16.65/5.13	YES
17.29/5.29	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
17.29/5.29	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
17.29/5.29	
17.29/5.29	
17.29/5.29	Termination w.r.t. Q of the given QTRS could be proven:
17.29/5.29	
17.29/5.29	(0) QTRS
17.29/5.29	(1) DependencyPairsProof [EQUIVALENT, 13 ms]
17.29/5.29	(2) QDP
17.29/5.29	(3) DependencyGraphProof [EQUIVALENT, 8 ms]
17.29/5.29	(4) QDP
17.29/5.29	(5) QDPOrderProof [EQUIVALENT, 37 ms]
17.29/5.29	(6) QDP
17.29/5.29	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
17.29/5.29	(8) YES
17.29/5.29	
17.29/5.29	
17.29/5.29	----------------------------------------
17.29/5.29	
17.29/5.29	(0)
17.29/5.29	Obligation:
17.29/5.29	Q restricted rewrite system:
17.29/5.29	The TRS R consists of the following rules:
17.29/5.29	
17.29/5.29	   a(b(x1)) -> b(a(a(a(x1))))
17.29/5.29	   b(a(x1)) -> a(a(x1))
17.29/5.29	   a(a(x1)) -> a(c(b(x1)))
17.29/5.29	
17.29/5.29	Q is empty.
17.29/5.29	
17.29/5.29	----------------------------------------
17.29/5.29	
17.29/5.29	(1) DependencyPairsProof (EQUIVALENT)
17.29/5.29	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
17.29/5.29	----------------------------------------
17.29/5.29	
17.29/5.29	(2)
17.29/5.29	Obligation:
17.29/5.29	Q DP problem:
17.29/5.29	The TRS P consists of the following rules:
17.29/5.29	
17.29/5.29	   A(b(x1)) -> B(a(a(a(x1))))
17.29/5.29	   A(b(x1)) -> A(a(a(x1)))
17.29/5.29	   A(b(x1)) -> A(a(x1))
17.29/5.29	   A(b(x1)) -> A(x1)
17.29/5.29	   B(a(x1)) -> A(a(x1))
17.29/5.29	   A(a(x1)) -> A(c(b(x1)))
17.29/5.29	   A(a(x1)) -> B(x1)
17.29/5.29	
17.29/5.29	The TRS R consists of the following rules:
17.29/5.29	
17.29/5.29	   a(b(x1)) -> b(a(a(a(x1))))
17.29/5.29	   b(a(x1)) -> a(a(x1))
17.29/5.29	   a(a(x1)) -> a(c(b(x1)))
17.29/5.29	
17.29/5.29	Q is empty.
17.29/5.29	We have to consider all minimal (P,Q,R)-chains.
17.29/5.29	----------------------------------------
17.29/5.29	
17.29/5.29	(3) DependencyGraphProof (EQUIVALENT)
17.29/5.29	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
17.29/5.29	----------------------------------------
17.29/5.29	
17.29/5.29	(4)
17.29/5.29	Obligation:
17.29/5.29	Q DP problem:
17.29/5.29	The TRS P consists of the following rules:
17.29/5.29	
17.29/5.29	   B(a(x1)) -> A(a(x1))
17.29/5.29	   A(b(x1)) -> B(a(a(a(x1))))
17.29/5.29	   A(b(x1)) -> A(a(a(x1)))
17.29/5.29	   A(b(x1)) -> A(a(x1))
17.29/5.29	   A(b(x1)) -> A(x1)
17.29/5.29	   A(a(x1)) -> B(x1)
17.29/5.29	
17.29/5.29	The TRS R consists of the following rules:
17.29/5.29	
17.29/5.29	   a(b(x1)) -> b(a(a(a(x1))))
17.29/5.29	   b(a(x1)) -> a(a(x1))
17.29/5.29	   a(a(x1)) -> a(c(b(x1)))
17.29/5.29	
17.29/5.29	Q is empty.
17.29/5.29	We have to consider all minimal (P,Q,R)-chains.
17.29/5.29	----------------------------------------
17.29/5.29	
17.29/5.29	(5) QDPOrderProof (EQUIVALENT)
17.29/5.29	We use the reduction pair processor [LPAR04,JAR06].
17.29/5.29	
17.29/5.29	
17.29/5.29	The following pairs can be oriented strictly and are deleted.
17.29/5.29	
17.29/5.29	   A(b(x1)) -> B(a(a(a(x1))))
17.29/5.29	   A(b(x1)) -> A(a(a(x1)))
17.29/5.29	   A(b(x1)) -> A(a(x1))
17.29/5.29	   A(b(x1)) -> A(x1)
17.29/5.29	The remaining pairs can at least be oriented weakly.
17.29/5.29	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
17.29/5.29	
17.29/5.29	POL( A_1(x_1) ) = x_1 + 2
17.29/5.29	POL( B_1(x_1) ) = x_1 + 2
17.29/5.29	POL( b_1(x_1) ) = 2x_1 + 1
17.29/5.29	POL( a_1(x_1) ) = x_1
17.29/5.29	POL( c_1(x_1) ) = max{0, -2}
17.29/5.29	
17.29/5.29	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
17.29/5.29	
17.29/5.29	   b(a(x1)) -> a(a(x1))
17.29/5.29	   a(b(x1)) -> b(a(a(a(x1))))
17.29/5.29	   a(a(x1)) -> a(c(b(x1)))
17.29/5.29	
17.29/5.29	
17.29/5.29	----------------------------------------
17.29/5.29	
17.29/5.29	(6)
17.29/5.29	Obligation:
17.29/5.29	Q DP problem:
17.29/5.29	The TRS P consists of the following rules:
17.29/5.29	
17.29/5.29	   B(a(x1)) -> A(a(x1))
17.29/5.29	   A(a(x1)) -> B(x1)
17.29/5.29	
17.29/5.29	The TRS R consists of the following rules:
17.29/5.29	
17.29/5.29	   a(b(x1)) -> b(a(a(a(x1))))
17.29/5.29	   b(a(x1)) -> a(a(x1))
17.29/5.29	   a(a(x1)) -> a(c(b(x1)))
17.29/5.29	
17.29/5.29	Q is empty.
17.29/5.29	We have to consider all minimal (P,Q,R)-chains.
17.29/5.29	----------------------------------------
17.29/5.29	
17.29/5.29	(7) QDPSizeChangeProof (EQUIVALENT)
17.29/5.29	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
17.29/5.29	
17.29/5.29	From the DPs we obtained the following set of size-change graphs:
17.29/5.29	*A(a(x1)) -> B(x1)
17.29/5.29	The graph contains the following edges 1 > 1
17.29/5.29	
17.29/5.29	
17.29/5.29	*B(a(x1)) -> A(a(x1))
17.29/5.29	The graph contains the following edges 1 >= 1
17.29/5.29	
17.29/5.29	
17.29/5.29	----------------------------------------
17.29/5.29	
17.29/5.29	(8)
17.29/5.29	YES
17.60/5.45	EOF