3.41/1.62	YES
3.41/1.63	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.41/1.63	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.41/1.63	
3.41/1.63	
3.41/1.63	Termination w.r.t. Q of the given QTRS could be proven:
3.41/1.63	
3.41/1.63	(0) QTRS
3.41/1.63	(1) DependencyPairsProof [EQUIVALENT, 0 ms]
3.41/1.63	(2) QDP
3.41/1.63	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
3.41/1.63	(4) QDP
3.41/1.63	(5) UsableRulesProof [EQUIVALENT, 0 ms]
3.41/1.63	(6) QDP
3.41/1.63	(7) QReductionProof [EQUIVALENT, 0 ms]
3.41/1.63	(8) QDP
3.41/1.63	(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.41/1.63	(10) YES
3.41/1.63	
3.41/1.63	
3.41/1.63	----------------------------------------
3.41/1.63	
3.41/1.63	(0)
3.41/1.63	Obligation:
3.41/1.63	Q restricted rewrite system:
3.41/1.63	The TRS R consists of the following rules:
3.41/1.63	
3.41/1.63	   f(g(x), s(0)) -> f(g(x), g(x))
3.41/1.63	   g(s(x)) -> s(g(x))
3.41/1.63	   g(0) -> 0
3.41/1.63	
3.41/1.63	The set Q consists of the following terms:
3.41/1.63	
3.41/1.63	   f(g(x0), s(0))
3.41/1.63	   g(s(x0))
3.41/1.63	   g(0)
3.41/1.63	
3.41/1.63	
3.41/1.63	----------------------------------------
3.41/1.63	
3.41/1.63	(1) DependencyPairsProof (EQUIVALENT)
3.41/1.63	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
3.41/1.63	----------------------------------------
3.41/1.63	
3.41/1.63	(2)
3.41/1.63	Obligation:
3.41/1.63	Q DP problem:
3.41/1.63	The TRS P consists of the following rules:
3.41/1.63	
3.41/1.63	   F(g(x), s(0)) -> F(g(x), g(x))
3.41/1.63	   G(s(x)) -> G(x)
3.41/1.63	
3.41/1.63	The TRS R consists of the following rules:
3.41/1.63	
3.41/1.63	   f(g(x), s(0)) -> f(g(x), g(x))
3.41/1.63	   g(s(x)) -> s(g(x))
3.41/1.63	   g(0) -> 0
3.41/1.63	
3.41/1.63	The set Q consists of the following terms:
3.41/1.63	
3.41/1.63	   f(g(x0), s(0))
3.41/1.63	   g(s(x0))
3.41/1.63	   g(0)
3.41/1.63	
3.41/1.63	We have to consider all minimal (P,Q,R)-chains.
3.41/1.63	----------------------------------------
3.41/1.63	
3.41/1.63	(3) DependencyGraphProof (EQUIVALENT)
3.41/1.63	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
3.41/1.63	----------------------------------------
3.41/1.63	
3.41/1.63	(4)
3.41/1.63	Obligation:
3.41/1.63	Q DP problem:
3.41/1.63	The TRS P consists of the following rules:
3.41/1.63	
3.41/1.63	   G(s(x)) -> G(x)
3.41/1.63	
3.41/1.63	The TRS R consists of the following rules:
3.41/1.63	
3.41/1.63	   f(g(x), s(0)) -> f(g(x), g(x))
3.41/1.63	   g(s(x)) -> s(g(x))
3.41/1.63	   g(0) -> 0
3.41/1.63	
3.41/1.63	The set Q consists of the following terms:
3.41/1.63	
3.41/1.63	   f(g(x0), s(0))
3.41/1.63	   g(s(x0))
3.41/1.63	   g(0)
3.41/1.63	
3.41/1.63	We have to consider all minimal (P,Q,R)-chains.
3.41/1.63	----------------------------------------
3.41/1.63	
3.41/1.63	(5) UsableRulesProof (EQUIVALENT)
3.41/1.63	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.
3.41/1.63	----------------------------------------
3.41/1.63	
3.41/1.63	(6)
3.41/1.63	Obligation:
3.41/1.63	Q DP problem:
3.41/1.63	The TRS P consists of the following rules:
3.41/1.63	
3.41/1.63	   G(s(x)) -> G(x)
3.41/1.63	
3.41/1.63	R is empty.
3.41/1.63	The set Q consists of the following terms:
3.41/1.63	
3.41/1.63	   f(g(x0), s(0))
3.41/1.63	   g(s(x0))
3.41/1.63	   g(0)
3.41/1.63	
3.41/1.63	We have to consider all minimal (P,Q,R)-chains.
3.41/1.63	----------------------------------------
3.41/1.63	
3.41/1.63	(7) QReductionProof (EQUIVALENT)
3.41/1.63	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
3.41/1.63	
3.41/1.63	   f(g(x0), s(0))
3.41/1.63	   g(s(x0))
3.41/1.63	   g(0)
3.41/1.63	
3.41/1.63	
3.41/1.63	----------------------------------------
3.41/1.63	
3.41/1.63	(8)
3.41/1.63	Obligation:
3.41/1.63	Q DP problem:
3.41/1.63	The TRS P consists of the following rules:
3.41/1.63	
3.41/1.63	   G(s(x)) -> G(x)
3.41/1.63	
3.41/1.63	R is empty.
3.41/1.63	Q is empty.
3.41/1.63	We have to consider all minimal (P,Q,R)-chains.
3.41/1.63	----------------------------------------
3.41/1.63	
3.41/1.63	(9) QDPSizeChangeProof (EQUIVALENT)
3.41/1.63	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. 
3.41/1.63	
3.41/1.63	From the DPs we obtained the following set of size-change graphs:
3.41/1.63	*G(s(x)) -> G(x)
3.41/1.63	The graph contains the following edges 1 > 1
3.41/1.63	
3.41/1.63	
3.41/1.63	----------------------------------------
3.41/1.63	
3.41/1.63	(10)
3.41/1.63	YES
3.41/1.65	EOF