31.45/8.85	YES
31.45/8.87	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
31.45/8.87	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
31.45/8.87	
31.45/8.87	
31.45/8.87	Termination w.r.t. Q of the given QTRS could be proven:
31.45/8.87	
31.45/8.87	(0) QTRS
31.45/8.87	(1) QTRS Reverse [EQUIVALENT, 0 ms]
31.45/8.87	(2) QTRS
31.45/8.87	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
31.45/8.87	(4) QDP
31.45/8.87	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
31.45/8.87	(6) QDP
31.45/8.87	(7) QDPOrderProof [EQUIVALENT, 37 ms]
31.45/8.87	(8) QDP
31.45/8.87	(9) QDPOrderProof [EQUIVALENT, 92 ms]
31.45/8.87	(10) QDP
31.45/8.87	(11) PisEmptyProof [EQUIVALENT, 0 ms]
31.45/8.87	(12) YES
31.45/8.87	
31.45/8.87	
31.45/8.87	----------------------------------------
31.45/8.87	
31.45/8.87	(0)
31.45/8.87	Obligation:
31.45/8.87	Q restricted rewrite system:
31.45/8.87	The TRS R consists of the following rules:
31.45/8.87	
31.45/8.87	   a(b(x1)) -> x1
31.45/8.87	   a(c(x1)) -> c(c(b(c(x1))))
31.45/8.87	   b(c(x1)) -> a(b(x1))
31.45/8.87	
31.45/8.87	Q is empty.
31.45/8.87	
31.45/8.87	----------------------------------------
31.45/8.87	
31.45/8.87	(1) QTRS Reverse (EQUIVALENT)
31.45/8.87	We applied the QTRS Reverse Processor [REVERSE].
31.45/8.87	----------------------------------------
31.45/8.87	
31.45/8.87	(2)
31.45/8.87	Obligation:
31.45/8.87	Q restricted rewrite system:
31.45/8.87	The TRS R consists of the following rules:
31.45/8.87	
31.45/8.87	   b(a(x1)) -> x1
31.45/8.87	   c(a(x1)) -> c(b(c(c(x1))))
31.45/8.87	   c(b(x1)) -> b(a(x1))
31.45/8.87	
31.45/8.87	Q is empty.
31.45/8.87	
31.45/8.87	----------------------------------------
31.45/8.87	
31.45/8.87	(3) DependencyPairsProof (EQUIVALENT)
31.45/8.87	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
31.45/8.87	----------------------------------------
31.45/8.87	
31.45/8.87	(4)
31.45/8.87	Obligation:
31.45/8.87	Q DP problem:
31.45/8.87	The TRS P consists of the following rules:
31.45/8.87	
31.45/8.87	   C(a(x1)) -> C(b(c(c(x1))))
31.45/8.87	   C(a(x1)) -> B(c(c(x1)))
31.45/8.87	   C(a(x1)) -> C(c(x1))
31.45/8.87	   C(a(x1)) -> C(x1)
31.45/8.87	   C(b(x1)) -> B(a(x1))
31.45/8.87	
31.45/8.87	The TRS R consists of the following rules:
31.45/8.87	
31.45/8.87	   b(a(x1)) -> x1
31.45/8.87	   c(a(x1)) -> c(b(c(c(x1))))
31.45/8.87	   c(b(x1)) -> b(a(x1))
31.45/8.87	
31.45/8.87	Q is empty.
31.45/8.87	We have to consider all minimal (P,Q,R)-chains.
31.45/8.87	----------------------------------------
31.45/8.87	
31.45/8.87	(5) DependencyGraphProof (EQUIVALENT)
31.45/8.87	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
31.45/8.87	----------------------------------------
31.45/8.87	
31.45/8.87	(6)
31.45/8.87	Obligation:
31.45/8.87	Q DP problem:
31.45/8.87	The TRS P consists of the following rules:
31.45/8.87	
31.45/8.87	   C(a(x1)) -> C(c(x1))
31.45/8.87	   C(a(x1)) -> C(b(c(c(x1))))
31.45/8.87	   C(a(x1)) -> C(x1)
31.45/8.87	
31.45/8.87	The TRS R consists of the following rules:
31.45/8.87	
31.45/8.87	   b(a(x1)) -> x1
31.45/8.87	   c(a(x1)) -> c(b(c(c(x1))))
31.45/8.87	   c(b(x1)) -> b(a(x1))
31.45/8.87	
31.45/8.87	Q is empty.
31.45/8.87	We have to consider all minimal (P,Q,R)-chains.
31.45/8.87	----------------------------------------
31.45/8.87	
31.45/8.87	(7) QDPOrderProof (EQUIVALENT)
31.45/8.87	We use the reduction pair processor [LPAR04,JAR06].
31.45/8.87	
31.45/8.87	
31.45/8.87	The following pairs can be oriented strictly and are deleted.
31.45/8.87	
31.45/8.87	   C(a(x1)) -> C(x1)
31.45/8.87	The remaining pairs can at least be oriented weakly.
31.45/8.87	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
31.45/8.87	
31.45/8.87	POL( C_1(x_1) ) = 2x_1 + 2
31.45/8.87	POL( c_1(x_1) ) = x_1 + 2
31.45/8.87	POL( a_1(x_1) ) = x_1 + 2
31.45/8.87	POL( b_1(x_1) ) = max{0, x_1 - 2}
31.45/8.87	
31.45/8.87	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
31.45/8.87	
31.45/8.87	   c(a(x1)) -> c(b(c(c(x1))))
31.45/8.87	   c(b(x1)) -> b(a(x1))
31.45/8.87	   b(a(x1)) -> x1
31.45/8.87	
31.45/8.87	
31.45/8.87	----------------------------------------
31.45/8.87	
31.45/8.87	(8)
31.45/8.87	Obligation:
31.45/8.87	Q DP problem:
31.45/8.87	The TRS P consists of the following rules:
31.45/8.87	
31.45/8.87	   C(a(x1)) -> C(c(x1))
31.45/8.87	   C(a(x1)) -> C(b(c(c(x1))))
31.45/8.87	
31.45/8.87	The TRS R consists of the following rules:
31.45/8.87	
31.45/8.87	   b(a(x1)) -> x1
31.45/8.88	   c(a(x1)) -> c(b(c(c(x1))))
31.45/8.88	   c(b(x1)) -> b(a(x1))
31.45/8.88	
31.45/8.88	Q is empty.
31.45/8.88	We have to consider all minimal (P,Q,R)-chains.
31.45/8.88	----------------------------------------
31.45/8.88	
31.45/8.88	(9) QDPOrderProof (EQUIVALENT)
31.45/8.88	We use the reduction pair processor [LPAR04,JAR06].
31.45/8.88	
31.45/8.88	
31.45/8.88	The following pairs can be oriented strictly and are deleted.
31.45/8.88	
31.45/8.88	   C(a(x1)) -> C(c(x1))
31.45/8.88	   C(a(x1)) -> C(b(c(c(x1))))
31.45/8.88	The remaining pairs can at least be oriented weakly.
31.45/8.88	Used ordering:  Polynomial interpretation [POLO,RATPOLO]:
31.45/8.88	
31.45/8.88	   POL(C(x_1)) = [3]x_1
31.45/8.88	   POL(a(x_1)) = [2] + [2]x_1
31.45/8.88	   POL(b(x_1)) = [7/4] + [1/2]x_1
31.45/8.88	   POL(c(x_1)) = [2]x_1
31.45/8.88	The value of delta used in the strict ordering is 3/4.
31.45/8.88	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
31.45/8.88	
31.45/8.88	   c(a(x1)) -> c(b(c(c(x1))))
31.45/8.88	   c(b(x1)) -> b(a(x1))
31.45/8.88	   b(a(x1)) -> x1
31.45/8.88	
31.45/8.88	
31.45/8.88	----------------------------------------
31.45/8.88	
31.45/8.88	(10)
31.45/8.88	Obligation:
31.45/8.88	Q DP problem:
31.45/8.88	P is empty.
31.45/8.88	The TRS R consists of the following rules:
31.45/8.88	
31.45/8.88	   b(a(x1)) -> x1
31.45/8.88	   c(a(x1)) -> c(b(c(c(x1))))
31.45/8.88	   c(b(x1)) -> b(a(x1))
31.45/8.88	
31.45/8.88	Q is empty.
31.45/8.88	We have to consider all minimal (P,Q,R)-chains.
31.45/8.88	----------------------------------------
31.45/8.88	
31.45/8.88	(11) PisEmptyProof (EQUIVALENT)
31.45/8.88	The TRS P is empty. Hence, there is no (P,Q,R) chain.
31.45/8.88	----------------------------------------
31.45/8.88	
31.45/8.88	(12)
31.45/8.88	YES
31.83/8.97	EOF