25.20/7.44	YES
25.20/7.47	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
25.20/7.47	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
25.20/7.47	
25.20/7.47	
25.20/7.47	Termination w.r.t. Q of the given QTRS could be proven:
25.20/7.47	
25.20/7.47	(0) QTRS
25.20/7.47	(1) DependencyPairsProof [EQUIVALENT, 12 ms]
25.20/7.47	(2) QDP
25.20/7.47	(3) QDPOrderProof [EQUIVALENT, 21 ms]
25.20/7.47	(4) QDP
25.20/7.47	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
25.20/7.47	(6) QDP
25.20/7.47	(7) QDPOrderProof [EQUIVALENT, 15 ms]
25.20/7.47	(8) QDP
25.20/7.47	(9) PisEmptyProof [EQUIVALENT, 0 ms]
25.20/7.47	(10) YES
25.20/7.47	
25.20/7.47	
25.20/7.47	----------------------------------------
25.20/7.47	
25.20/7.47	(0)
25.20/7.47	Obligation:
25.20/7.47	Q restricted rewrite system:
25.20/7.47	The TRS R consists of the following rules:
25.20/7.47	
25.20/7.47	   b(a(a(x1))) -> a(b(c(x1)))
25.20/7.47	   c(a(x1)) -> a(c(x1))
25.20/7.47	   c(b(x1)) -> b(a(x1))
25.20/7.47	   a(a(x1)) -> a(b(a(x1)))
25.20/7.47	
25.20/7.47	Q is empty.
25.20/7.47	
25.20/7.47	----------------------------------------
25.20/7.47	
25.20/7.47	(1) DependencyPairsProof (EQUIVALENT)
25.20/7.47	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
25.20/7.47	----------------------------------------
25.20/7.47	
25.20/7.47	(2)
25.20/7.47	Obligation:
25.20/7.47	Q DP problem:
25.20/7.47	The TRS P consists of the following rules:
25.20/7.47	
25.20/7.47	   B(a(a(x1))) -> A(b(c(x1)))
25.20/7.47	   B(a(a(x1))) -> B(c(x1))
25.20/7.47	   B(a(a(x1))) -> C(x1)
25.20/7.47	   C(a(x1)) -> A(c(x1))
25.20/7.47	   C(a(x1)) -> C(x1)
25.20/7.47	   C(b(x1)) -> B(a(x1))
25.20/7.47	   C(b(x1)) -> A(x1)
25.20/7.47	   A(a(x1)) -> A(b(a(x1)))
25.20/7.47	   A(a(x1)) -> B(a(x1))
25.20/7.47	
25.20/7.47	The TRS R consists of the following rules:
25.20/7.47	
25.20/7.47	   b(a(a(x1))) -> a(b(c(x1)))
25.20/7.47	   c(a(x1)) -> a(c(x1))
25.20/7.47	   c(b(x1)) -> b(a(x1))
25.20/7.47	   a(a(x1)) -> a(b(a(x1)))
25.20/7.47	
25.20/7.47	Q is empty.
25.20/7.47	We have to consider all minimal (P,Q,R)-chains.
25.20/7.47	----------------------------------------
25.20/7.47	
25.20/7.47	(3) QDPOrderProof (EQUIVALENT)
25.20/7.47	We use the reduction pair processor [LPAR04,JAR06].
25.20/7.47	
25.20/7.47	
25.20/7.47	The following pairs can be oriented strictly and are deleted.
25.20/7.47	
25.20/7.47	   B(a(a(x1))) -> A(b(c(x1)))
25.20/7.47	   B(a(a(x1))) -> B(c(x1))
25.20/7.47	   B(a(a(x1))) -> C(x1)
25.20/7.47	   C(a(x1)) -> A(c(x1))
25.20/7.47	   C(a(x1)) -> C(x1)
25.20/7.47	   C(b(x1)) -> A(x1)
25.20/7.47	The remaining pairs can at least be oriented weakly.
25.20/7.47	Used ordering:  Polynomial interpretation [POLO]:
25.20/7.47	
25.20/7.47	   POL(A(x_1)) = x_1
25.20/7.47	   POL(B(x_1)) = x_1
25.20/7.47	   POL(C(x_1)) = 1 + x_1
25.20/7.47	   POL(a(x_1)) = 1 + x_1
25.20/7.47	   POL(b(x_1)) = x_1
25.20/7.47	   POL(c(x_1)) = 1 + x_1
25.20/7.47	
25.20/7.47	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
25.20/7.47	
25.20/7.47	   c(a(x1)) -> a(c(x1))
25.20/7.47	   c(b(x1)) -> b(a(x1))
25.20/7.47	   b(a(a(x1))) -> a(b(c(x1)))
25.20/7.47	   a(a(x1)) -> a(b(a(x1)))
25.20/7.47	
25.20/7.47	
25.20/7.47	----------------------------------------
25.20/7.47	
25.20/7.47	(4)
25.20/7.47	Obligation:
25.20/7.47	Q DP problem:
25.20/7.47	The TRS P consists of the following rules:
25.20/7.47	
25.20/7.47	   C(b(x1)) -> B(a(x1))
25.20/7.47	   A(a(x1)) -> A(b(a(x1)))
25.20/7.47	   A(a(x1)) -> B(a(x1))
25.20/7.47	
25.20/7.47	The TRS R consists of the following rules:
25.20/7.47	
25.20/7.47	   b(a(a(x1))) -> a(b(c(x1)))
25.20/7.47	   c(a(x1)) -> a(c(x1))
25.20/7.47	   c(b(x1)) -> b(a(x1))
25.20/7.47	   a(a(x1)) -> a(b(a(x1)))
25.20/7.47	
25.20/7.47	Q is empty.
25.20/7.47	We have to consider all minimal (P,Q,R)-chains.
25.20/7.47	----------------------------------------
25.20/7.47	
25.20/7.47	(5) DependencyGraphProof (EQUIVALENT)
25.20/7.47	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
25.20/7.47	----------------------------------------
25.20/7.47	
25.20/7.47	(6)
25.20/7.47	Obligation:
25.20/7.47	Q DP problem:
25.20/7.47	The TRS P consists of the following rules:
25.20/7.47	
25.20/7.47	   A(a(x1)) -> A(b(a(x1)))
25.20/7.47	
25.20/7.47	The TRS R consists of the following rules:
25.20/7.47	
25.20/7.47	   b(a(a(x1))) -> a(b(c(x1)))
25.20/7.47	   c(a(x1)) -> a(c(x1))
25.20/7.47	   c(b(x1)) -> b(a(x1))
25.20/7.47	   a(a(x1)) -> a(b(a(x1)))
25.20/7.47	
25.20/7.47	Q is empty.
25.20/7.47	We have to consider all minimal (P,Q,R)-chains.
25.20/7.47	----------------------------------------
25.20/7.47	
25.20/7.47	(7) QDPOrderProof (EQUIVALENT)
25.20/7.47	We use the reduction pair processor [LPAR04,JAR06].
25.20/7.47	
25.20/7.47	
25.20/7.47	The following pairs can be oriented strictly and are deleted.
25.20/7.47	
25.20/7.47	   A(a(x1)) -> A(b(a(x1)))
25.20/7.47	The remaining pairs can at least be oriented weakly.
25.20/7.47	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
25.20/7.47	
25.20/7.47	POL( A_1(x_1) ) = max{0, x_1 - 1}
25.20/7.47	POL( a_1(x_1) ) = x_1 + 2
25.20/7.47	POL( b_1(x_1) ) = max{0, x_1 - 2}
25.20/7.47	POL( c_1(x_1) ) = x_1 + 2
25.20/7.47	
25.20/7.47	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
25.20/7.47	
25.20/7.47	   a(a(x1)) -> a(b(a(x1)))
25.20/7.47	   b(a(a(x1))) -> a(b(c(x1)))
25.20/7.47	   c(a(x1)) -> a(c(x1))
25.20/7.47	   c(b(x1)) -> b(a(x1))
25.20/7.47	
25.20/7.47	
25.20/7.47	----------------------------------------
25.20/7.47	
25.20/7.47	(8)
25.20/7.47	Obligation:
25.20/7.47	Q DP problem:
25.20/7.47	P is empty.
25.20/7.47	The TRS R consists of the following rules:
25.20/7.47	
25.20/7.47	   b(a(a(x1))) -> a(b(c(x1)))
25.20/7.47	   c(a(x1)) -> a(c(x1))
25.20/7.47	   c(b(x1)) -> b(a(x1))
25.20/7.47	   a(a(x1)) -> a(b(a(x1)))
25.20/7.47	
25.20/7.47	Q is empty.
25.20/7.47	We have to consider all minimal (P,Q,R)-chains.
25.20/7.47	----------------------------------------
25.20/7.47	
25.20/7.47	(9) PisEmptyProof (EQUIVALENT)
25.20/7.47	The TRS P is empty. Hence, there is no (P,Q,R) chain.
25.20/7.47	----------------------------------------
25.20/7.47	
25.20/7.47	(10)
25.20/7.47	YES
25.41/7.53	EOF