14.47/4.62	YES
14.47/4.65	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
14.47/4.65	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
14.47/4.65	
14.47/4.65	
14.47/4.65	Termination w.r.t. Q of the given QTRS could be proven:
14.47/4.65	
14.47/4.65	(0) QTRS
14.47/4.65	(1) QTRS Reverse [EQUIVALENT, 0 ms]
14.47/4.65	(2) QTRS
14.47/4.65	(3) DependencyPairsProof [EQUIVALENT, 22 ms]
14.47/4.65	(4) QDP
14.47/4.65	(5) QDPOrderProof [EQUIVALENT, 21 ms]
14.47/4.65	(6) QDP
14.47/4.65	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
14.47/4.65	(8) AND
14.47/4.65	    (9) QDP
14.47/4.65	        (10) UsableRulesProof [EQUIVALENT, 0 ms]
14.47/4.65	        (11) QDP
14.47/4.65	        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
14.47/4.65	        (13) YES
14.47/4.65	    (14) QDP
14.47/4.65	        (15) UsableRulesProof [EQUIVALENT, 0 ms]
14.47/4.65	        (16) QDP
14.47/4.65	        (17) QDPSizeChangeProof [EQUIVALENT, 0 ms]
14.47/4.65	        (18) YES
14.47/4.65	
14.47/4.65	
14.47/4.65	----------------------------------------
14.47/4.65	
14.47/4.65	(0)
14.47/4.65	Obligation:
14.47/4.65	Q restricted rewrite system:
14.47/4.65	The TRS R consists of the following rules:
14.47/4.65	
14.47/4.65	   a(b(x1)) -> b(b(a(x1)))
14.47/4.65	   b(c(x1)) -> c(b(b(x1)))
14.47/4.65	   c(a(x1)) -> a(c(c(x1)))
14.47/4.65	
14.47/4.65	Q is empty.
14.47/4.65	
14.47/4.65	----------------------------------------
14.47/4.65	
14.47/4.65	(1) QTRS Reverse (EQUIVALENT)
14.47/4.65	We applied the QTRS Reverse Processor [REVERSE].
14.47/4.65	----------------------------------------
14.47/4.65	
14.47/4.65	(2)
14.47/4.65	Obligation:
14.47/4.65	Q restricted rewrite system:
14.47/4.65	The TRS R consists of the following rules:
14.47/4.65	
14.47/4.65	   b(a(x1)) -> a(b(b(x1)))
14.47/4.65	   c(b(x1)) -> b(b(c(x1)))
14.47/4.65	   a(c(x1)) -> c(c(a(x1)))
14.47/4.65	
14.47/4.65	Q is empty.
14.47/4.65	
14.47/4.65	----------------------------------------
14.47/4.65	
14.47/4.65	(3) DependencyPairsProof (EQUIVALENT)
14.47/4.65	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
14.47/4.65	----------------------------------------
14.47/4.65	
14.47/4.65	(4)
14.47/4.65	Obligation:
14.47/4.65	Q DP problem:
14.47/4.65	The TRS P consists of the following rules:
14.47/4.65	
14.47/4.65	   B(a(x1)) -> A(b(b(x1)))
14.47/4.65	   B(a(x1)) -> B(b(x1))
14.47/4.65	   B(a(x1)) -> B(x1)
14.47/4.65	   C(b(x1)) -> B(b(c(x1)))
14.47/4.65	   C(b(x1)) -> B(c(x1))
14.47/4.65	   C(b(x1)) -> C(x1)
14.47/4.65	   A(c(x1)) -> C(c(a(x1)))
14.47/4.65	   A(c(x1)) -> C(a(x1))
14.47/4.65	   A(c(x1)) -> A(x1)
14.47/4.65	
14.47/4.65	The TRS R consists of the following rules:
14.47/4.65	
14.47/4.65	   b(a(x1)) -> a(b(b(x1)))
14.47/4.65	   c(b(x1)) -> b(b(c(x1)))
14.47/4.65	   a(c(x1)) -> c(c(a(x1)))
14.47/4.65	
14.47/4.65	Q is empty.
14.47/4.65	We have to consider all minimal (P,Q,R)-chains.
14.47/4.65	----------------------------------------
14.47/4.65	
14.47/4.65	(5) QDPOrderProof (EQUIVALENT)
14.47/4.65	We use the reduction pair processor [LPAR04,JAR06].
14.47/4.65	
14.47/4.65	
14.47/4.65	The following pairs can be oriented strictly and are deleted.
14.47/4.65	
14.47/4.65	   B(a(x1)) -> A(b(b(x1)))
14.47/4.65	   B(a(x1)) -> B(b(x1))
14.47/4.65	   B(a(x1)) -> B(x1)
14.47/4.65	The remaining pairs can at least be oriented weakly.
14.47/4.65	Used ordering:  Polynomial interpretation [POLO]:
14.47/4.65	
14.47/4.65	   POL(A(x_1)) = 0
14.47/4.65	   POL(B(x_1)) = x_1
14.47/4.65	   POL(C(x_1)) = 0
14.47/4.65	   POL(a(x_1)) = 1 + x_1
14.47/4.65	   POL(b(x_1)) = x_1
14.47/4.65	   POL(c(x_1)) = 0
14.47/4.65	
14.47/4.65	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
14.47/4.65	
14.47/4.65	   a(c(x1)) -> c(c(a(x1)))
14.47/4.65	   c(b(x1)) -> b(b(c(x1)))
14.47/4.65	   b(a(x1)) -> a(b(b(x1)))
14.47/4.65	
14.47/4.65	
14.47/4.65	----------------------------------------
14.47/4.65	
14.47/4.65	(6)
14.47/4.65	Obligation:
14.47/4.65	Q DP problem:
14.47/4.65	The TRS P consists of the following rules:
14.47/4.65	
14.47/4.65	   C(b(x1)) -> B(b(c(x1)))
14.47/4.65	   C(b(x1)) -> B(c(x1))
14.47/4.65	   C(b(x1)) -> C(x1)
14.47/4.65	   A(c(x1)) -> C(c(a(x1)))
14.47/4.65	   A(c(x1)) -> C(a(x1))
14.47/4.65	   A(c(x1)) -> A(x1)
14.47/4.65	
14.47/4.65	The TRS R consists of the following rules:
14.47/4.65	
14.47/4.65	   b(a(x1)) -> a(b(b(x1)))
14.47/4.65	   c(b(x1)) -> b(b(c(x1)))
14.47/4.65	   a(c(x1)) -> c(c(a(x1)))
14.47/4.65	
14.47/4.65	Q is empty.
14.47/4.65	We have to consider all minimal (P,Q,R)-chains.
14.47/4.65	----------------------------------------
14.47/4.65	
14.47/4.65	(7) DependencyGraphProof (EQUIVALENT)
14.47/4.65	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 4 less nodes.
14.47/4.65	----------------------------------------
14.47/4.65	
14.47/4.65	(8)
14.47/4.65	Complex Obligation (AND)
14.47/4.65	
14.47/4.65	----------------------------------------
14.47/4.65	
14.47/4.65	(9)
14.47/4.65	Obligation:
14.47/4.65	Q DP problem:
14.47/4.65	The TRS P consists of the following rules:
14.47/4.65	
14.47/4.65	   C(b(x1)) -> C(x1)
14.47/4.65	
14.47/4.65	The TRS R consists of the following rules:
14.47/4.65	
14.47/4.65	   b(a(x1)) -> a(b(b(x1)))
14.47/4.65	   c(b(x1)) -> b(b(c(x1)))
14.47/4.65	   a(c(x1)) -> c(c(a(x1)))
14.47/4.65	
14.47/4.65	Q is empty.
14.47/4.65	We have to consider all minimal (P,Q,R)-chains.
14.47/4.65	----------------------------------------
14.47/4.65	
14.47/4.65	(10) UsableRulesProof (EQUIVALENT)
14.47/4.65	We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
14.47/4.65	----------------------------------------
14.47/4.65	
14.47/4.65	(11)
14.47/4.65	Obligation:
14.47/4.65	Q DP problem:
14.47/4.65	The TRS P consists of the following rules:
14.47/4.65	
14.47/4.65	   C(b(x1)) -> C(x1)
14.47/4.65	
14.47/4.65	R is empty.
14.47/4.65	Q is empty.
14.47/4.65	We have to consider all minimal (P,Q,R)-chains.
14.47/4.65	----------------------------------------
14.47/4.65	
14.47/4.65	(12) QDPSizeChangeProof (EQUIVALENT)
14.47/4.65	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. 
14.47/4.65	
14.47/4.65	From the DPs we obtained the following set of size-change graphs:
14.47/4.65	*C(b(x1)) -> C(x1)
14.47/4.65	The graph contains the following edges 1 > 1
14.47/4.65	
14.47/4.65	
14.47/4.65	----------------------------------------
14.47/4.65	
14.47/4.65	(13)
14.47/4.65	YES
14.47/4.65	
14.47/4.65	----------------------------------------
14.47/4.65	
14.47/4.65	(14)
14.47/4.65	Obligation:
14.47/4.65	Q DP problem:
14.47/4.65	The TRS P consists of the following rules:
14.47/4.65	
14.47/4.65	   A(c(x1)) -> A(x1)
14.47/4.65	
14.47/4.65	The TRS R consists of the following rules:
14.47/4.65	
14.47/4.65	   b(a(x1)) -> a(b(b(x1)))
14.47/4.65	   c(b(x1)) -> b(b(c(x1)))
14.47/4.65	   a(c(x1)) -> c(c(a(x1)))
14.47/4.65	
14.47/4.65	Q is empty.
14.47/4.65	We have to consider all minimal (P,Q,R)-chains.
14.47/4.65	----------------------------------------
14.47/4.65	
14.47/4.65	(15) UsableRulesProof (EQUIVALENT)
14.47/4.65	We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
14.47/4.65	----------------------------------------
14.47/4.65	
14.47/4.65	(16)
14.47/4.65	Obligation:
14.47/4.65	Q DP problem:
14.47/4.65	The TRS P consists of the following rules:
14.47/4.65	
14.47/4.65	   A(c(x1)) -> A(x1)
14.47/4.65	
14.47/4.65	R is empty.
14.47/4.65	Q is empty.
14.47/4.65	We have to consider all minimal (P,Q,R)-chains.
14.47/4.65	----------------------------------------
14.47/4.65	
14.47/4.65	(17) QDPSizeChangeProof (EQUIVALENT)
14.47/4.65	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. 
14.47/4.65	
14.47/4.65	From the DPs we obtained the following set of size-change graphs:
14.47/4.65	*A(c(x1)) -> A(x1)
14.47/4.65	The graph contains the following edges 1 > 1
14.47/4.65	
14.47/4.65	
14.47/4.65	----------------------------------------
14.47/4.65	
14.47/4.65	(18)
14.47/4.65	YES
14.69/4.72	EOF