42.53/11.82	YES
42.85/11.90	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
42.85/11.90	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
42.85/11.90	
42.85/11.90	
42.85/11.90	Termination w.r.t. Q of the given QTRS could be proven:
42.85/11.90	
42.85/11.90	(0) QTRS
42.85/11.90	(1) DependencyPairsProof [EQUIVALENT, 9 ms]
42.85/11.90	(2) QDP
42.85/11.90	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
42.85/11.90	(4) QDP
42.85/11.90	(5) QDPOrderProof [EQUIVALENT, 40 ms]
42.85/11.90	(6) QDP
42.85/11.90	(7) QDPOrderProof [EQUIVALENT, 41 ms]
42.85/11.90	(8) QDP
42.85/11.90	(9) PisEmptyProof [EQUIVALENT, 0 ms]
42.85/11.90	(10) YES
42.85/11.90	
42.85/11.90	
42.85/11.90	----------------------------------------
42.85/11.90	
42.85/11.90	(0)
42.85/11.90	Obligation:
42.85/11.90	Q restricted rewrite system:
42.85/11.90	The TRS R consists of the following rules:
42.85/11.90	
42.85/11.90	   a(b(x1)) -> x1
42.85/11.90	   a(c(x1)) -> b(c(c(b(a(a(x1))))))
42.85/11.90	   b(c(x1)) -> x1
42.85/11.90	
42.85/11.90	Q is empty.
42.85/11.90	
42.85/11.90	----------------------------------------
42.85/11.90	
42.85/11.90	(1) DependencyPairsProof (EQUIVALENT)
42.85/11.90	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
42.85/11.90	----------------------------------------
42.85/11.90	
42.85/11.90	(2)
42.85/11.90	Obligation:
42.85/11.90	Q DP problem:
42.85/11.90	The TRS P consists of the following rules:
42.85/11.90	
42.85/11.90	   A(c(x1)) -> B(c(c(b(a(a(x1))))))
42.85/11.90	   A(c(x1)) -> B(a(a(x1)))
42.85/11.90	   A(c(x1)) -> A(a(x1))
42.85/11.90	   A(c(x1)) -> A(x1)
42.85/11.90	
42.85/11.90	The TRS R consists of the following rules:
42.85/11.90	
42.85/11.90	   a(b(x1)) -> x1
42.85/11.90	   a(c(x1)) -> b(c(c(b(a(a(x1))))))
42.85/11.90	   b(c(x1)) -> x1
42.85/11.90	
42.85/11.90	Q is empty.
42.85/11.90	We have to consider all minimal (P,Q,R)-chains.
42.85/11.90	----------------------------------------
42.85/11.90	
42.85/11.90	(3) DependencyGraphProof (EQUIVALENT)
42.85/11.90	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
42.85/11.90	----------------------------------------
42.85/11.90	
42.85/11.90	(4)
42.85/11.90	Obligation:
42.85/11.90	Q DP problem:
42.85/11.90	The TRS P consists of the following rules:
42.85/11.90	
42.85/11.90	   A(c(x1)) -> A(x1)
42.85/11.90	   A(c(x1)) -> A(a(x1))
42.85/11.90	
42.85/11.90	The TRS R consists of the following rules:
42.85/11.90	
42.85/11.90	   a(b(x1)) -> x1
42.85/11.90	   a(c(x1)) -> b(c(c(b(a(a(x1))))))
42.85/11.90	   b(c(x1)) -> x1
42.85/11.90	
42.85/11.90	Q is empty.
42.85/11.90	We have to consider all minimal (P,Q,R)-chains.
42.85/11.90	----------------------------------------
42.85/11.90	
42.85/11.90	(5) QDPOrderProof (EQUIVALENT)
42.85/11.90	We use the reduction pair processor [LPAR04,JAR06].
42.85/11.90	
42.85/11.90	
42.85/11.90	The following pairs can be oriented strictly and are deleted.
42.85/11.90	
42.85/11.90	   A(c(x1)) -> A(x1)
42.85/11.90	The remaining pairs can at least be oriented weakly.
42.85/11.90	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
42.85/11.90	
42.85/11.90	POL( A_1(x_1) ) = max{0, x_1 - 1}
42.85/11.90	POL( a_1(x_1) ) = x_1 + 2
42.85/11.90	POL( b_1(x_1) ) = max{0, x_1 - 2}
42.85/11.90	POL( c_1(x_1) ) = x_1 + 2
42.85/11.90	
42.85/11.90	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
42.85/11.90	
42.85/11.90	   a(b(x1)) -> x1
42.85/11.90	   a(c(x1)) -> b(c(c(b(a(a(x1))))))
42.85/11.90	   b(c(x1)) -> x1
42.85/11.90	
42.85/11.90	
42.85/11.90	----------------------------------------
42.85/11.90	
42.85/11.90	(6)
42.85/11.90	Obligation:
42.85/11.90	Q DP problem:
42.85/11.90	The TRS P consists of the following rules:
42.85/11.90	
42.85/11.90	   A(c(x1)) -> A(a(x1))
42.85/11.90	
42.85/11.90	The TRS R consists of the following rules:
42.85/11.90	
42.85/11.90	   a(b(x1)) -> x1
42.85/11.90	   a(c(x1)) -> b(c(c(b(a(a(x1))))))
42.85/11.90	   b(c(x1)) -> x1
42.85/11.90	
42.85/11.90	Q is empty.
42.85/11.90	We have to consider all minimal (P,Q,R)-chains.
42.85/11.90	----------------------------------------
42.85/11.90	
42.85/11.90	(7) QDPOrderProof (EQUIVALENT)
42.85/11.90	We use the reduction pair processor [LPAR04,JAR06].
42.85/11.90	
42.85/11.90	
42.85/11.90	The following pairs can be oriented strictly and are deleted.
42.85/11.90	
42.85/11.90	   A(c(x1)) -> A(a(x1))
42.85/11.90	The remaining pairs can at least be oriented weakly.
42.85/11.90	Used ordering:  Polynomial interpretation [POLO,RATPOLO]:
42.85/11.90	
42.85/11.90	   POL(A(x_1)) = [4]x_1
42.85/11.90	   POL(a(x_1)) = [4]x_1
42.85/11.90	   POL(b(x_1)) = [3/2] + [1/4]x_1
42.85/11.90	   POL(c(x_1)) = [4] + [4]x_1
42.85/11.90	The value of delta used in the strict ordering is 16.
42.85/11.90	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
42.85/11.90	
42.85/11.90	   a(b(x1)) -> x1
42.85/11.90	   a(c(x1)) -> b(c(c(b(a(a(x1))))))
42.85/11.90	   b(c(x1)) -> x1
42.85/11.90	
42.85/11.90	
42.85/11.90	----------------------------------------
42.85/11.90	
42.85/11.90	(8)
42.85/11.90	Obligation:
42.85/11.90	Q DP problem:
42.85/11.90	P is empty.
42.85/11.90	The TRS R consists of the following rules:
42.85/11.90	
42.85/11.90	   a(b(x1)) -> x1
42.85/11.90	   a(c(x1)) -> b(c(c(b(a(a(x1))))))
42.85/11.90	   b(c(x1)) -> x1
42.85/11.90	
42.85/11.90	Q is empty.
42.85/11.90	We have to consider all minimal (P,Q,R)-chains.
42.85/11.90	----------------------------------------
42.85/11.90	
42.85/11.90	(9) PisEmptyProof (EQUIVALENT)
42.85/11.90	The TRS P is empty. Hence, there is no (P,Q,R) chain.
42.85/11.90	----------------------------------------
42.85/11.90	
42.85/11.90	(10)
42.85/11.90	YES
43.11/11.99	EOF