6.37/2.70	MAYBE
6.37/2.71	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
6.37/2.71	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.37/2.71	
6.37/2.71	
6.37/2.71	Quasi decreasingness of the given CTRS could not be shown:
6.37/2.71	
6.37/2.71	(0) CTRS
6.37/2.71	(1) CTRSToQTRSProof [SOUND, 0 ms]
6.37/2.71	(2) QTRS
6.37/2.71	(3) Overlay + Local Confluence [EQUIVALENT, 0 ms]
6.37/2.71	(4) QTRS
6.37/2.71	(5) DependencyPairsProof [EQUIVALENT, 0 ms]
6.37/2.71	(6) QDP
6.37/2.71	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
6.37/2.71	(8) QDP
6.37/2.71	(9) UsableRulesProof [EQUIVALENT, 0 ms]
6.37/2.71	(10) QDP
6.37/2.71	(11) QReductionProof [EQUIVALENT, 0 ms]
6.37/2.71	(12) QDP
6.37/2.71	(13) TransformationProof [EQUIVALENT, 0 ms]
6.37/2.71	(14) QDP
6.37/2.71	(15) NonTerminationLoopProof [COMPLETE, 0 ms]
6.37/2.71	(16) NO
6.37/2.71	
6.37/2.71	
6.37/2.71	----------------------------------------
6.37/2.71	
6.37/2.71	(0)
6.37/2.71	Obligation:
6.37/2.71	Conditional term rewrite system:
6.37/2.71	The TRS R consists of the following rules:
6.37/2.71	
6.37/2.71	   g(a) -> c(b)
6.37/2.71	   b -> f(a)
6.37/2.71	
6.37/2.71	The conditional TRS C consists of the following conditional rules:
6.37/2.71	
6.37/2.71	   f(x) -> y <= g(x) -> c(y)
6.37/2.71	
6.37/2.71	
6.37/2.71	----------------------------------------
6.37/2.71	
6.37/2.71	(1) CTRSToQTRSProof (SOUND)
6.37/2.71	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
6.37/2.71	----------------------------------------
6.37/2.71	
6.37/2.71	(2)
6.37/2.71	Obligation:
6.37/2.71	Q restricted rewrite system:
6.37/2.71	The TRS R consists of the following rules:
6.37/2.71	
6.37/2.71	   f(x) -> U1(g(x))
6.37/2.71	   U1(c(y)) -> y
6.37/2.71	   g(a) -> c(b)
6.37/2.71	   b -> f(a)
6.37/2.71	
6.37/2.71	Q is empty.
6.37/2.71	
6.37/2.71	----------------------------------------
6.37/2.71	
6.37/2.71	(3) Overlay + Local Confluence (EQUIVALENT)
6.37/2.71	The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
6.37/2.71	----------------------------------------
6.37/2.71	
6.37/2.71	(4)
6.37/2.71	Obligation:
6.37/2.71	Q restricted rewrite system:
6.37/2.71	The TRS R consists of the following rules:
6.37/2.71	
6.37/2.71	   f(x) -> U1(g(x))
6.37/2.71	   U1(c(y)) -> y
6.37/2.71	   g(a) -> c(b)
6.37/2.71	   b -> f(a)
6.37/2.71	
6.37/2.71	The set Q consists of the following terms:
6.37/2.71	
6.37/2.71	   f(x0)
6.37/2.71	   U1(c(x0))
6.37/2.71	   g(a)
6.37/2.71	   b
6.37/2.71	
6.37/2.71	
6.37/2.71	----------------------------------------
6.37/2.71	
6.37/2.71	(5) DependencyPairsProof (EQUIVALENT)
6.37/2.71	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
6.37/2.71	----------------------------------------
6.37/2.71	
6.37/2.71	(6)
6.37/2.71	Obligation:
6.37/2.71	Q DP problem:
6.37/2.71	The TRS P consists of the following rules:
6.37/2.71	
6.37/2.71	   F(x) -> U1^1(g(x))
6.37/2.71	   F(x) -> G(x)
6.37/2.71	   G(a) -> B
6.37/2.71	   B -> F(a)
6.37/2.71	
6.37/2.71	The TRS R consists of the following rules:
6.37/2.71	
6.37/2.71	   f(x) -> U1(g(x))
6.37/2.71	   U1(c(y)) -> y
6.37/2.71	   g(a) -> c(b)
6.37/2.71	   b -> f(a)
6.37/2.71	
6.37/2.71	The set Q consists of the following terms:
6.37/2.71	
6.37/2.71	   f(x0)
6.37/2.71	   U1(c(x0))
6.37/2.71	   g(a)
6.37/2.71	   b
6.37/2.71	
6.37/2.71	We have to consider all minimal (P,Q,R)-chains.
6.37/2.71	----------------------------------------
6.37/2.71	
6.37/2.71	(7) DependencyGraphProof (EQUIVALENT)
6.37/2.71	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
6.37/2.71	----------------------------------------
6.37/2.71	
6.37/2.71	(8)
6.37/2.71	Obligation:
6.37/2.71	Q DP problem:
6.37/2.71	The TRS P consists of the following rules:
6.37/2.71	
6.37/2.71	   F(x) -> G(x)
6.37/2.71	   G(a) -> B
6.37/2.71	   B -> F(a)
6.37/2.71	
6.37/2.71	The TRS R consists of the following rules:
6.37/2.71	
6.37/2.71	   f(x) -> U1(g(x))
6.37/2.71	   U1(c(y)) -> y
6.37/2.71	   g(a) -> c(b)
6.37/2.71	   b -> f(a)
6.37/2.71	
6.37/2.71	The set Q consists of the following terms:
6.37/2.71	
6.37/2.71	   f(x0)
6.37/2.71	   U1(c(x0))
6.37/2.71	   g(a)
6.37/2.71	   b
6.37/2.71	
6.37/2.71	We have to consider all minimal (P,Q,R)-chains.
6.37/2.71	----------------------------------------
6.37/2.71	
6.37/2.71	(9) UsableRulesProof (EQUIVALENT)
6.37/2.71	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.
6.37/2.71	----------------------------------------
6.37/2.71	
6.37/2.71	(10)
6.37/2.71	Obligation:
6.37/2.71	Q DP problem:
6.37/2.71	The TRS P consists of the following rules:
6.37/2.71	
6.37/2.71	   F(x) -> G(x)
6.37/2.71	   G(a) -> B
6.37/2.71	   B -> F(a)
6.37/2.71	
6.37/2.71	R is empty.
6.37/2.71	The set Q consists of the following terms:
6.37/2.71	
6.37/2.71	   f(x0)
6.37/2.71	   U1(c(x0))
6.37/2.71	   g(a)
6.37/2.71	   b
6.37/2.71	
6.37/2.71	We have to consider all minimal (P,Q,R)-chains.
6.37/2.71	----------------------------------------
6.37/2.71	
6.37/2.71	(11) QReductionProof (EQUIVALENT)
6.37/2.71	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
6.37/2.71	
6.37/2.71	   f(x0)
6.37/2.71	   U1(c(x0))
6.37/2.71	   g(a)
6.37/2.71	   b
6.37/2.71	
6.37/2.71	
6.37/2.71	----------------------------------------
6.37/2.71	
6.37/2.71	(12)
6.37/2.71	Obligation:
6.37/2.71	Q DP problem:
6.37/2.71	The TRS P consists of the following rules:
6.37/2.71	
6.37/2.71	   F(x) -> G(x)
6.37/2.71	   G(a) -> B
6.37/2.71	   B -> F(a)
6.37/2.71	
6.37/2.71	R is empty.
6.37/2.71	Q is empty.
6.37/2.71	We have to consider all minimal (P,Q,R)-chains.
6.37/2.71	----------------------------------------
6.37/2.71	
6.37/2.71	(13) TransformationProof (EQUIVALENT)
6.37/2.71	By instantiating [LPAR04] the rule F(x) -> G(x) we obtained the following new rules [LPAR04]:
6.37/2.71	
6.37/2.71	   (F(a) -> G(a),F(a) -> G(a))
6.37/2.71	
6.37/2.71	
6.37/2.71	----------------------------------------
6.37/2.71	
6.37/2.71	(14)
6.37/2.71	Obligation:
6.37/2.71	Q DP problem:
6.37/2.71	The TRS P consists of the following rules:
6.37/2.71	
6.37/2.71	   G(a) -> B
6.37/2.71	   B -> F(a)
6.37/2.71	   F(a) -> G(a)
6.37/2.71	
6.37/2.71	R is empty.
6.37/2.71	Q is empty.
6.37/2.71	We have to consider all minimal (P,Q,R)-chains.
6.37/2.71	----------------------------------------
6.37/2.71	
6.37/2.71	(15) NonTerminationLoopProof (COMPLETE)
6.37/2.71	We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
6.37/2.71	Found a loop by narrowing to the left:
6.37/2.71	
6.37/2.71	s = B evaluates to  t =B
6.37/2.71	
6.37/2.71	Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
6.37/2.71	* Matcher: [ ]
6.37/2.71	* Semiunifier: [ ]
6.37/2.71	
6.37/2.71	--------------------------------------------------------------------------------
6.37/2.71	Rewriting sequence
6.37/2.71	
6.37/2.71	B -> F(a)
6.37/2.71	with rule B -> F(a) at position [] and matcher [ ]
6.37/2.71	
6.37/2.71	F(a) -> G(a)
6.37/2.71	with rule F(a) -> G(a) at position [] and matcher [ ]
6.37/2.71	
6.37/2.71	G(a) -> B
6.37/2.71	with rule G(a) -> B
6.37/2.71	
6.37/2.71	Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence
6.37/2.71	
6.37/2.71	
6.37/2.71	All these steps are and every following step will be a correct step w.r.t to Q.
6.37/2.71	
6.37/2.71	
6.37/2.71	
6.37/2.71	
6.37/2.71	----------------------------------------
6.37/2.71	
6.37/2.71	(16)
6.37/2.71	NO
6.37/2.75	EOF