2.90/1.56	WORST_CASE(NON_POLY, ?)
2.90/1.57	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
2.90/1.57	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.90/1.57	
2.90/1.57	
2.90/1.57	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
2.90/1.57	
2.90/1.57	(0) CpxTRS
2.90/1.57	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
2.90/1.57	(2) TRS for Loop Detection
2.90/1.57	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
2.90/1.57	(4) BEST
2.90/1.57	    (5) proven lower bound
2.90/1.57	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
2.90/1.57	        (7) BOUNDS(n^1, INF)
2.90/1.57	    (8) TRS for Loop Detection
2.90/1.57	        (9) InfiniteLowerBoundProof [FINISHED, 0 ms]
2.90/1.57	        (10) BOUNDS(INF, INF)
2.90/1.57	
2.90/1.57	
2.90/1.57	----------------------------------------
2.90/1.57	
2.90/1.57	(0)
2.90/1.57	Obligation:
2.90/1.57	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
2.90/1.57	
2.90/1.57	
2.90/1.57	The TRS R consists of the following rules:
2.90/1.57	
2.90/1.57	   and(true, X) -> X
2.90/1.57	   and(false, Y) -> false
2.90/1.57	   if(true, X, Y) -> X
2.90/1.57	   if(false, X, Y) -> Y
2.90/1.57	   add(0, X) -> X
2.90/1.57	   add(s(X), Y) -> s(add(X, Y))
2.90/1.57	   first(0, X) -> nil
2.90/1.57	   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
2.90/1.57	   from(X) -> cons(X, from(s(X)))
2.90/1.57	
2.90/1.57	S is empty.
2.90/1.57	Rewrite Strategy: INNERMOST
2.90/1.57	----------------------------------------
2.90/1.57	
2.90/1.57	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
2.90/1.57	Transformed a relative TRS into a decreasing-loop problem.
2.90/1.57	----------------------------------------
2.90/1.57	
2.90/1.57	(2)
2.90/1.57	Obligation:
2.90/1.57	Analyzing the following TRS for decreasing loops:
2.90/1.57	
2.90/1.57	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
2.90/1.57	
2.90/1.57	
2.90/1.57	The TRS R consists of the following rules:
2.90/1.57	
2.90/1.57	   and(true, X) -> X
2.90/1.57	   and(false, Y) -> false
2.90/1.57	   if(true, X, Y) -> X
2.90/1.57	   if(false, X, Y) -> Y
2.90/1.57	   add(0, X) -> X
2.90/1.57	   add(s(X), Y) -> s(add(X, Y))
2.90/1.57	   first(0, X) -> nil
2.90/1.57	   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
2.90/1.57	   from(X) -> cons(X, from(s(X)))
2.90/1.57	
2.90/1.57	S is empty.
2.90/1.57	Rewrite Strategy: INNERMOST
2.90/1.57	----------------------------------------
2.90/1.57	
2.90/1.57	(3) DecreasingLoopProof (LOWER BOUND(ID))
2.90/1.57	The following loop(s) give(s) rise to the lower bound Omega(n^1):
2.90/1.57	
2.90/1.57	The rewrite sequence
2.90/1.57	
2.90/1.57	first(s(X), cons(Y, Z)) ->^+ cons(Y, first(X, Z))
2.90/1.57	
2.90/1.57	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
2.90/1.57	
2.90/1.57	The pumping substitution is [X / s(X), Z / cons(Y, Z)].
2.90/1.57	
2.90/1.57	The result substitution is [ ].
2.90/1.57	
2.90/1.57	
2.90/1.57	
2.90/1.57	
2.90/1.57	----------------------------------------
2.90/1.57	
2.90/1.57	(4)
2.90/1.57	Complex Obligation (BEST)
2.90/1.57	
2.90/1.57	----------------------------------------
2.90/1.57	
2.90/1.57	(5)
2.90/1.57	Obligation:
2.90/1.57	Proved the lower bound n^1 for the following obligation:
2.90/1.57	
2.90/1.57	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
2.90/1.57	
2.90/1.57	
2.90/1.57	The TRS R consists of the following rules:
2.90/1.57	
2.90/1.57	   and(true, X) -> X
2.90/1.57	   and(false, Y) -> false
2.90/1.57	   if(true, X, Y) -> X
2.90/1.57	   if(false, X, Y) -> Y
2.90/1.57	   add(0, X) -> X
2.90/1.57	   add(s(X), Y) -> s(add(X, Y))
2.90/1.57	   first(0, X) -> nil
2.90/1.57	   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
2.90/1.57	   from(X) -> cons(X, from(s(X)))
2.90/1.57	
2.90/1.57	S is empty.
2.90/1.57	Rewrite Strategy: INNERMOST
2.90/1.57	----------------------------------------
2.90/1.57	
2.90/1.57	(6) LowerBoundPropagationProof (FINISHED)
2.90/1.57	Propagated lower bound.
2.90/1.57	----------------------------------------
2.90/1.57	
2.90/1.57	(7)
2.90/1.57	BOUNDS(n^1, INF)
2.90/1.57	
2.90/1.57	----------------------------------------
2.90/1.57	
2.90/1.57	(8)
2.90/1.57	Obligation:
2.90/1.57	Analyzing the following TRS for decreasing loops:
2.90/1.57	
2.90/1.57	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
2.90/1.57	
2.90/1.57	
2.90/1.57	The TRS R consists of the following rules:
2.90/1.57	
2.90/1.57	   and(true, X) -> X
2.90/1.57	   and(false, Y) -> false
2.90/1.57	   if(true, X, Y) -> X
2.90/1.57	   if(false, X, Y) -> Y
2.90/1.57	   add(0, X) -> X
2.90/1.57	   add(s(X), Y) -> s(add(X, Y))
2.90/1.57	   first(0, X) -> nil
2.90/1.57	   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
2.90/1.57	   from(X) -> cons(X, from(s(X)))
2.90/1.57	
2.90/1.57	S is empty.
2.90/1.57	Rewrite Strategy: INNERMOST
2.90/1.57	----------------------------------------
2.90/1.57	
2.90/1.57	(9) InfiniteLowerBoundProof (FINISHED)
2.90/1.57	The following loop proves infinite runtime complexity:
2.90/1.57	
2.90/1.57	The rewrite sequence
2.90/1.57	
2.90/1.57	from(X) ->^+ cons(X, from(s(X)))
2.90/1.57	
2.90/1.57	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
2.90/1.57	
2.90/1.57	The pumping substitution is [ ].
2.90/1.57	
2.90/1.57	The result substitution is [X / s(X)].
2.90/1.57	
2.90/1.57	
2.90/1.57	
2.90/1.57	
2.90/1.57	----------------------------------------
2.90/1.57	
2.90/1.57	(10)
2.90/1.57	BOUNDS(INF, INF)
3.07/1.59	EOF