3.44/1.68	WORST_CASE(NON_POLY, ?)
3.44/1.69	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.44/1.69	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.44/1.69	
3.44/1.69	
3.44/1.69	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.44/1.69	
3.44/1.69	(0) CpxTRS
3.44/1.69	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.44/1.69	(2) TRS for Loop Detection
3.44/1.69	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.44/1.69	(4) BEST
3.44/1.69	    (5) proven lower bound
3.44/1.69	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.44/1.69	        (7) BOUNDS(n^1, INF)
3.44/1.69	    (8) TRS for Loop Detection
3.44/1.69	        (9) InfiniteLowerBoundProof [FINISHED, 7 ms]
3.44/1.69	        (10) BOUNDS(INF, INF)
3.44/1.69	
3.44/1.69	
3.44/1.69	----------------------------------------
3.44/1.69	
3.44/1.69	(0)
3.44/1.69	Obligation:
3.44/1.69	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.44/1.69	
3.44/1.69	
3.44/1.69	The TRS R consists of the following rules:
3.44/1.69	
3.44/1.69	   inc(Cons(x, xs)) -> Cons(Cons(Nil, Nil), inc(xs))
3.44/1.69	   nestinc(Nil) -> number17(Nil)
3.44/1.69	   nestinc(Cons(x, xs)) -> nestinc(inc(Cons(x, xs)))
3.44/1.69	   inc(Nil) -> Cons(Nil, Nil)
3.44/1.69	   number17(x) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))
3.44/1.69	   goal(x) -> nestinc(x)
3.44/1.69	
3.44/1.69	S is empty.
3.44/1.69	Rewrite Strategy: INNERMOST
3.44/1.69	----------------------------------------
3.44/1.69	
3.44/1.69	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.44/1.69	Transformed a relative TRS into a decreasing-loop problem.
3.44/1.69	----------------------------------------
3.44/1.69	
3.44/1.69	(2)
3.44/1.69	Obligation:
3.44/1.69	Analyzing the following TRS for decreasing loops:
3.44/1.69	
3.44/1.69	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.44/1.69	
3.44/1.69	
3.44/1.69	The TRS R consists of the following rules:
3.44/1.69	
3.44/1.69	   inc(Cons(x, xs)) -> Cons(Cons(Nil, Nil), inc(xs))
3.44/1.69	   nestinc(Nil) -> number17(Nil)
3.44/1.69	   nestinc(Cons(x, xs)) -> nestinc(inc(Cons(x, xs)))
3.44/1.69	   inc(Nil) -> Cons(Nil, Nil)
3.44/1.69	   number17(x) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))
3.44/1.69	   goal(x) -> nestinc(x)
3.44/1.69	
3.44/1.69	S is empty.
3.44/1.69	Rewrite Strategy: INNERMOST
3.44/1.69	----------------------------------------
3.44/1.69	
3.44/1.69	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.44/1.69	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.44/1.69	
3.44/1.69	The rewrite sequence
3.44/1.69	
3.44/1.69	inc(Cons(x, xs)) ->^+ Cons(Cons(Nil, Nil), inc(xs))
3.44/1.69	
3.44/1.69	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
3.44/1.69	
3.44/1.69	The pumping substitution is [xs / Cons(x, xs)].
3.44/1.69	
3.44/1.69	The result substitution is [ ].
3.44/1.69	
3.44/1.69	
3.44/1.69	
3.44/1.69	
3.44/1.69	----------------------------------------
3.44/1.69	
3.44/1.69	(4)
3.44/1.69	Complex Obligation (BEST)
3.44/1.69	
3.44/1.69	----------------------------------------
3.44/1.69	
3.44/1.69	(5)
3.44/1.69	Obligation:
3.44/1.69	Proved the lower bound n^1 for the following obligation:
3.44/1.69	
3.44/1.69	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.44/1.69	
3.44/1.69	
3.44/1.69	The TRS R consists of the following rules:
3.44/1.69	
3.44/1.69	   inc(Cons(x, xs)) -> Cons(Cons(Nil, Nil), inc(xs))
3.44/1.69	   nestinc(Nil) -> number17(Nil)
3.44/1.69	   nestinc(Cons(x, xs)) -> nestinc(inc(Cons(x, xs)))
3.44/1.69	   inc(Nil) -> Cons(Nil, Nil)
3.44/1.69	   number17(x) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))
3.44/1.69	   goal(x) -> nestinc(x)
3.44/1.69	
3.44/1.69	S is empty.
3.44/1.69	Rewrite Strategy: INNERMOST
3.44/1.69	----------------------------------------
3.44/1.69	
3.44/1.69	(6) LowerBoundPropagationProof (FINISHED)
3.44/1.69	Propagated lower bound.
3.44/1.69	----------------------------------------
3.44/1.69	
3.44/1.69	(7)
3.44/1.69	BOUNDS(n^1, INF)
3.44/1.69	
3.44/1.69	----------------------------------------
3.44/1.69	
3.44/1.69	(8)
3.44/1.69	Obligation:
3.44/1.69	Analyzing the following TRS for decreasing loops:
3.44/1.69	
3.44/1.69	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.44/1.69	
3.44/1.69	
3.44/1.69	The TRS R consists of the following rules:
3.44/1.69	
3.44/1.69	   inc(Cons(x, xs)) -> Cons(Cons(Nil, Nil), inc(xs))
3.44/1.69	   nestinc(Nil) -> number17(Nil)
3.44/1.69	   nestinc(Cons(x, xs)) -> nestinc(inc(Cons(x, xs)))
3.44/1.69	   inc(Nil) -> Cons(Nil, Nil)
3.44/1.69	   number17(x) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))
3.44/1.69	   goal(x) -> nestinc(x)
3.44/1.69	
3.44/1.69	S is empty.
3.44/1.69	Rewrite Strategy: INNERMOST
3.44/1.69	----------------------------------------
3.44/1.69	
3.44/1.69	(9) InfiniteLowerBoundProof (FINISHED)
3.44/1.69	The following loop proves infinite runtime complexity:
3.44/1.69	
3.44/1.69	The rewrite sequence
3.44/1.69	
3.44/1.69	nestinc(Cons(x, xs)) ->^+ nestinc(Cons(Cons(Nil, Nil), inc(xs)))
3.44/1.69	
3.44/1.69	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
3.44/1.69	
3.44/1.69	The pumping substitution is [ ].
3.44/1.69	
3.44/1.69	The result substitution is [x / Cons(Nil, Nil), xs / inc(xs)].
3.44/1.69	
3.44/1.69	
3.44/1.69	
3.44/1.69	
3.44/1.69	----------------------------------------
3.44/1.69	
3.44/1.69	(10)
3.44/1.69	BOUNDS(INF, INF)
3.88/1.73	EOF