47.10/12.98	YES
49.08/13.45	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
49.08/13.45	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
49.08/13.45	
49.08/13.45	
49.08/13.45	Termination w.r.t. Q of the given QTRS could be proven:
49.08/13.45	
49.08/13.45	(0) QTRS
49.08/13.45	(1) QTRSRRRProof [EQUIVALENT, 38 ms]
49.08/13.45	(2) QTRS
49.08/13.45	(3) QTRSRRRProof [EQUIVALENT, 2 ms]
49.08/13.45	(4) QTRS
49.08/13.45	(5) DependencyPairsProof [EQUIVALENT, 3 ms]
49.08/13.45	(6) QDP
49.08/13.45	(7) QDPOrderProof [EQUIVALENT, 73 ms]
49.08/13.45	(8) QDP
49.08/13.45	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
49.08/13.45	(10) AND
49.08/13.45	    (11) QDP
49.08/13.45	        (12) UsableRulesProof [EQUIVALENT, 0 ms]
49.08/13.45	        (13) QDP
49.08/13.45	        (14) QDPSizeChangeProof [EQUIVALENT, 0 ms]
49.08/13.45	        (15) YES
49.08/13.45	    (16) QDP
49.08/13.45	        (17) MRRProof [EQUIVALENT, 0 ms]
49.08/13.45	        (18) QDP
49.08/13.45	        (19) MRRProof [EQUIVALENT, 0 ms]
49.08/13.45	        (20) QDP
49.08/13.45	        (21) MNOCProof [EQUIVALENT, 0 ms]
49.08/13.45	        (22) QDP
49.08/13.45	        (23) QDPOrderProof [EQUIVALENT, 0 ms]
49.08/13.45	        (24) QDP
49.08/13.45	        (25) PisEmptyProof [EQUIVALENT, 0 ms]
49.08/13.45	        (26) YES
49.08/13.45	
49.08/13.45	
49.08/13.45	----------------------------------------
49.08/13.45	
49.08/13.45	(0)
49.08/13.45	Obligation:
49.08/13.45	Q restricted rewrite system:
49.08/13.45	The TRS R consists of the following rules:
49.08/13.45	
49.08/13.45	   P(x1) -> Q(Q(p(x1)))
49.08/13.45	   p(p(x1)) -> q(q(x1))
49.08/13.45	   p(Q(Q(x1))) -> Q(Q(p(x1)))
49.08/13.45	   Q(p(q(x1))) -> q(p(Q(x1)))
49.08/13.45	   q(q(p(x1))) -> p(q(q(x1)))
49.08/13.45	   q(Q(x1)) -> x1
49.08/13.45	   Q(q(x1)) -> x1
49.08/13.45	   p(P(x1)) -> x1
49.08/13.45	   P(p(x1)) -> x1
49.08/13.45	
49.08/13.45	Q is empty.
49.08/13.45	
49.08/13.45	----------------------------------------
49.08/13.45	
49.08/13.45	(1) QTRSRRRProof (EQUIVALENT)
49.08/13.45	Used ordering:
49.08/13.45	Polynomial interpretation [POLO]:
49.08/13.45	
49.08/13.45	   POL(P(x_1)) = 2 + x_1
49.08/13.45	   POL(Q(x_1)) = 1 + x_1
49.08/13.45	   POL(p(x_1)) = x_1
49.08/13.45	   POL(q(x_1)) = x_1
49.08/13.45	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
49.08/13.45	
49.08/13.45	   q(Q(x1)) -> x1
49.08/13.45	   Q(q(x1)) -> x1
49.08/13.45	   p(P(x1)) -> x1
49.08/13.45	   P(p(x1)) -> x1
49.08/13.45	
49.08/13.45	
49.08/13.45	
49.08/13.45	
49.08/13.45	----------------------------------------
49.08/13.45	
49.08/13.45	(2)
49.08/13.45	Obligation:
49.08/13.45	Q restricted rewrite system:
49.08/13.45	The TRS R consists of the following rules:
49.08/13.45	
49.08/13.45	   P(x1) -> Q(Q(p(x1)))
49.08/13.45	   p(p(x1)) -> q(q(x1))
49.08/13.45	   p(Q(Q(x1))) -> Q(Q(p(x1)))
49.08/13.45	   Q(p(q(x1))) -> q(p(Q(x1)))
49.08/13.45	   q(q(p(x1))) -> p(q(q(x1)))
49.08/13.45	
49.08/13.45	Q is empty.
49.08/13.45	
49.08/13.45	----------------------------------------
49.08/13.45	
49.08/13.45	(3) QTRSRRRProof (EQUIVALENT)
49.08/13.45	Used ordering:
49.08/13.45	Polynomial interpretation [POLO]:
49.08/13.45	
49.08/13.45	   POL(P(x_1)) = 2 + x_1
49.08/13.45	   POL(Q(x_1)) = x_1
49.08/13.45	   POL(p(x_1)) = 1 + x_1
49.08/13.45	   POL(q(x_1)) = x_1
49.08/13.45	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
49.08/13.45	
49.08/13.45	   P(x1) -> Q(Q(p(x1)))
49.08/13.45	   p(p(x1)) -> q(q(x1))
49.08/13.45	
49.08/13.45	
49.08/13.45	
49.08/13.45	
49.08/13.45	----------------------------------------
49.08/13.45	
49.08/13.45	(4)
49.08/13.45	Obligation:
49.08/13.45	Q restricted rewrite system:
49.08/13.45	The TRS R consists of the following rules:
49.08/13.45	
49.08/13.45	   p(Q(Q(x1))) -> Q(Q(p(x1)))
49.08/13.45	   Q(p(q(x1))) -> q(p(Q(x1)))
49.08/13.45	   q(q(p(x1))) -> p(q(q(x1)))
49.08/13.45	
49.08/13.45	Q is empty.
49.08/13.45	
49.08/13.45	----------------------------------------
49.08/13.45	
49.08/13.45	(5) DependencyPairsProof (EQUIVALENT)
49.08/13.45	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
49.08/13.45	----------------------------------------
49.08/13.45	
49.08/13.45	(6)
49.08/13.45	Obligation:
49.08/13.45	Q DP problem:
49.08/13.45	The TRS P consists of the following rules:
49.08/13.45	
49.08/13.45	   P(Q(Q(x1))) -> Q^1(Q(p(x1)))
49.08/13.45	   P(Q(Q(x1))) -> Q^1(p(x1))
49.08/13.45	   P(Q(Q(x1))) -> P(x1)
49.08/13.45	   Q^1(p(q(x1))) -> Q^2(p(Q(x1)))
49.08/13.45	   Q^1(p(q(x1))) -> P(Q(x1))
49.08/13.45	   Q^1(p(q(x1))) -> Q^1(x1)
49.08/13.45	   Q^2(q(p(x1))) -> P(q(q(x1)))
49.08/13.45	   Q^2(q(p(x1))) -> Q^2(q(x1))
49.08/13.45	   Q^2(q(p(x1))) -> Q^2(x1)
49.08/13.45	
49.08/13.45	The TRS R consists of the following rules:
49.08/13.45	
49.08/13.45	   p(Q(Q(x1))) -> Q(Q(p(x1)))
49.08/13.45	   Q(p(q(x1))) -> q(p(Q(x1)))
49.08/13.45	   q(q(p(x1))) -> p(q(q(x1)))
49.08/13.45	
49.08/13.45	Q is empty.
49.08/13.45	We have to consider all minimal (P,Q,R)-chains.
49.08/13.45	----------------------------------------
49.08/13.45	
49.08/13.45	(7) QDPOrderProof (EQUIVALENT)
49.08/13.45	We use the reduction pair processor [LPAR04,JAR06].
49.08/13.45	
49.08/13.45	
49.08/13.45	The following pairs can be oriented strictly and are deleted.
49.08/13.45	
49.08/13.45	   P(Q(Q(x1))) -> Q^1(Q(p(x1)))
49.08/13.45	   P(Q(Q(x1))) -> Q^1(p(x1))
49.08/13.45	   Q^1(p(q(x1))) -> Q^2(p(Q(x1)))
49.08/13.45	   Q^1(p(q(x1))) -> P(Q(x1))
49.08/13.45	   Q^1(p(q(x1))) -> Q^1(x1)
49.08/13.45	   Q^2(q(p(x1))) -> Q^2(x1)
49.08/13.45	The remaining pairs can at least be oriented weakly.
49.08/13.45	Used ordering:  Polynomial interpretation [POLO]:
49.08/13.45	
49.08/13.45	   POL(P(x_1)) = 1 + 2*x_1
49.08/13.45	   POL(Q(x_1)) = 2*x_1
49.08/13.45	   POL(Q^1(x_1)) = 4*x_1
49.08/13.45	   POL(Q^2(x_1)) = 3 + 4*x_1
49.08/13.45	   POL(p(x_1)) = x_1
49.08/13.45	   POL(q(x_1)) = 1 + 2*x_1
49.08/13.45	
49.08/13.45	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
49.08/13.45	
49.08/13.45	   Q(p(q(x1))) -> q(p(Q(x1)))
49.08/13.45	   q(q(p(x1))) -> p(q(q(x1)))
49.08/13.45	   p(Q(Q(x1))) -> Q(Q(p(x1)))
49.08/13.45	
49.08/13.45	
49.08/13.45	----------------------------------------
49.08/13.45	
49.08/13.45	(8)
49.08/13.45	Obligation:
49.08/13.45	Q DP problem:
49.08/13.45	The TRS P consists of the following rules:
49.08/13.45	
49.08/13.45	   P(Q(Q(x1))) -> P(x1)
49.08/13.45	   Q^2(q(p(x1))) -> P(q(q(x1)))
49.08/13.45	   Q^2(q(p(x1))) -> Q^2(q(x1))
49.08/13.45	
49.08/13.45	The TRS R consists of the following rules:
49.08/13.45	
49.08/13.45	   p(Q(Q(x1))) -> Q(Q(p(x1)))
49.08/13.45	   Q(p(q(x1))) -> q(p(Q(x1)))
49.08/13.45	   q(q(p(x1))) -> p(q(q(x1)))
49.08/13.45	
49.08/13.45	Q is empty.
49.08/13.45	We have to consider all minimal (P,Q,R)-chains.
49.08/13.45	----------------------------------------
49.08/13.45	
49.08/13.45	(9) DependencyGraphProof (EQUIVALENT)
49.08/13.45	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
49.08/13.45	----------------------------------------
49.08/13.45	
49.08/13.45	(10)
49.08/13.45	Complex Obligation (AND)
49.08/13.45	
49.08/13.45	----------------------------------------
49.08/13.45	
49.08/13.45	(11)
49.08/13.45	Obligation:
49.08/13.45	Q DP problem:
49.08/13.45	The TRS P consists of the following rules:
49.08/13.45	
49.08/13.45	   P(Q(Q(x1))) -> P(x1)
49.08/13.45	
49.08/13.45	The TRS R consists of the following rules:
49.08/13.45	
49.08/13.45	   p(Q(Q(x1))) -> Q(Q(p(x1)))
49.08/13.45	   Q(p(q(x1))) -> q(p(Q(x1)))
49.08/13.45	   q(q(p(x1))) -> p(q(q(x1)))
49.08/13.45	
49.08/13.45	Q is empty.
49.08/13.45	We have to consider all minimal (P,Q,R)-chains.
49.08/13.45	----------------------------------------
49.08/13.45	
49.08/13.45	(12) UsableRulesProof (EQUIVALENT)
49.08/13.45	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.
49.08/13.45	----------------------------------------
49.08/13.45	
49.08/13.45	(13)
49.08/13.45	Obligation:
49.08/13.45	Q DP problem:
49.08/13.45	The TRS P consists of the following rules:
49.08/13.45	
49.08/13.45	   P(Q(Q(x1))) -> P(x1)
49.08/13.45	
49.08/13.45	R is empty.
49.08/13.45	Q is empty.
49.08/13.45	We have to consider all minimal (P,Q,R)-chains.
49.08/13.45	----------------------------------------
49.08/13.45	
49.08/13.45	(14) QDPSizeChangeProof (EQUIVALENT)
49.08/13.45	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. 
49.08/13.45	
49.08/13.45	From the DPs we obtained the following set of size-change graphs:
49.08/13.45	*P(Q(Q(x1))) -> P(x1)
49.08/13.45	The graph contains the following edges 1 > 1
49.08/13.45	
49.08/13.45	
49.08/13.45	----------------------------------------
49.08/13.45	
49.08/13.45	(15)
49.08/13.45	YES
49.08/13.45	
49.08/13.45	----------------------------------------
49.08/13.45	
49.08/13.45	(16)
49.08/13.45	Obligation:
49.08/13.45	Q DP problem:
49.08/13.45	The TRS P consists of the following rules:
49.08/13.45	
49.08/13.45	   Q^2(q(p(x1))) -> Q^2(q(x1))
49.08/13.45	
49.08/13.45	The TRS R consists of the following rules:
49.08/13.45	
49.08/13.45	   p(Q(Q(x1))) -> Q(Q(p(x1)))
49.08/13.45	   Q(p(q(x1))) -> q(p(Q(x1)))
49.08/13.45	   q(q(p(x1))) -> p(q(q(x1)))
49.08/13.45	
49.08/13.45	Q is empty.
49.08/13.45	We have to consider all minimal (P,Q,R)-chains.
49.08/13.45	----------------------------------------
49.08/13.45	
49.08/13.45	(17) MRRProof (EQUIVALENT)
49.08/13.45	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.
49.08/13.45	
49.08/13.45	
49.08/13.45	Strictly oriented rules of the TRS R:
49.08/13.45	
49.08/13.45	   Q(p(q(x1))) -> q(p(Q(x1)))
49.08/13.45	
49.08/13.45	Used ordering: Polynomial interpretation [POLO]:
49.08/13.45	
49.08/13.45	   POL(Q(x_1)) = 2*x_1
49.08/13.45	   POL(Q^2(x_1)) = x_1
49.08/13.45	   POL(p(x_1)) = x_1
49.08/13.45	   POL(q(x_1)) = 2 + 2*x_1
49.08/13.45	
49.08/13.45	
49.08/13.45	----------------------------------------
49.08/13.45	
49.08/13.45	(18)
49.08/13.45	Obligation:
49.08/13.45	Q DP problem:
49.08/13.45	The TRS P consists of the following rules:
49.08/13.45	
49.08/13.45	   Q^2(q(p(x1))) -> Q^2(q(x1))
49.08/13.45	
49.08/13.45	The TRS R consists of the following rules:
49.08/13.45	
49.08/13.45	   p(Q(Q(x1))) -> Q(Q(p(x1)))
49.08/13.45	   q(q(p(x1))) -> p(q(q(x1)))
49.08/13.45	
49.08/13.45	Q is empty.
49.08/13.45	We have to consider all minimal (P,Q,R)-chains.
49.08/13.45	----------------------------------------
49.08/13.45	
49.08/13.45	(19) MRRProof (EQUIVALENT)
49.08/13.45	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.
49.08/13.45	
49.08/13.45	
49.08/13.45	Strictly oriented rules of the TRS R:
49.08/13.45	
49.08/13.45	   p(Q(Q(x1))) -> Q(Q(p(x1)))
49.08/13.45	
49.08/13.45	Used ordering: Polynomial interpretation [POLO]:
49.08/13.45	
49.08/13.45	   POL(Q(x_1)) = 2 + 2*x_1
49.08/13.45	   POL(Q^2(x_1)) = x_1
49.08/13.45	   POL(p(x_1)) = 2*x_1
49.08/13.45	   POL(q(x_1)) = 2*x_1
49.08/13.45	
49.08/13.45	
49.08/13.45	----------------------------------------
49.08/13.45	
49.08/13.45	(20)
49.08/13.45	Obligation:
49.08/13.45	Q DP problem:
49.08/13.45	The TRS P consists of the following rules:
49.08/13.45	
49.08/13.45	   Q^2(q(p(x1))) -> Q^2(q(x1))
49.08/13.45	
49.08/13.45	The TRS R consists of the following rules:
49.08/13.45	
49.08/13.45	   q(q(p(x1))) -> p(q(q(x1)))
49.08/13.45	
49.08/13.45	Q is empty.
49.08/13.45	We have to consider all minimal (P,Q,R)-chains.
49.08/13.45	----------------------------------------
49.08/13.45	
49.08/13.45	(21) MNOCProof (EQUIVALENT)
49.08/13.45	We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.
49.08/13.45	----------------------------------------
49.08/13.45	
49.08/13.45	(22)
49.08/13.45	Obligation:
49.08/13.45	Q DP problem:
49.08/13.45	The TRS P consists of the following rules:
49.08/13.45	
49.08/13.45	   Q^2(q(p(x1))) -> Q^2(q(x1))
49.08/13.45	
49.08/13.45	The TRS R consists of the following rules:
49.08/13.45	
49.08/13.45	   q(q(p(x1))) -> p(q(q(x1)))
49.08/13.45	
49.08/13.45	The set Q consists of the following terms:
49.08/13.45	
49.08/13.45	   q(q(p(x0)))
49.08/13.45	
49.08/13.45	We have to consider all minimal (P,Q,R)-chains.
49.08/13.45	----------------------------------------
49.08/13.45	
49.08/13.45	(23) QDPOrderProof (EQUIVALENT)
49.08/13.45	We use the reduction pair processor [LPAR04,JAR06].
49.08/13.45	
49.08/13.45	
49.08/13.45	The following pairs can be oriented strictly and are deleted.
49.08/13.45	
49.08/13.45	   Q^2(q(p(x1))) -> Q^2(q(x1))
49.08/13.45	The remaining pairs can at least be oriented weakly.
49.08/13.45	Used ordering:  Polynomial interpretation [POLO]:
49.08/13.45	
49.08/13.45	   POL(Q^2(x_1)) = x_1
49.08/13.45	   POL(p(x_1)) = 1 + x_1
49.08/13.45	   POL(q(x_1)) = x_1
49.08/13.45	
49.08/13.45	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
49.08/13.45	
49.08/13.45	   q(q(p(x1))) -> p(q(q(x1)))
49.08/13.45	
49.08/13.45	
49.08/13.45	----------------------------------------
49.08/13.45	
49.08/13.45	(24)
49.08/13.45	Obligation:
49.08/13.45	Q DP problem:
49.08/13.45	P is empty.
49.08/13.45	The TRS R consists of the following rules:
49.08/13.45	
49.08/13.45	   q(q(p(x1))) -> p(q(q(x1)))
49.08/13.45	
49.08/13.45	The set Q consists of the following terms:
49.08/13.45	
49.08/13.45	   q(q(p(x0)))
49.08/13.45	
49.08/13.45	We have to consider all minimal (P,Q,R)-chains.
49.08/13.45	----------------------------------------
49.08/13.45	
49.08/13.45	(25) PisEmptyProof (EQUIVALENT)
49.08/13.45	The TRS P is empty. Hence, there is no (P,Q,R) chain.
49.08/13.45	----------------------------------------
49.08/13.45	
49.08/13.45	(26)
49.08/13.45	YES
49.30/13.56	EOF