10.41/3.58	YES
10.96/3.67	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
10.96/3.67	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.96/3.67	
10.96/3.67	
10.96/3.67	Termination w.r.t. Q of the given QTRS could be proven:
10.96/3.67	
10.96/3.67	(0) QTRS
10.96/3.67	(1) DependencyPairsProof [EQUIVALENT, 11 ms]
10.96/3.67	(2) QDP
10.96/3.67	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
10.96/3.67	(4) AND
10.96/3.67	    (5) QDP
10.96/3.67	        (6) UsableRulesProof [EQUIVALENT, 0 ms]
10.96/3.67	        (7) QDP
10.96/3.67	        (8) MNOCProof [EQUIVALENT, 0 ms]
10.96/3.67	        (9) QDP
10.96/3.67	        (10) QDPOrderProof [EQUIVALENT, 3 ms]
10.96/3.67	        (11) QDP
10.96/3.67	        (12) QDPOrderProof [EQUIVALENT, 6 ms]
10.96/3.67	        (13) QDP
10.96/3.67	        (14) PisEmptyProof [EQUIVALENT, 0 ms]
10.96/3.67	        (15) YES
10.96/3.67	    (16) QDP
10.96/3.67	        (17) UsableRulesProof [EQUIVALENT, 0 ms]
10.96/3.67	        (18) QDP
10.96/3.67	        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.96/3.67	        (20) YES
10.96/3.67	
10.96/3.67	
10.96/3.67	----------------------------------------
10.96/3.67	
10.96/3.67	(0)
10.96/3.67	Obligation:
10.96/3.67	Q restricted rewrite system:
10.96/3.67	The TRS R consists of the following rules:
10.96/3.67	
10.96/3.67	   a(b(x1)) -> b(c(a(x1)))
10.96/3.67	   b(c(x1)) -> c(b(b(x1)))
10.96/3.67	   b(a(x1)) -> a(c(b(x1)))
10.96/3.67	
10.96/3.67	Q is empty.
10.96/3.67	
10.96/3.67	----------------------------------------
10.96/3.67	
10.96/3.67	(1) DependencyPairsProof (EQUIVALENT)
10.96/3.67	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
10.96/3.67	----------------------------------------
10.96/3.67	
10.96/3.67	(2)
10.96/3.67	Obligation:
10.96/3.67	Q DP problem:
10.96/3.67	The TRS P consists of the following rules:
10.96/3.67	
10.96/3.67	   A(b(x1)) -> B(c(a(x1)))
10.96/3.67	   A(b(x1)) -> A(x1)
10.96/3.67	   B(c(x1)) -> B(b(x1))
10.96/3.67	   B(c(x1)) -> B(x1)
10.96/3.67	   B(a(x1)) -> A(c(b(x1)))
10.96/3.67	   B(a(x1)) -> B(x1)
10.96/3.67	
10.96/3.67	The TRS R consists of the following rules:
10.96/3.67	
10.96/3.67	   a(b(x1)) -> b(c(a(x1)))
10.96/3.67	   b(c(x1)) -> c(b(b(x1)))
10.96/3.67	   b(a(x1)) -> a(c(b(x1)))
10.96/3.67	
10.96/3.67	Q is empty.
10.96/3.67	We have to consider all minimal (P,Q,R)-chains.
10.96/3.67	----------------------------------------
10.96/3.67	
10.96/3.67	(3) DependencyGraphProof (EQUIVALENT)
10.96/3.67	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 2 less nodes.
10.96/3.67	----------------------------------------
10.96/3.67	
10.96/3.67	(4)
10.96/3.67	Complex Obligation (AND)
10.96/3.67	
10.96/3.67	----------------------------------------
10.96/3.67	
10.96/3.67	(5)
10.96/3.67	Obligation:
10.96/3.67	Q DP problem:
10.96/3.67	The TRS P consists of the following rules:
10.96/3.67	
10.96/3.67	   B(c(x1)) -> B(x1)
10.96/3.67	   B(c(x1)) -> B(b(x1))
10.96/3.67	   B(a(x1)) -> B(x1)
10.96/3.67	
10.96/3.67	The TRS R consists of the following rules:
10.96/3.67	
10.96/3.67	   a(b(x1)) -> b(c(a(x1)))
10.96/3.67	   b(c(x1)) -> c(b(b(x1)))
10.96/3.67	   b(a(x1)) -> a(c(b(x1)))
10.96/3.67	
10.96/3.67	Q is empty.
10.96/3.67	We have to consider all minimal (P,Q,R)-chains.
10.96/3.67	----------------------------------------
10.96/3.67	
10.96/3.67	(6) UsableRulesProof (EQUIVALENT)
10.96/3.67	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.
10.96/3.67	----------------------------------------
10.96/3.67	
10.96/3.67	(7)
10.96/3.67	Obligation:
10.96/3.67	Q DP problem:
10.96/3.67	The TRS P consists of the following rules:
10.96/3.67	
10.96/3.67	   B(c(x1)) -> B(x1)
10.96/3.67	   B(c(x1)) -> B(b(x1))
10.96/3.67	   B(a(x1)) -> B(x1)
10.96/3.67	
10.96/3.67	The TRS R consists of the following rules:
10.96/3.67	
10.96/3.67	   b(c(x1)) -> c(b(b(x1)))
10.96/3.67	   b(a(x1)) -> a(c(b(x1)))
10.96/3.67	
10.96/3.67	Q is empty.
10.96/3.67	We have to consider all minimal (P,Q,R)-chains.
10.96/3.67	----------------------------------------
10.96/3.67	
10.96/3.67	(8) MNOCProof (EQUIVALENT)
10.96/3.67	We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.
10.96/3.67	----------------------------------------
10.96/3.67	
10.96/3.67	(9)
10.96/3.67	Obligation:
10.96/3.67	Q DP problem:
10.96/3.67	The TRS P consists of the following rules:
10.96/3.67	
10.96/3.67	   B(c(x1)) -> B(x1)
10.96/3.67	   B(c(x1)) -> B(b(x1))
10.96/3.67	   B(a(x1)) -> B(x1)
10.96/3.67	
10.96/3.67	The TRS R consists of the following rules:
10.96/3.67	
10.96/3.67	   b(c(x1)) -> c(b(b(x1)))
10.96/3.67	   b(a(x1)) -> a(c(b(x1)))
10.96/3.67	
10.96/3.67	The set Q consists of the following terms:
10.96/3.67	
10.96/3.67	   b(c(x0))
10.96/3.67	   b(a(x0))
10.96/3.67	
10.96/3.67	We have to consider all minimal (P,Q,R)-chains.
10.96/3.67	----------------------------------------
10.96/3.67	
10.96/3.67	(10) QDPOrderProof (EQUIVALENT)
10.96/3.67	We use the reduction pair processor [LPAR04,JAR06].
10.96/3.67	
10.96/3.67	
10.96/3.67	The following pairs can be oriented strictly and are deleted.
10.96/3.67	
10.96/3.67	   B(a(x1)) -> B(x1)
10.96/3.67	The remaining pairs can at least be oriented weakly.
10.96/3.67	Used ordering:  Polynomial interpretation [POLO]:
10.96/3.67	
10.96/3.67	   POL(B(x_1)) = x_1
10.96/3.67	   POL(a(x_1)) = 1 + x_1
10.96/3.67	   POL(b(x_1)) = x_1
10.96/3.67	   POL(c(x_1)) = x_1
10.96/3.67	
10.96/3.67	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
10.96/3.67	
10.96/3.67	   b(c(x1)) -> c(b(b(x1)))
10.96/3.67	   b(a(x1)) -> a(c(b(x1)))
10.96/3.67	
10.96/3.67	
10.96/3.67	----------------------------------------
10.96/3.67	
10.96/3.67	(11)
10.96/3.67	Obligation:
10.96/3.67	Q DP problem:
10.96/3.67	The TRS P consists of the following rules:
10.96/3.67	
10.96/3.67	   B(c(x1)) -> B(x1)
10.96/3.67	   B(c(x1)) -> B(b(x1))
10.96/3.67	
10.96/3.67	The TRS R consists of the following rules:
10.96/3.67	
10.96/3.67	   b(c(x1)) -> c(b(b(x1)))
10.96/3.67	   b(a(x1)) -> a(c(b(x1)))
10.96/3.67	
10.96/3.67	The set Q consists of the following terms:
10.96/3.67	
10.96/3.67	   b(c(x0))
10.96/3.67	   b(a(x0))
10.96/3.67	
10.96/3.67	We have to consider all minimal (P,Q,R)-chains.
10.96/3.67	----------------------------------------
10.96/3.67	
10.96/3.67	(12) QDPOrderProof (EQUIVALENT)
10.96/3.67	We use the reduction pair processor [LPAR04,JAR06].
10.96/3.67	
10.96/3.67	
10.96/3.67	The following pairs can be oriented strictly and are deleted.
10.96/3.67	
10.96/3.67	   B(c(x1)) -> B(x1)
10.96/3.67	   B(c(x1)) -> B(b(x1))
10.96/3.67	The remaining pairs can at least be oriented weakly.
10.96/3.67	Used ordering:  Polynomial interpretation [POLO]:
10.96/3.67	
10.96/3.67	   POL(B(x_1)) = x_1
10.96/3.67	   POL(a(x_1)) = 1
10.96/3.67	   POL(b(x_1)) = x_1
10.96/3.67	   POL(c(x_1)) = 1 + x_1
10.96/3.67	
10.96/3.67	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
10.96/3.67	
10.96/3.67	   b(c(x1)) -> c(b(b(x1)))
10.96/3.67	   b(a(x1)) -> a(c(b(x1)))
10.96/3.67	
10.96/3.67	
10.96/3.67	----------------------------------------
10.96/3.67	
10.96/3.67	(13)
10.96/3.67	Obligation:
10.96/3.67	Q DP problem:
10.96/3.67	P is empty.
10.96/3.67	The TRS R consists of the following rules:
10.96/3.67	
10.96/3.67	   b(c(x1)) -> c(b(b(x1)))
10.96/3.67	   b(a(x1)) -> a(c(b(x1)))
10.96/3.67	
10.96/3.67	The set Q consists of the following terms:
10.96/3.67	
10.96/3.67	   b(c(x0))
10.96/3.67	   b(a(x0))
10.96/3.67	
10.96/3.67	We have to consider all minimal (P,Q,R)-chains.
10.96/3.67	----------------------------------------
10.96/3.67	
10.96/3.67	(14) PisEmptyProof (EQUIVALENT)
10.96/3.67	The TRS P is empty. Hence, there is no (P,Q,R) chain.
10.96/3.67	----------------------------------------
10.96/3.67	
10.96/3.67	(15)
10.96/3.67	YES
10.96/3.67	
10.96/3.67	----------------------------------------
10.96/3.67	
10.96/3.67	(16)
10.96/3.67	Obligation:
10.96/3.67	Q DP problem:
10.96/3.67	The TRS P consists of the following rules:
10.96/3.67	
10.96/3.67	   A(b(x1)) -> A(x1)
10.96/3.67	
10.96/3.67	The TRS R consists of the following rules:
10.96/3.67	
10.96/3.67	   a(b(x1)) -> b(c(a(x1)))
10.96/3.67	   b(c(x1)) -> c(b(b(x1)))
10.96/3.67	   b(a(x1)) -> a(c(b(x1)))
10.96/3.67	
10.96/3.67	Q is empty.
10.96/3.67	We have to consider all minimal (P,Q,R)-chains.
10.96/3.67	----------------------------------------
10.96/3.67	
10.96/3.67	(17) UsableRulesProof (EQUIVALENT)
10.96/3.67	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.
10.96/3.67	----------------------------------------
10.96/3.67	
10.96/3.67	(18)
10.96/3.67	Obligation:
10.96/3.67	Q DP problem:
10.96/3.67	The TRS P consists of the following rules:
10.96/3.67	
10.96/3.67	   A(b(x1)) -> A(x1)
10.96/3.67	
10.96/3.67	R is empty.
10.96/3.67	Q is empty.
10.96/3.67	We have to consider all minimal (P,Q,R)-chains.
10.96/3.67	----------------------------------------
10.96/3.67	
10.96/3.67	(19) QDPSizeChangeProof (EQUIVALENT)
10.96/3.67	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. 
10.96/3.67	
10.96/3.67	From the DPs we obtained the following set of size-change graphs:
10.96/3.67	*A(b(x1)) -> A(x1)
10.96/3.67	The graph contains the following edges 1 > 1
10.96/3.67	
10.96/3.67	
10.96/3.67	----------------------------------------
10.96/3.67	
10.96/3.67	(20)
10.96/3.67	YES
11.22/3.77	EOF