20.31/6.29	YES
20.31/6.30	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
20.31/6.30	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
20.31/6.30	
20.31/6.30	
20.31/6.30	Termination w.r.t. Q of the given QTRS could be proven:
20.31/6.30	
20.31/6.30	(0) QTRS
20.31/6.30	(1) QTRS Reverse [EQUIVALENT, 0 ms]
20.31/6.30	(2) QTRS
20.31/6.30	(3) QTRSRRRProof [EQUIVALENT, 22 ms]
20.31/6.30	(4) QTRS
20.31/6.30	(5) DependencyPairsProof [EQUIVALENT, 27 ms]
20.31/6.30	(6) QDP
20.31/6.30	(7) DependencyGraphProof [EQUIVALENT, 2 ms]
20.31/6.30	(8) AND
20.31/6.30	    (9) QDP
20.31/6.30	        (10) UsableRulesProof [EQUIVALENT, 0 ms]
20.31/6.30	        (11) QDP
20.31/6.30	        (12) MRRProof [EQUIVALENT, 11 ms]
20.31/6.30	        (13) QDP
20.31/6.30	        (14) PisEmptyProof [EQUIVALENT, 0 ms]
20.31/6.30	        (15) YES
20.31/6.30	    (16) QDP
20.31/6.30	        (17) UsableRulesProof [EQUIVALENT, 1 ms]
20.31/6.30	        (18) QDP
20.31/6.30	        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
20.31/6.30	        (20) YES
20.31/6.30	
20.31/6.30	
20.31/6.30	----------------------------------------
20.31/6.30	
20.31/6.30	(0)
20.31/6.30	Obligation:
20.31/6.30	Q restricted rewrite system:
20.31/6.30	The TRS R consists of the following rules:
20.31/6.30	
20.31/6.30	   1(1(x1)) -> 4(3(x1))
20.31/6.30	   1(2(x1)) -> 2(1(x1))
20.31/6.30	   2(2(x1)) -> 1(1(1(x1)))
20.31/6.30	   3(3(x1)) -> 5(6(x1))
20.31/6.30	   3(4(x1)) -> 1(1(x1))
20.31/6.30	   4(4(x1)) -> 3(x1)
20.31/6.30	   5(5(x1)) -> 6(2(x1))
20.31/6.30	   5(6(x1)) -> 1(2(x1))
20.31/6.30	   6(6(x1)) -> 2(1(x1))
20.31/6.30	
20.31/6.30	Q is empty.
20.31/6.30	
20.31/6.30	----------------------------------------
20.31/6.30	
20.31/6.30	(1) QTRS Reverse (EQUIVALENT)
20.31/6.30	We applied the QTRS Reverse Processor [REVERSE].
20.31/6.30	----------------------------------------
20.31/6.30	
20.31/6.30	(2)
20.31/6.30	Obligation:
20.31/6.30	Q restricted rewrite system:
20.31/6.30	The TRS R consists of the following rules:
20.31/6.30	
20.31/6.30	   1(1(x1)) -> 3(4(x1))
20.31/6.30	   2(1(x1)) -> 1(2(x1))
20.31/6.30	   2(2(x1)) -> 1(1(1(x1)))
20.31/6.30	   3(3(x1)) -> 6(5(x1))
20.31/6.30	   4(3(x1)) -> 1(1(x1))
20.31/6.30	   4(4(x1)) -> 3(x1)
20.31/6.30	   5(5(x1)) -> 2(6(x1))
20.31/6.30	   6(5(x1)) -> 2(1(x1))
20.31/6.30	   6(6(x1)) -> 1(2(x1))
20.31/6.30	
20.31/6.30	Q is empty.
20.31/6.30	
20.31/6.30	----------------------------------------
20.31/6.30	
20.31/6.30	(3) QTRSRRRProof (EQUIVALENT)
20.31/6.30	Used ordering:
20.31/6.30	Polynomial interpretation [POLO]:
20.31/6.30	
20.31/6.30	   POL(1(x_1)) = 85 + x_1
20.31/6.30	   POL(2(x_1)) = 128 + x_1
20.31/6.30	   POL(3(x_1)) = 113 + x_1
20.31/6.30	   POL(4(x_1)) = 57 + x_1
20.31/6.30	   POL(5(x_1)) = 118 + x_1
20.31/6.30	   POL(6(x_1)) = 107 + x_1
20.31/6.30	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
20.31/6.30	
20.31/6.30	   2(2(x1)) -> 1(1(1(x1)))
20.31/6.30	   3(3(x1)) -> 6(5(x1))
20.31/6.30	   4(4(x1)) -> 3(x1)
20.31/6.30	   5(5(x1)) -> 2(6(x1))
20.31/6.30	   6(5(x1)) -> 2(1(x1))
20.31/6.30	   6(6(x1)) -> 1(2(x1))
20.31/6.30	
20.31/6.30	
20.31/6.30	
20.31/6.30	
20.31/6.30	----------------------------------------
20.31/6.30	
20.31/6.30	(4)
20.31/6.30	Obligation:
20.31/6.30	Q restricted rewrite system:
20.31/6.30	The TRS R consists of the following rules:
20.31/6.30	
20.31/6.30	   1(1(x1)) -> 3(4(x1))
20.31/6.30	   2(1(x1)) -> 1(2(x1))
20.31/6.30	   4(3(x1)) -> 1(1(x1))
20.31/6.30	
20.31/6.30	Q is empty.
20.31/6.30	
20.31/6.30	----------------------------------------
20.31/6.30	
20.31/6.30	(5) DependencyPairsProof (EQUIVALENT)
20.31/6.30	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
20.31/6.30	----------------------------------------
20.31/6.30	
20.31/6.30	(6)
20.31/6.30	Obligation:
20.31/6.30	Q DP problem:
20.31/6.30	The TRS P consists of the following rules:
20.31/6.30	
20.31/6.30	   1^1(1(x1)) -> 4^1(x1)
20.31/6.30	   2^1(1(x1)) -> 1^1(2(x1))
20.31/6.30	   2^1(1(x1)) -> 2^1(x1)
20.31/6.30	   4^1(3(x1)) -> 1^1(1(x1))
20.31/6.30	   4^1(3(x1)) -> 1^1(x1)
20.31/6.30	
20.31/6.30	The TRS R consists of the following rules:
20.31/6.30	
20.31/6.30	   1(1(x1)) -> 3(4(x1))
20.31/6.30	   2(1(x1)) -> 1(2(x1))
20.31/6.30	   4(3(x1)) -> 1(1(x1))
20.31/6.30	
20.31/6.30	Q is empty.
20.31/6.30	We have to consider all minimal (P,Q,R)-chains.
20.31/6.30	----------------------------------------
20.31/6.30	
20.31/6.30	(7) DependencyGraphProof (EQUIVALENT)
20.31/6.30	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
20.31/6.30	----------------------------------------
20.31/6.30	
20.31/6.30	(8)
20.31/6.30	Complex Obligation (AND)
20.31/6.30	
20.31/6.30	----------------------------------------
20.31/6.30	
20.31/6.30	(9)
20.31/6.30	Obligation:
20.31/6.30	Q DP problem:
20.31/6.30	The TRS P consists of the following rules:
20.31/6.30	
20.31/6.30	   4^1(3(x1)) -> 1^1(1(x1))
20.31/6.30	   1^1(1(x1)) -> 4^1(x1)
20.31/6.30	   4^1(3(x1)) -> 1^1(x1)
20.31/6.30	
20.31/6.30	The TRS R consists of the following rules:
20.31/6.30	
20.31/6.30	   1(1(x1)) -> 3(4(x1))
20.31/6.30	   2(1(x1)) -> 1(2(x1))
20.31/6.30	   4(3(x1)) -> 1(1(x1))
20.31/6.30	
20.31/6.30	Q is empty.
20.31/6.30	We have to consider all minimal (P,Q,R)-chains.
20.31/6.30	----------------------------------------
20.31/6.30	
20.31/6.30	(10) UsableRulesProof (EQUIVALENT)
20.31/6.30	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.
20.31/6.30	----------------------------------------
20.31/6.30	
20.31/6.30	(11)
20.31/6.30	Obligation:
20.31/6.30	Q DP problem:
20.31/6.30	The TRS P consists of the following rules:
20.31/6.30	
20.31/6.30	   4^1(3(x1)) -> 1^1(1(x1))
20.31/6.30	   1^1(1(x1)) -> 4^1(x1)
20.31/6.30	   4^1(3(x1)) -> 1^1(x1)
20.31/6.30	
20.31/6.30	The TRS R consists of the following rules:
20.31/6.30	
20.31/6.30	   1(1(x1)) -> 3(4(x1))
20.31/6.30	   4(3(x1)) -> 1(1(x1))
20.31/6.30	
20.31/6.30	Q is empty.
20.31/6.30	We have to consider all minimal (P,Q,R)-chains.
20.31/6.30	----------------------------------------
20.31/6.30	
20.31/6.30	(12) MRRProof (EQUIVALENT)
20.31/6.30	By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
20.31/6.30	
20.31/6.30	Strictly oriented dependency pairs:
20.31/6.30	
20.31/6.30	   4^1(3(x1)) -> 1^1(1(x1))
20.31/6.30	   1^1(1(x1)) -> 4^1(x1)
20.31/6.30	   4^1(3(x1)) -> 1^1(x1)
20.31/6.30	
20.31/6.30	Strictly oriented rules of the TRS R:
20.31/6.30	
20.31/6.30	   1(1(x1)) -> 3(4(x1))
20.31/6.30	   4(3(x1)) -> 1(1(x1))
20.31/6.30	
20.31/6.30	Used ordering: Polynomial interpretation [POLO]:
20.31/6.30	
20.31/6.30	   POL(1(x_1)) = 1 + 2*x_1
20.31/6.30	   POL(1^1(x_1)) = x_1
20.31/6.30	   POL(3(x_1)) = 2 + 2*x_1
20.31/6.30	   POL(4(x_1)) = 2*x_1
20.31/6.30	   POL(4^1(x_1)) = 2*x_1
20.31/6.30	
20.31/6.30	
20.31/6.30	----------------------------------------
20.31/6.30	
20.31/6.30	(13)
20.31/6.30	Obligation:
20.31/6.30	Q DP problem:
20.31/6.30	P is empty.
20.31/6.30	R is empty.
20.31/6.30	Q is empty.
20.31/6.30	We have to consider all minimal (P,Q,R)-chains.
20.31/6.30	----------------------------------------
20.31/6.30	
20.31/6.30	(14) PisEmptyProof (EQUIVALENT)
20.31/6.30	The TRS P is empty. Hence, there is no (P,Q,R) chain.
20.31/6.30	----------------------------------------
20.31/6.30	
20.31/6.30	(15)
20.31/6.30	YES
20.31/6.30	
20.31/6.30	----------------------------------------
20.31/6.30	
20.31/6.30	(16)
20.31/6.30	Obligation:
20.31/6.30	Q DP problem:
20.31/6.30	The TRS P consists of the following rules:
20.31/6.30	
20.31/6.30	   2^1(1(x1)) -> 2^1(x1)
20.31/6.30	
20.31/6.30	The TRS R consists of the following rules:
20.31/6.30	
20.31/6.30	   1(1(x1)) -> 3(4(x1))
20.31/6.30	   2(1(x1)) -> 1(2(x1))
20.31/6.30	   4(3(x1)) -> 1(1(x1))
20.31/6.30	
20.31/6.30	Q is empty.
20.31/6.30	We have to consider all minimal (P,Q,R)-chains.
20.31/6.30	----------------------------------------
20.31/6.30	
20.31/6.30	(17) UsableRulesProof (EQUIVALENT)
20.31/6.30	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.
20.31/6.30	----------------------------------------
20.31/6.30	
20.31/6.30	(18)
20.31/6.30	Obligation:
20.31/6.30	Q DP problem:
20.31/6.30	The TRS P consists of the following rules:
20.31/6.30	
20.31/6.30	   2^1(1(x1)) -> 2^1(x1)
20.31/6.30	
20.31/6.30	R is empty.
20.31/6.30	Q is empty.
20.31/6.30	We have to consider all minimal (P,Q,R)-chains.
20.31/6.30	----------------------------------------
20.31/6.30	
20.31/6.30	(19) QDPSizeChangeProof (EQUIVALENT)
20.31/6.30	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. 
20.31/6.30	
20.31/6.30	From the DPs we obtained the following set of size-change graphs:
20.31/6.30	*2^1(1(x1)) -> 2^1(x1)
20.31/6.30	The graph contains the following edges 1 > 1
20.31/6.30	
20.31/6.30	
20.31/6.30	----------------------------------------
20.31/6.30	
20.31/6.30	(20)
20.31/6.30	YES
20.62/6.83	EOF