.61/11.26	(7) QDPOrderProof [EQUIVALENT, 129 ms]
40.61/11.26	(8) QDP
40.61/11.26	(9) PisEmptyProof [EQUIVALENT, 0 ms]
40.61/11.26	(10) YES
40.61/11.26	
40.61/11.26	
40.61/11.26	----------------------------------------
40.61/11.26	
40.61/11.26	(0)
40.61/11.26	Obligation:
40.61/11.26	Q restricted rewrite system:
40.61/11.26	The TRS R consists of the following rules:
40.61/11.26	
40.61/11.26	   a(b(x1)) -> x1
40.61/11.26	   a(c(x1)) -> c(b(b(c(a(a(x1))))))
40.61/11.26	   b(c(x1)) -> x1
40.61/11.26	
40.61/11.26	Q is empty.
40.61/11.26	
40.61/11.26	----------------------------------------
40.61/11.26	
40.61/11.26	(1) DependencyPairsProof (EQUIVALENT)
40.61/11.26	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
40.61/11.26	----------------------------------------
40.61/11.26	
40.61/11.26	(2)
40.61/11.26	Obligation:
40.61/11.26	Q DP problem:
40.61/11.26	The TRS P consists of the following rules:
40.61/11.26	
40.61/11.26	   A(c(x1)) -> B(b(c(a(a(x1)))))
40.61/11.26	   A(c(x1)) -> B(c(a(a(x1))))
40.61/11.26	   A(c(x1)) -> A(a(x1))
40.61/11.26	   A(c(x1)) -> A(x1)
40.61/11.26	
40.61/11.26	The TRS R consists of the following rules:
40.61/11.26	
40.61/11.26	   a(b(x1)) -> x1
40.61/11.26	   a(c(x1)) -> c(b(b(c(a(a(x1))))))
40.61/11.26	   b(c(x1)) -> x1
40.61/11.26	
40.61/11.26	Q is empty.
40.61/11.26	We have to consider all minimal (P,Q,R)-chains.
40.61/11.26	----------------------------------------
40.61/11.26	
40.61/11.26	(3) DependencyGraphProof (EQUIVALENT)
40.61/11.26	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
40.61/11.26	----------------------------------------
40.61/11.26	
40.61/11.26	(4)
40.61/11.26	Obligation:
40.61/11.26	Q DP problem:
40.61/11.26	The TRS P consists of the following rules:
40.61/11.26	
40.61/11.26	   A(c(x1)) -> A(x1)
40.61/11.26	   A(c(x1)) -> A(a(x1))
40.61/11.26	
40.61/11.26	The TRS R consists of the following rules:
40.61/11.26	
40.61/11.26	   a(b(x1)) -> x1
40.61/11.26	   a(c(x1)) -> c(b(b(c(a(a(x1))))))
40.61/11.26	   b(c(x1)) -> x1
40.61/11.26	
40.61/11.26	Q is empty.
40.61/11.26	We have to consider all minimal (P,Q,R)-chains.
40.61/11.26	----------------------------------------
40.61/11.26	
40.61/11.26	(5) QDPOrderProof (EQUIVALENT)
40.61/11.26	We use the reduction pair processor [LPAR04,JAR06].
40.61/11.26	
40.61/11.26	
40.61/11.26	The following pairs can be oriented strictly and are deleted.
40.61/11.26	
40.61/11.26	   A(c(x1)) -> A(x1)
40.61/11.26	The remaining pairs can at least be oriented weakly.
40.61/11.26	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
40.61/11.26	
40.61/11.26	POL( A_1(x_1) ) = 2x_1
40.61/11.26	POL( a_1(x_1) ) = x_1 + 2
40.61/11.26	POL( b_1(x_1) ) = max{0, x_1 - 2}
40.61/11.26	POL( c_1(x_1) ) = x_1 + 2
40.61/11.26	
40.61/11.26	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
40.61/11.26	
40.61/11.26	   a(b(x1)) -> x1
40.61/11.26	   a(c(x1)) -> c(b(b(c(a(a(x1))))))
40.61/11.26	   b(c(x1)) -> x1
40.61/11.26	
40.61/11.26	
40.61/11.26	----------------------------------------
40.61/11.26	
40.61/11.26	(6)
40.61/11.26	Obligation:
40.61/11.26	Q DP problem:
40.61/11.26	The TRS P consists of the following rules:
40.61/11.26	
40.61/11.26	   A(c(x1)) -> A(a(x1))
40.61/11.26	
40.61/11.26	The TRS R consists of the following rules:
40.61/11.26	
40.61/11.26	   a(b(x1)) -> x1
40.61/11.26	   a(c(x1)) -> c(b(b(c(a(a(x1))))))
40.61/11.26	   b(c(x1)) -> x1
40.61/11.26	
40.61/11.26	Q is empty.
40.61/11.26	We have to consider all minimal (P,Q,R)-chains.
40.61/11.26	----------------------------------------
40.61/11.26	
40.61/11.26	(7) QDPOrderProof (EQUIVALENT)
40.61/11.26	We use the reduction pair processor [LPAR04,JAR06].
40.61/11.26	
40.61/11.26	
40.61/11.26	The following pairs can be oriented strictly and are deleted.
40.61/11.26	
40.61/11.26	   A(c(x1)) -> A(a(x1))
40.61/11.26	The remaining pairs can at least be oriented weakly.
40.61/11.26	Used ordering:  Polynomial interpretation [POLO,RATPOLO]:
40.61/11.26	
40.61/11.26	   POL(A(x_1)) = [4]x_1
40.61/11.26	   POL(a(x_1)) = [4]x_1
40.61/11.26	   POL(b(x_1)) = [1/4]x_1
40.61/11.26	   POL(c(x_1)) = [1/4] + [4]x_1
40.61/11.26	The value of delta used in the strict ordering is 1.
40.61/11.26	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
40.61/11.26	
40.61/11.26	   a(b(x1)) -> x1
40.61/11.26	   a(c(x1)) -> c(b(b(c(a(a(x1))))))
40.61/11.26	   b(c(x1)) -> x1
40.61/11.26	
40.61/11.26	
40.61/11.26	----------------------------------------
40.61/11.26	
40.61/11.26	(8)
40.61/11.26	Obligation:
40.61/11.26	Q DP problem:
40.61/11.26	P is empty.
40.61/11.26	The TRS R consists of the following rules:
40.61/11.26	
40.61/11.26	   a(b(x1)) -> x1
40.61/11.26	   a(c(x1)) -> c(b(b(c(a(a(x1))))))
40.61/11.26	   b(c(x1)) -> x1
40.61/11.26	
40.61/11.26	Q is empty.
40.61/11.26	We have to consider all minimal (P,Q,R)-chains.
40.61/11.26	----------------------------------------
40.61/11.26	
40.61/11.26	(9) PisEmptyProof (EQUIVALENT)
40.61/11.26	The TRS P is empty. Hence, there is no (P,Q,R) chain.
40.61/11.26	----------------------------------------
40.61/11.26	
40.61/11.26	(10)
40.61/11.26	YES
40.77/11.34	EOF