3.40/2.22	WORST_CASE(NON_POLY, ?)
3.40/2.23	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.40/2.23	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.40/2.23	
3.40/2.23	
3.40/2.23	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.40/2.23	
3.40/2.23	(0) CpxTRS
3.40/2.23	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.40/2.23	(2) TRS for Loop Detection
3.40/2.23	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.40/2.23	(4) BEST
3.40/2.23	    (5) proven lower bound
3.40/2.23	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.40/2.23	        (7) BOUNDS(n^1, INF)
3.40/2.23	    (8) TRS for Loop Detection
3.40/2.23	        (9) InfiniteLowerBoundProof [FINISHED, 0 ms]
3.40/2.23	        (10) BOUNDS(INF, INF)
3.40/2.23	
3.40/2.23	
3.40/2.23	----------------------------------------
3.40/2.23	
3.40/2.23	(0)
3.40/2.23	Obligation:
3.40/2.23	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.40/2.23	
3.40/2.23	
3.40/2.23	The TRS R consists of the following rules:
3.40/2.23	
3.40/2.23	   f(X) -> cons(X, f(g(X)))
3.40/2.23	   g(0) -> s(0)
3.40/2.23	   g(s(X)) -> s(s(g(X)))
3.40/2.23	   sel(0, cons(X, Y)) -> X
3.40/2.23	   sel(s(X), cons(Y, Z)) -> sel(X, Z)
3.40/2.23	
3.40/2.23	S is empty.
3.40/2.23	Rewrite Strategy: FULL
3.40/2.23	----------------------------------------
3.40/2.23	
3.40/2.23	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.40/2.23	Transformed a relative TRS into a decreasing-loop problem.
3.40/2.23	----------------------------------------
3.40/2.23	
3.40/2.23	(2)
3.40/2.23	Obligation:
3.40/2.23	Analyzing the following TRS for decreasing loops:
3.40/2.23	
3.40/2.23	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.40/2.23	
3.40/2.23	
3.40/2.23	The TRS R consists of the following rules:
3.40/2.23	
3.40/2.23	   f(X) -> cons(X, f(g(X)))
3.40/2.23	   g(0) -> s(0)
3.40/2.23	   g(s(X)) -> s(s(g(X)))
3.40/2.23	   sel(0, cons(X, Y)) -> X
3.40/2.23	   sel(s(X), cons(Y, Z)) -> sel(X, Z)
3.40/2.23	
3.40/2.23	S is empty.
3.40/2.23	Rewrite Strategy: FULL
3.40/2.23	----------------------------------------
3.40/2.23	
3.40/2.23	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.40/2.23	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.40/2.23	
3.40/2.23	The rewrite sequence
3.40/2.23	
3.40/2.23	sel(s(X), cons(Y, Z)) ->^+ sel(X, Z)
3.40/2.23	
3.40/2.23	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
3.40/2.23	
3.40/2.23	The pumping substitution is [X / s(X), Z / cons(Y, Z)].
3.40/2.23	
3.40/2.23	The result substitution is [ ].
3.40/2.23	
3.40/2.23	
3.40/2.23	
3.40/2.23	
3.40/2.23	----------------------------------------
3.40/2.23	
3.40/2.23	(4)
3.40/2.23	Complex Obligation (BEST)
3.40/2.23	
3.40/2.23	----------------------------------------
3.40/2.23	
3.40/2.23	(5)
3.40/2.23	Obligation:
3.40/2.23	Proved the lower bound n^1 for the following obligation:
3.40/2.23	
3.40/2.23	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.40/2.23	
3.40/2.23	
3.40/2.23	The TRS R consists of the following rules:
3.40/2.23	
3.40/2.23	   f(X) -> cons(X, f(g(X)))
3.40/2.23	   g(0) -> s(0)
3.40/2.23	   g(s(X)) -> s(s(g(X)))
3.40/2.23	   sel(0, cons(X, Y)) -> X
3.40/2.23	   sel(s(X), cons(Y, Z)) -> sel(X, Z)
3.40/2.23	
3.40/2.23	S is empty.
3.40/2.23	Rewrite Strategy: FULL
3.40/2.23	----------------------------------------
3.40/2.23	
3.40/2.23	(6) LowerBoundPropagationProof (FINISHED)
3.40/2.23	Propagated lower bound.
3.40/2.23	----------------------------------------
3.40/2.23	
3.40/2.23	(7)
3.40/2.23	BOUNDS(n^1, INF)
3.40/2.23	
3.40/2.23	----------------------------------------
3.40/2.23	
3.40/2.23	(8)
3.40/2.23	Obligation:
3.40/2.23	Analyzing the following TRS for decreasing loops:
3.40/2.23	
3.40/2.23	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.40/2.23	
3.40/2.23	
3.40/2.23	The TRS R consists of the following rules:
3.40/2.23	
3.40/2.23	   f(X) -> cons(X, f(g(X)))
3.40/2.23	   g(0) -> s(0)
3.40/2.23	   g(s(X)) -> s(s(g(X)))
3.40/2.23	   sel(0, cons(X, Y)) -> X
3.40/2.23	   sel(s(X), cons(Y, Z)) -> sel(X, Z)
3.40/2.23	
3.40/2.23	S is empty.
3.40/2.23	Rewrite Strategy: FULL
3.40/2.23	----------------------------------------
3.40/2.23	
3.40/2.23	(9) InfiniteLowerBoundProof (FINISHED)
3.40/2.23	The following loop proves infinite runtime complexity:
3.40/2.23	
3.40/2.23	The rewrite sequence
3.40/2.23	
3.40/2.23	f(X) ->^+ cons(X, f(g(X)))
3.40/2.23	
3.40/2.23	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
3.40/2.23	
3.40/2.23	The pumping substitution is [ ].
3.40/2.23	
3.40/2.23	The result substitution is [X / g(X)].
3.40/2.23	
3.40/2.23	
3.40/2.23	
3.40/2.23	
3.40/2.23	----------------------------------------
3.40/2.23	
3.40/2.23	(10)
3.40/2.23	BOUNDS(INF, INF)
3.40/2.25	EOF