13.35/5.28	YES
13.35/5.29	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
13.35/5.29	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
13.35/5.29	
13.35/5.29	
13.35/5.29	Termination w.r.t. Q of the given QTRS could be proven:
13.35/5.29	
13.35/5.29	(0) QTRS
13.35/5.29	(1) QTRSRRRProof [EQUIVALENT, 39 ms]
13.35/5.29	(2) QTRS
13.35/5.29	(3) Overlay + Local Confluence [EQUIVALENT, 3 ms]
13.35/5.29	(4) QTRS
13.35/5.29	(5) DependencyPairsProof [EQUIVALENT, 4 ms]
13.35/5.29	(6) QDP
13.35/5.29	(7) DependencyGraphProof [EQUIVALENT, 3 ms]
13.35/5.29	(8) QDP
13.35/5.29	(9) UsableRulesProof [EQUIVALENT, 0 ms]
13.35/5.29	(10) QDP
13.35/5.29	(11) QReductionProof [EQUIVALENT, 0 ms]
13.35/5.29	(12) QDP
13.35/5.29	(13) QDPOrderProof [EQUIVALENT, 9 ms]
13.35/5.29	(14) QDP
13.35/5.29	(15) PisEmptyProof [EQUIVALENT, 0 ms]
13.35/5.29	(16) YES
13.35/5.29	
13.35/5.29	
13.35/5.29	----------------------------------------
13.35/5.29	
13.35/5.29	(0)
13.35/5.29	Obligation:
13.35/5.29	Q restricted rewrite system:
13.35/5.29	The TRS R consists of the following rules:
13.35/5.29	
13.35/5.29	   f(s(x1)) -> s(s(f(p(s(x1)))))
13.35/5.29	   f(0(x1)) -> 0(x1)
13.35/5.29	   p(s(x1)) -> x1
13.35/5.29	
13.35/5.29	Q is empty.
13.35/5.29	
13.35/5.29	----------------------------------------
13.35/5.29	
13.35/5.29	(1) QTRSRRRProof (EQUIVALENT)
13.35/5.29	Used ordering:
13.35/5.29	Polynomial interpretation [POLO]:
13.35/5.29	
13.35/5.29	   POL(0(x_1)) = x_1
13.35/5.29	   POL(f(x_1)) = 1 + x_1
13.35/5.29	   POL(p(x_1)) = x_1
13.35/5.29	   POL(s(x_1)) = x_1
13.35/5.29	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
13.35/5.29	
13.35/5.29	   f(0(x1)) -> 0(x1)
13.35/5.29	
13.35/5.29	
13.35/5.29	
13.35/5.29	
13.35/5.29	----------------------------------------
13.35/5.29	
13.35/5.29	(2)
13.35/5.29	Obligation:
13.35/5.29	Q restricted rewrite system:
13.35/5.29	The TRS R consists of the following rules:
13.35/5.29	
13.35/5.29	   f(s(x1)) -> s(s(f(p(s(x1)))))
13.35/5.29	   p(s(x1)) -> x1
13.35/5.29	
13.35/5.29	Q is empty.
13.35/5.29	
13.35/5.29	----------------------------------------
13.35/5.29	
13.35/5.29	(3) Overlay + Local Confluence (EQUIVALENT)
13.35/5.29	The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
13.35/5.29	----------------------------------------
13.35/5.29	
13.35/5.29	(4)
13.35/5.29	Obligation:
13.35/5.29	Q restricted rewrite system:
13.35/5.29	The TRS R consists of the following rules:
13.35/5.29	
13.35/5.29	   f(s(x1)) -> s(s(f(p(s(x1)))))
13.35/5.29	   p(s(x1)) -> x1
13.35/5.29	
13.35/5.29	The set Q consists of the following terms:
13.35/5.29	
13.35/5.29	   f(s(x0))
13.35/5.29	   p(s(x0))
13.35/5.29	
13.35/5.29	
13.35/5.29	----------------------------------------
13.35/5.29	
13.35/5.29	(5) DependencyPairsProof (EQUIVALENT)
13.35/5.29	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
13.35/5.29	----------------------------------------
13.35/5.29	
13.35/5.29	(6)
13.35/5.29	Obligation:
13.35/5.29	Q DP problem:
13.35/5.29	The TRS P consists of the following rules:
13.35/5.29	
13.35/5.29	   F(s(x1)) -> F(p(s(x1)))
13.35/5.29	   F(s(x1)) -> P(s(x1))
13.35/5.29	
13.35/5.29	The TRS R consists of the following rules:
13.35/5.29	
13.35/5.29	   f(s(x1)) -> s(s(f(p(s(x1)))))
13.35/5.29	   p(s(x1)) -> x1
13.35/5.29	
13.35/5.29	The set Q consists of the following terms:
13.35/5.29	
13.35/5.29	   f(s(x0))
13.35/5.29	   p(s(x0))
13.35/5.29	
13.35/5.29	We have to consider all minimal (P,Q,R)-chains.
13.35/5.29	----------------------------------------
13.35/5.29	
13.35/5.29	(7) DependencyGraphProof (EQUIVALENT)
13.35/5.29	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
13.35/5.29	----------------------------------------
13.35/5.29	
13.35/5.29	(8)
13.35/5.29	Obligation:
13.35/5.29	Q DP problem:
13.35/5.29	The TRS P consists of the following rules:
13.35/5.29	
13.35/5.29	   F(s(x1)) -> F(p(s(x1)))
13.35/5.29	
13.35/5.29	The TRS R consists of the following rules:
13.35/5.29	
13.35/5.29	   f(s(x1)) -> s(s(f(p(s(x1)))))
13.35/5.29	   p(s(x1)) -> x1
13.35/5.29	
13.35/5.29	The set Q consists of the following terms:
13.35/5.29	
13.35/5.29	   f(s(x0))
13.35/5.29	   p(s(x0))
13.35/5.29	
13.35/5.29	We have to consider all minimal (P,Q,R)-chains.
13.35/5.29	----------------------------------------
13.35/5.29	
13.35/5.29	(9) UsableRulesProof (EQUIVALENT)
13.35/5.29	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
13.35/5.29	----------------------------------------
13.35/5.29	
13.35/5.29	(10)
13.35/5.29	Obligation:
13.35/5.29	Q DP problem:
13.35/5.29	The TRS P consists of the following rules:
13.35/5.29	
13.35/5.29	   F(s(x1)) -> F(p(s(x1)))
13.35/5.29	
13.35/5.29	The TRS R consists of the following rules:
13.35/5.29	
13.35/5.29	   p(s(x1)) -> x1
13.35/5.29	
13.35/5.29	The set Q consists of the following terms:
13.35/5.29	
13.35/5.29	   f(s(x0))
13.35/5.29	   p(s(x0))
13.35/5.29	
13.35/5.29	We have to consider all minimal (P,Q,R)-chains.
13.35/5.29	----------------------------------------
13.35/5.29	
13.35/5.29	(11) QReductionProof (EQUIVALENT)
13.35/5.29	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
13.35/5.29	
13.35/5.29	   f(s(x0))
13.35/5.29	
13.35/5.29	
13.35/5.29	----------------------------------------
13.35/5.29	
13.35/5.29	(12)
13.35/5.29	Obligation:
13.35/5.29	Q DP problem:
13.35/5.29	The TRS P consists of the following rules:
13.35/5.29	
13.35/5.29	   F(s(x1)) -> F(p(s(x1)))
13.35/5.29	
13.35/5.29	The TRS R consists of the following rules:
13.35/5.29	
13.35/5.29	   p(s(x1)) -> x1
13.35/5.29	
13.35/5.29	The set Q consists of the following terms:
13.35/5.29	
13.35/5.29	   p(s(x0))
13.35/5.29	
13.35/5.29	We have to consider all minimal (P,Q,R)-chains.
13.35/5.29	----------------------------------------
13.35/5.29	
13.35/5.29	(13) QDPOrderProof (EQUIVALENT)
13.35/5.29	We use the reduction pair processor [LPAR04,JAR06].
13.35/5.29	
13.35/5.29	
13.35/5.29	The following pairs can be oriented strictly and are deleted.
13.35/5.29	
13.35/5.29	   F(s(x1)) -> F(p(s(x1)))
13.35/5.29	The remaining pairs can at least be oriented weakly.
13.35/5.29	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
13.35/5.29	
13.35/5.29	POL( F_1(x_1) ) = x_1
13.35/5.29	POL( p_1(x_1) ) = max{0, x_1 - 1}
13.35/5.29	POL( s_1(x_1) ) = x_1 + 1
13.35/5.29	
13.35/5.29	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
13.35/5.29	
13.35/5.29	   p(s(x1)) -> x1
13.35/5.29	
13.35/5.29	
13.35/5.29	----------------------------------------
13.35/5.29	
13.35/5.29	(14)
13.35/5.29	Obligation:
13.35/5.29	Q DP problem:
13.35/5.29	P is empty.
13.35/5.29	The TRS R consists of the following rules:
13.35/5.29	
13.35/5.29	   p(s(x1)) -> x1
13.35/5.29	
13.35/5.29	The set Q consists of the following terms:
13.35/5.29	
13.35/5.29	   p(s(x0))
13.35/5.29	
13.35/5.29	We have to consider all minimal (P,Q,R)-chains.
13.35/5.29	----------------------------------------
13.35/5.29	
13.35/5.29	(15) PisEmptyProof (EQUIVALENT)
13.35/5.29	The TRS P is empty. Hence, there is no (P,Q,R) chain.
13.35/5.29	----------------------------------------
13.35/5.29	
13.35/5.29	(16)
13.35/5.29	YES
13.88/7.10	EOF