39.94/11.09	YES
40.61/11.26	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
40.61/11.26	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
40.61/11.26	
40.61/11.26	
40.61/11.26	Termination w.r.t. Q of the given QTRS could be proven:
40.61/11.26	
40.61/11.26	(0) QTRS
40.61/11.26	(1) DependencyPairsProof [EQUIVALENT, 4 ms]
40.61/11.26	(2) QDP
40.61/11.26	(3) QDPOrderProof [EQUIVALENT, 34 ms]
40.61/11.26	(4) QDP
40.61/11.26	(5) DependencyGraphProof [EQUIVALENT, 1 ms]
40.61/11.26	(6) QDP
40.61/11.26	(7) QDPOrderProof [EQUIVALENT, 38 ms]
40.61/11.26	(8) QDP
40.61/11.26	(9) PisEmptyProof [EQUIVALENT, 0 ms]
40.61/11.26	(10) YES
40.61/11.26	
40.61/11.26	
40.61/11.26	----------------------------------------
40.61/11.26	
40.61/11.26	(0)
40.61/11.26	Obligation:
40.61/11.26	Q restricted rewrite system:
40.61/11.26	The TRS R consists of the following rules:
40.61/11.26	
40.61/11.26	   b(a(a(x1))) -> a(b(c(x1)))
40.61/11.26	   c(a(x1)) -> a(c(x1))
40.61/11.26	   c(b(x1)) -> b(a(x1))
40.61/11.26	   a(a(b(x1))) -> d(b(a(x1)))
40.61/11.26	   a(d(x1)) -> d(a(x1))
40.61/11.26	   b(d(x1)) -> a(b(x1))
40.61/11.26	   a(a(x1)) -> a(b(a(x1)))
40.61/11.26	
40.61/11.26	Q is empty.
40.61/11.26	
40.61/11.26	----------------------------------------
40.61/11.26	
40.61/11.26	(1) DependencyPairsProof (EQUIVALENT)
40.61/11.26	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
40.61/11.26	----------------------------------------
40.61/11.26	
40.61/11.26	(2)
40.61/11.26	Obligation:
40.61/11.26	Q DP problem:
40.61/11.26	The TRS P consists of the following rules:
40.61/11.26	
40.61/11.26	   B(a(a(x1))) -> A(b(c(x1)))
40.61/11.26	   B(a(a(x1))) -> B(c(x1))
40.61/11.26	   B(a(a(x1))) -> C(x1)
40.61/11.26	   C(a(x1)) -> A(c(x1))
40.61/11.26	   C(a(x1)) -> C(x1)
40.61/11.26	   C(b(x1)) -> B(a(x1))
40.61/11.26	   C(b(x1)) -> A(x1)
40.61/11.26	   A(a(b(x1))) -> B(a(x1))
40.61/11.26	   A(a(b(x1))) -> A(x1)
40.61/11.26	   A(d(x1)) -> A(x1)
40.61/11.26	   B(d(x1)) -> A(b(x1))
40.61/11.26	   B(d(x1)) -> B(x1)
40.61/11.26	   A(a(x1)) -> A(b(a(x1)))
40.61/11.26	   A(a(x1)) -> B(a(x1))
40.61/11.26	
40.61/11.26	The TRS R consists of the following rules:
40.61/11.26	
40.61/11.26	   b(a(a(x1))) -> a(b(c(x1)))
40.61/11.26	   c(a(x1)) -> a(c(x1))
40.61/11.26	   c(b(x1)) -> b(a(x1))
40.61/11.26	   a(a(b(x1))) -> d(b(a(x1)))
40.61/11.26	   a(d(x1)) -> d(a(x1))
40.61/11.26	   b(d(x1)) -> a(b(x1))
40.61/11.26	   a(a(x1)) -> a(b(a(x1)))
40.61/11.26	
40.61/11.26	Q is empty.
40.61/11.26	We have to consider all minimal (P,Q,R)-chains.
40.61/11.26	----------------------------------------
40.61/11.26	
40.61/11.26	(3) QDPOrderProof (EQUIVALENT)
40.61/11.26	We use the reduction pair processor [LPAR04,JAR06].
40.61/11.26	
40.61/11.26	
40.61/11.26	The following pairs can be oriented strictly and are deleted.
40.61/11.26	
40.61/11.26	   B(a(a(x1))) -> A(b(c(x1)))
40.61/11.26	   B(a(a(x1))) -> B(c(x1))
40.61/11.26	   B(a(a(x1))) -> C(x1)
40.61/11.26	   C(a(x1)) -> A(c(x1))
40.61/11.26	   C(a(x1)) -> C(x1)
40.61/11.26	   C(b(x1)) -> A(x1)
40.61/11.26	   A(a(b(x1))) -> A(x1)
40.61/11.26	   A(d(x1)) -> A(x1)
40.61/11.26	   B(d(x1)) -> A(b(x1))
40.61/11.26	   B(d(x1)) -> B(x1)
40.61/11.26	The remaining pairs can at least be oriented weakly.
40.61/11.26	Used ordering:  Polynomial interpretation [POLO]:
40.61/11.26	
40.61/11.26	   POL(A(x_1)) = x_1
40.61/11.26	   POL(B(x_1)) = x_1
40.61/11.26	   POL(C(x_1)) = 1 + x_1
40.61/11.26	   POL(a(x_1)) = 1 + x_1
40.61/11.26	   POL(b(x_1)) = x_1
40.61/11.26	   POL(c(x_1)) = 1 + x_1
40.61/11.26	   POL(d(x_1)) = 1 + x_1
40.61/11.26	
40.61/11.26	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
40.61/11.26	
40.61/11.26	   c(a(x1)) -> a(c(x1))
40.61/11.26	   c(b(x1)) -> b(a(x1))
40.61/11.26	   b(a(a(x1))) -> a(b(c(x1)))
40.61/11.26	   b(d(x1)) -> a(b(x1))
40.61/11.26	   a(a(b(x1))) -> d(b(a(x1)))
40.61/11.26	   a(d(x1)) -> d(a(x1))
40.61/11.26	   a(a(x1)) -> a(b(a(x1)))
40.61/11.26	
40.61/11.26	
40.61/11.26	----------------------------------------
40.61/11.26	
40.61/11.26	(4)
40.61/11.26	Obligation:
40.61/11.26	Q DP problem:
40.61/11.26	The TRS P consists of the following rules:
40.61/11.26	
40.61/11.26	   C(b(x1)) -> B(a(x1))
40.61/11.26	   A(a(b(x1))) -> B(a(x1))
40.61/11.26	   A(a(x1)) -> A(b(a(x1)))
40.61/11.26	   A(a(x1)) -> B(a(x1))
40.61/11.26	
40.61/11.26	The TRS R consists of the following rules:
40.61/11.26	
40.61/11.26	   b(a(a(x1))) -> a(b(c(x1)))
40.61/11.26	   c(a(x1)) -> a(c(x1))
40.61/11.26	   c(b(x1)) -> b(a(x1))
40.61/11.26	   a(a(b(x1))) -> d(b(a(x1)))
40.61/11.26	   a(d(x1)) -> d(a(x1))
40.61/11.26	   b(d(x1)) -> a(b(x1))
40.61/11.26	   a(a(x1)) -> a(b(a(x1)))
40.61/11.26	
40.61/11.26	Q is empty.
40.61/11.26	We have to consider all minimal (P,Q,R)-chains.
40.61/11.26	----------------------------------------
40.61/11.26	
40.61/11.26	(5) DependencyGraphProof (EQUIVALENT)
40.61/11.26	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
40.61/11.26	----------------------------------------
40.61/11.26	
40.61/11.26	(6)
40.61/11.26	Obligation:
40.61/11.26	Q DP problem:
40.61/11.26	The TRS P consists of the following rules:
40.61/11.26	
40.61/11.26	   A(a(x1)) -> A(b(a(x1)))
40.61/11.26	
40.61/11.26	The TRS R consists of the following rules:
40.61/11.26	
40.61/11.26	   b(a(a(x1))) -> a(b(c(x1)))
40.61/11.26	   c(a(x1)) -> a(c(x1))
40.61/11.26	   c(b(x1)) -> b(a(x1))
40.61/11.26	   a(a(b(x1))) -> d(b(a(x1)))
40.61/11.26	   a(d(x1)) -> d(a(x1))
40.61/11.26	   b(d(x1)) -> a(b(x1))
40.61/11.26	   a(a(x1)) -> a(b(a(x1)))
40.61/11.26	
40.61/11.26	Q is empty.
40.61/11.26	We have to consider all minimal (P,Q,R)-chains.
40.61/11.26	----------------------------------------
40.61/11.26	
40.61/11.26	(7) QDPOrderProof (EQUIVALENT)
40.61/11.26	We use the reduction pair processor [LPAR04,JAR06].
40.61/11.26	
40.61/11.26	
40.61/11.26	The following pairs can be oriented strictly and are deleted.
40.61/11.26	
40.61/11.26	   A(a(x1)) -> A(b(a(x1)))
40.61/11.26	The remaining pairs can at least be oriented weakly.
40.61/11.26	Used ordering:  Polynomial interpretation [POLO,RATPOLO]:
40.61/11.27	
40.61/11.27	   POL(A(x_1)) = [1/4]x_1
40.61/11.27	   POL(a(x_1)) = [2] + [2]x_1
40.61/11.27	   POL(b(x_1)) = [1/2]x_1
40.61/11.27	   POL(c(x_1)) = [1] + [2]x_1
40.61/11.27	   POL(d(x_1)) = [4] + [2]x_1
40.61/11.27	The value of delta used in the strict ordering is 1/4.
40.61/11.27	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
40.61/11.27	
40.61/11.27	   a(a(b(x1))) -> d(b(a(x1)))
40.61/11.27	   a(d(x1)) -> d(a(x1))
40.61/11.27	   a(a(x1)) -> a(b(a(x1)))
40.61/11.27	   b(a(a(x1))) -> a(b(c(x1)))
40.61/11.27	   b(d(x1)) -> a(b(x1))
40.61/11.27	   c(a(x1)) -> a(c(x1))
40.61/11.27	   c(b(x1)) -> b(a(x1))
40.61/11.27	
40.61/11.27	
40.61/11.27	----------------------------------------
40.61/11.27	
40.61/11.27	(8)
40.61/11.27	Obligation:
40.61/11.27	Q DP problem:
40.61/11.27	P is empty.
40.61/11.27	The TRS R consists of the following rules:
40.61/11.27	
40.61/11.27	   b(a(a(x1))) -> a(b(c(x1)))
40.61/11.27	   c(a(x1)) -> a(c(x1))
40.61/11.27	   c(b(x1)) -> b(a(x1))
40.61/11.27	   a(a(b(x1))) -> d(b(a(x1)))
40.61/11.27	   a(d(x1)) -> d(a(x1))
40.61/11.27	   b(d(x1)) -> a(b(x1))
40.61/11.27	   a(a(x1)) -> a(b(a(x1)))
40.61/11.27	
40.61/11.27	Q is empty.
40.61/11.27	We have to consider all minimal (P,Q,R)-chains.
40.61/11.27	----------------------------------------
40.61/11.27	
40.61/11.27	(9) PisEmptyProof (EQUIVALENT)
40.61/11.27	The TRS P is empty. Hence, there is no (P,Q,R) chain.
40.61/11.27	----------------------------------------
40.61/11.27	
40.61/11.27	(10)
40.61/11.27	YES
40.61/11.32	EOF