40.87/11.56	YES
40.87/11.60	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
40.87/11.60	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
40.87/11.60	
40.87/11.60	
40.87/11.60	Termination w.r.t. Q of the given QTRS could be proven:
40.87/11.60	
40.87/11.60	(0) QTRS
40.87/11.60	(1) QTRS Reverse [EQUIVALENT, 0 ms]
40.87/11.60	(2) QTRS
40.87/11.60	(3) DependencyPairsProof [EQUIVALENT, 1 ms]
40.87/11.60	(4) QDP
40.87/11.60	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
40.87/11.60	(6) QDP
40.87/11.60	(7) QDPOrderProof [EQUIVALENT, 35 ms]
40.87/11.60	(8) QDP
40.87/11.60	(9) QDPOrderProof [EQUIVALENT, 43 ms]
40.87/11.60	(10) QDP
40.87/11.60	(11) PisEmptyProof [EQUIVALENT, 0 ms]
40.87/11.60	(12) YES
40.87/11.60	
40.87/11.60	
40.87/11.60	----------------------------------------
40.87/11.60	
40.87/11.60	(0)
40.87/11.60	Obligation:
40.87/11.60	Q restricted rewrite system:
40.87/11.60	The TRS R consists of the following rules:
40.87/11.60	
40.87/11.60	   a(b(x1)) -> x1
40.87/11.60	   a(c(x1)) -> c(c(b(a(x1))))
40.87/11.60	   b(c(x1)) -> a(b(x1))
40.87/11.60	
40.87/11.60	Q is empty.
40.87/11.60	
40.87/11.60	----------------------------------------
40.87/11.60	
40.87/11.60	(1) QTRS Reverse (EQUIVALENT)
40.87/11.60	We applied the QTRS Reverse Processor [REVERSE].
40.87/11.60	----------------------------------------
40.87/11.60	
40.87/11.60	(2)
40.87/11.60	Obligation:
40.87/11.60	Q restricted rewrite system:
40.87/11.60	The TRS R consists of the following rules:
40.87/11.60	
40.87/11.60	   b(a(x1)) -> x1
40.87/11.60	   c(a(x1)) -> a(b(c(c(x1))))
40.87/11.60	   c(b(x1)) -> b(a(x1))
40.87/11.60	
40.87/11.60	Q is empty.
40.87/11.60	
40.87/11.60	----------------------------------------
40.87/11.60	
40.87/11.60	(3) DependencyPairsProof (EQUIVALENT)
40.87/11.60	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
40.87/11.60	----------------------------------------
40.87/11.60	
40.87/11.60	(4)
40.87/11.60	Obligation:
40.87/11.60	Q DP problem:
40.87/11.60	The TRS P consists of the following rules:
40.87/11.60	
40.87/11.60	   C(a(x1)) -> B(c(c(x1)))
40.87/11.60	   C(a(x1)) -> C(c(x1))
40.87/11.60	   C(a(x1)) -> C(x1)
40.87/11.60	   C(b(x1)) -> B(a(x1))
40.87/11.60	
40.87/11.60	The TRS R consists of the following rules:
40.87/11.60	
40.87/11.60	   b(a(x1)) -> x1
40.87/11.60	   c(a(x1)) -> a(b(c(c(x1))))
40.87/11.60	   c(b(x1)) -> b(a(x1))
40.87/11.60	
40.87/11.60	Q is empty.
40.87/11.60	We have to consider all minimal (P,Q,R)-chains.
40.87/11.60	----------------------------------------
40.87/11.60	
40.87/11.60	(5) DependencyGraphProof (EQUIVALENT)
40.87/11.60	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
40.87/11.60	----------------------------------------
40.87/11.60	
40.87/11.60	(6)
40.87/11.60	Obligation:
40.87/11.60	Q DP problem:
40.87/11.60	The TRS P consists of the following rules:
40.87/11.60	
40.87/11.60	   C(a(x1)) -> C(x1)
40.87/11.60	   C(a(x1)) -> C(c(x1))
40.87/11.60	
40.87/11.60	The TRS R consists of the following rules:
40.87/11.60	
40.87/11.60	   b(a(x1)) -> x1
40.87/11.60	   c(a(x1)) -> a(b(c(c(x1))))
40.87/11.60	   c(b(x1)) -> b(a(x1))
40.87/11.60	
40.87/11.60	Q is empty.
40.87/11.60	We have to consider all minimal (P,Q,R)-chains.
40.87/11.60	----------------------------------------
40.87/11.60	
40.87/11.60	(7) QDPOrderProof (EQUIVALENT)
40.87/11.60	We use the reduction pair processor [LPAR04,JAR06].
40.87/11.60	
40.87/11.60	
40.87/11.60	The following pairs can be oriented strictly and are deleted.
40.87/11.60	
40.87/11.60	   C(a(x1)) -> C(x1)
40.87/11.60	The remaining pairs can at least be oriented weakly.
40.87/11.60	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
40.87/11.60	
40.87/11.60	POL( C_1(x_1) ) = 2x_1
40.87/11.60	POL( c_1(x_1) ) = x_1 + 2
40.87/11.60	POL( a_1(x_1) ) = x_1 + 2
40.87/11.60	POL( b_1(x_1) ) = max{0, x_1 - 2}
40.87/11.60	
40.87/11.60	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
40.87/11.60	
40.87/11.60	   c(a(x1)) -> a(b(c(c(x1))))
40.87/11.60	   c(b(x1)) -> b(a(x1))
40.87/11.60	   b(a(x1)) -> x1
40.87/11.60	
40.87/11.60	
40.87/11.60	----------------------------------------
40.87/11.60	
40.87/11.60	(8)
40.87/11.60	Obligation:
40.87/11.60	Q DP problem:
40.87/11.60	The TRS P consists of the following rules:
40.87/11.60	
40.87/11.60	   C(a(x1)) -> C(c(x1))
40.87/11.60	
40.87/11.60	The TRS R consists of the following rules:
40.87/11.60	
40.87/11.60	   b(a(x1)) -> x1
40.87/11.60	   c(a(x1)) -> a(b(c(c(x1))))
40.87/11.60	   c(b(x1)) -> b(a(x1))
40.87/11.60	
40.87/11.60	Q is empty.
40.87/11.60	We have to consider all minimal (P,Q,R)-chains.
40.87/11.60	----------------------------------------
40.87/11.60	
40.87/11.60	(9) QDPOrderProof (EQUIVALENT)
40.87/11.60	We use the reduction pair processor [LPAR04,JAR06].
40.87/11.60	
40.87/11.60	
40.87/11.60	The following pairs can be oriented strictly and are deleted.
40.87/11.60	
40.87/11.60	   C(a(x1)) -> C(c(x1))
40.87/11.60	The remaining pairs can at least be oriented weakly.
40.87/11.60	Used ordering:  Polynomial interpretation [POLO,RATPOLO]:
40.87/11.60	
40.87/11.60	   POL(C(x_1)) = x_1
40.87/11.60	   POL(a(x_1)) = [1] + [4]x_1
40.87/11.60	   POL(b(x_1)) = [3/4] + [1/4]x_1
40.87/11.60	   POL(c(x_1)) = [4]x_1
40.87/11.60	The value of delta used in the strict ordering is 1.
40.87/11.60	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
40.87/11.60	
40.87/11.60	   c(a(x1)) -> a(b(c(c(x1))))
40.87/11.60	   c(b(x1)) -> b(a(x1))
40.87/11.60	   b(a(x1)) -> x1
40.87/11.60	
40.87/11.60	
40.87/11.60	----------------------------------------
40.87/11.60	
40.87/11.60	(10)
40.87/11.60	Obligation:
40.87/11.60	Q DP problem:
40.87/11.60	P is empty.
40.87/11.60	The TRS R consists of the following rules:
40.87/11.60	
40.87/11.60	   b(a(x1)) -> x1
40.87/11.60	   c(a(x1)) -> a(b(c(c(x1))))
40.87/11.60	   c(b(x1)) -> b(a(x1))
40.87/11.60	
40.87/11.60	Q is empty.
40.87/11.60	We have to consider all minimal (P,Q,R)-chains.
40.87/11.60	----------------------------------------
40.87/11.60	
40.87/11.60	(11) PisEmptyProof (EQUIVALENT)
40.87/11.60	The TRS P is empty. Hence, there is no (P,Q,R) chain.
40.87/11.60	----------------------------------------
40.87/11.60	
40.87/11.60	(12)
40.87/11.60	YES
41.38/11.78	EOF