32.57/9.29	YES
32.57/9.30	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
32.57/9.30	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
32.57/9.30	
32.57/9.30	
32.57/9.30	Termination w.r.t. Q of the given QTRS could be proven:
32.57/9.30	
32.57/9.30	(0) QTRS
32.57/9.30	(1) DependencyPairsProof [EQUIVALENT, 17 ms]
32.57/9.30	(2) QDP
32.57/9.30	(3) QDPOrderProof [EQUIVALENT, 34 ms]
32.57/9.30	(4) QDP
32.57/9.30	(5) QDPOrderProof [EQUIVALENT, 13 ms]
32.57/9.30	(6) QDP
32.57/9.30	(7) QDPOrderProof [EQUIVALENT, 29 ms]
32.57/9.30	(8) QDP
32.57/9.30	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
32.57/9.30	(10) TRUE
32.57/9.30	
32.57/9.30	
32.57/9.30	----------------------------------------
32.57/9.30	
32.57/9.30	(0)
32.57/9.30	Obligation:
32.57/9.30	Q restricted rewrite system:
32.57/9.30	The TRS R consists of the following rules:
32.57/9.30	
32.57/9.30	   a(a(b(x1))) -> b(a(x1))
32.57/9.30	   b(a(a(x1))) -> a(a(a(b(x1))))
32.57/9.30	   a(c(x1)) -> c(b(x1))
32.57/9.30	
32.57/9.30	Q is empty.
32.57/9.30	
32.57/9.30	----------------------------------------
32.57/9.30	
32.57/9.30	(1) DependencyPairsProof (EQUIVALENT)
32.57/9.30	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
32.57/9.30	----------------------------------------
32.57/9.30	
32.57/9.30	(2)
32.57/9.30	Obligation:
32.57/9.30	Q DP problem:
32.57/9.30	The TRS P consists of the following rules:
32.57/9.30	
32.57/9.30	   A(a(b(x1))) -> B(a(x1))
32.57/9.30	   A(a(b(x1))) -> A(x1)
32.57/9.30	   B(a(a(x1))) -> A(a(a(b(x1))))
32.57/9.30	   B(a(a(x1))) -> A(a(b(x1)))
32.57/9.30	   B(a(a(x1))) -> A(b(x1))
32.57/9.30	   B(a(a(x1))) -> B(x1)
32.57/9.30	   A(c(x1)) -> B(x1)
32.57/9.30	
32.57/9.30	The TRS R consists of the following rules:
32.57/9.30	
32.57/9.30	   a(a(b(x1))) -> b(a(x1))
32.57/9.30	   b(a(a(x1))) -> a(a(a(b(x1))))
32.57/9.30	   a(c(x1)) -> c(b(x1))
32.57/9.30	
32.57/9.30	Q is empty.
32.57/9.30	We have to consider all minimal (P,Q,R)-chains.
32.57/9.30	----------------------------------------
32.57/9.30	
32.57/9.30	(3) QDPOrderProof (EQUIVALENT)
32.57/9.30	We use the reduction pair processor [LPAR04,JAR06].
32.57/9.30	
32.57/9.30	
32.57/9.30	The following pairs can be oriented strictly and are deleted.
32.57/9.30	
32.57/9.30	   A(c(x1)) -> B(x1)
32.57/9.30	The remaining pairs can at least be oriented weakly.
32.57/9.30	Used ordering:  Polynomial interpretation [POLO]:
32.57/9.30	
32.57/9.30	   POL(A(x_1)) = x_1
32.57/9.30	   POL(B(x_1)) = x_1
32.57/9.30	   POL(a(x_1)) = x_1
32.57/9.30	   POL(b(x_1)) = x_1
32.57/9.30	   POL(c(x_1)) = 1 + x_1
32.57/9.30	
32.57/9.30	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
32.57/9.30	
32.57/9.30	   b(a(a(x1))) -> a(a(a(b(x1))))
32.57/9.30	   a(a(b(x1))) -> b(a(x1))
32.57/9.30	   a(c(x1)) -> c(b(x1))
32.57/9.30	
32.57/9.30	
32.57/9.30	----------------------------------------
32.57/9.30	
32.57/9.30	(4)
32.57/9.30	Obligation:
32.57/9.30	Q DP problem:
32.57/9.30	The TRS P consists of the following rules:
32.57/9.30	
32.57/9.30	   A(a(b(x1))) -> B(a(x1))
32.57/9.30	   A(a(b(x1))) -> A(x1)
32.57/9.30	   B(a(a(x1))) -> A(a(a(b(x1))))
32.57/9.30	   B(a(a(x1))) -> A(a(b(x1)))
32.57/9.30	   B(a(a(x1))) -> A(b(x1))
32.57/9.30	   B(a(a(x1))) -> B(x1)
32.57/9.30	
32.57/9.30	The TRS R consists of the following rules:
32.57/9.30	
32.57/9.30	   a(a(b(x1))) -> b(a(x1))
32.57/9.30	   b(a(a(x1))) -> a(a(a(b(x1))))
32.57/9.30	   a(c(x1)) -> c(b(x1))
32.57/9.30	
32.57/9.30	Q is empty.
32.57/9.30	We have to consider all minimal (P,Q,R)-chains.
32.57/9.30	----------------------------------------
32.57/9.30	
32.57/9.30	(5) QDPOrderProof (EQUIVALENT)
32.57/9.30	We use the reduction pair processor [LPAR04,JAR06].
32.57/9.30	
32.57/9.30	
32.57/9.30	The following pairs can be oriented strictly and are deleted.
32.57/9.30	
32.57/9.30	   A(a(b(x1))) -> A(x1)
32.57/9.30	The remaining pairs can at least be oriented weakly.
32.57/9.30	Used ordering:  Polynomial interpretation [POLO]:
32.57/9.30	
32.57/9.30	   POL(A(x_1)) = x_1
32.57/9.30	   POL(B(x_1)) = 1 + x_1
32.57/9.30	   POL(a(x_1)) = x_1
32.57/9.30	   POL(b(x_1)) = 1 + x_1
32.57/9.30	   POL(c(x_1)) = 1
32.57/9.30	
32.57/9.30	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
32.57/9.30	
32.57/9.30	   b(a(a(x1))) -> a(a(a(b(x1))))
32.57/9.30	   a(a(b(x1))) -> b(a(x1))
32.57/9.30	   a(c(x1)) -> c(b(x1))
32.57/9.30	
32.57/9.30	
32.57/9.30	----------------------------------------
32.57/9.30	
32.57/9.30	(6)
32.57/9.30	Obligation:
32.57/9.30	Q DP problem:
32.57/9.30	The TRS P consists of the following rules:
32.57/9.30	
32.57/9.30	   A(a(b(x1))) -> B(a(x1))
32.57/9.30	   B(a(a(x1))) -> A(a(a(b(x1))))
32.57/9.30	   B(a(a(x1))) -> A(a(b(x1)))
32.57/9.30	   B(a(a(x1))) -> A(b(x1))
32.57/9.30	   B(a(a(x1))) -> B(x1)
32.57/9.30	
32.57/9.30	The TRS R consists of the following rules:
32.57/9.30	
32.57/9.30	   a(a(b(x1))) -> b(a(x1))
32.57/9.30	   b(a(a(x1))) -> a(a(a(b(x1))))
32.57/9.30	   a(c(x1)) -> c(b(x1))
32.57/9.30	
32.57/9.30	Q is empty.
32.57/9.30	We have to consider all minimal (P,Q,R)-chains.
32.57/9.30	----------------------------------------
32.57/9.30	
32.57/9.30	(7) QDPOrderProof (EQUIVALENT)
32.57/9.30	We use the reduction pair processor [LPAR04,JAR06].
32.57/9.30	
32.57/9.30	
32.57/9.30	The following pairs can be oriented strictly and are deleted.
32.57/9.30	
32.57/9.30	   B(a(a(x1))) -> A(a(a(b(x1))))
32.57/9.30	   B(a(a(x1))) -> A(a(b(x1)))
32.57/9.30	   B(a(a(x1))) -> A(b(x1))
32.57/9.30	   B(a(a(x1))) -> B(x1)
32.57/9.30	The remaining pairs can at least be oriented weakly.
32.57/9.30	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
32.57/9.30	
32.57/9.30	POL( A_1(x_1) ) = max{0, x_1 - 1}
32.57/9.30	POL( B_1(x_1) ) = max{0, 2x_1 - 2}
32.57/9.30	POL( b_1(x_1) ) = 2x_1
32.57/9.30	POL( a_1(x_1) ) = x_1 + 1
32.57/9.30	POL( c_1(x_1) ) = max{0, -2}
32.57/9.30	
32.57/9.30	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
32.57/9.30	
32.57/9.30	   b(a(a(x1))) -> a(a(a(b(x1))))
32.57/9.30	   a(a(b(x1))) -> b(a(x1))
32.57/9.30	   a(c(x1)) -> c(b(x1))
32.57/9.30	
32.57/9.30	
32.57/9.30	----------------------------------------
32.57/9.30	
32.57/9.30	(8)
32.57/9.30	Obligation:
32.57/9.30	Q DP problem:
32.57/9.30	The TRS P consists of the following rules:
32.57/9.30	
32.57/9.30	   A(a(b(x1))) -> B(a(x1))
32.57/9.30	
32.57/9.30	The TRS R consists of the following rules:
32.57/9.30	
32.57/9.30	   a(a(b(x1))) -> b(a(x1))
32.57/9.30	   b(a(a(x1))) -> a(a(a(b(x1))))
32.57/9.30	   a(c(x1)) -> c(b(x1))
32.57/9.30	
32.57/9.30	Q is empty.
32.57/9.30	We have to consider all minimal (P,Q,R)-chains.
32.57/9.30	----------------------------------------
32.57/9.30	
32.57/9.30	(9) DependencyGraphProof (EQUIVALENT)
32.57/9.30	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
32.57/9.30	----------------------------------------
32.57/9.30	
32.57/9.30	(10)
32.57/9.30	TRUE
33.07/9.44	EOF