3.14/1.54	WORST_CASE(NON_POLY, ?)
3.14/1.55	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.14/1.55	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.14/1.55	
3.14/1.55	
3.14/1.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.14/1.55	
3.14/1.55	(0) CpxTRS
3.14/1.55	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.14/1.55	(2) TRS for Loop Detection
3.14/1.55	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.14/1.55	(4) BEST
3.14/1.55	    (5) proven lower bound
3.14/1.55	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.14/1.55	        (7) BOUNDS(n^1, INF)
3.14/1.55	    (8) TRS for Loop Detection
3.14/1.55	        (9) InfiniteLowerBoundProof [FINISHED, 0 ms]
3.14/1.55	        (10) BOUNDS(INF, INF)
3.14/1.55	
3.14/1.55	
3.14/1.55	----------------------------------------
3.14/1.55	
3.14/1.55	(0)
3.14/1.55	Obligation:
3.14/1.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.14/1.55	
3.14/1.55	
3.14/1.55	The TRS R consists of the following rules:
3.14/1.55	
3.14/1.55	   and(true, X) -> X
3.14/1.55	   and(false, Y) -> false
3.14/1.55	   if(true, X, Y) -> X
3.14/1.55	   if(false, X, Y) -> Y
3.14/1.55	   add(0, X) -> X
3.14/1.55	   add(s(X), Y) -> s(add(X, Y))
3.14/1.55	   first(0, X) -> nil
3.14/1.55	   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
3.14/1.55	   from(X) -> cons(X, from(s(X)))
3.14/1.55	
3.14/1.55	S is empty.
3.14/1.55	Rewrite Strategy: FULL
3.14/1.55	----------------------------------------
3.14/1.55	
3.14/1.55	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.14/1.55	Transformed a relative TRS into a decreasing-loop problem.
3.14/1.55	----------------------------------------
3.14/1.55	
3.14/1.55	(2)
3.14/1.55	Obligation:
3.14/1.55	Analyzing the following TRS for decreasing loops:
3.14/1.55	
3.14/1.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.14/1.55	
3.14/1.55	
3.14/1.55	The TRS R consists of the following rules:
3.14/1.55	
3.14/1.55	   and(true, X) -> X
3.14/1.55	   and(false, Y) -> false
3.14/1.55	   if(true, X, Y) -> X
3.14/1.55	   if(false, X, Y) -> Y
3.14/1.55	   add(0, X) -> X
3.14/1.55	   add(s(X), Y) -> s(add(X, Y))
3.14/1.55	   first(0, X) -> nil
3.14/1.55	   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
3.14/1.55	   from(X) -> cons(X, from(s(X)))
3.14/1.55	
3.14/1.55	S is empty.
3.14/1.55	Rewrite Strategy: FULL
3.14/1.55	----------------------------------------
3.14/1.55	
3.14/1.55	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.14/1.55	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.14/1.55	
3.14/1.55	The rewrite sequence
3.14/1.55	
3.14/1.55	first(s(X), cons(Y, Z)) ->^+ cons(Y, first(X, Z))
3.14/1.55	
3.14/1.55	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
3.14/1.55	
3.14/1.55	The pumping substitution is [X / s(X), Z / cons(Y, Z)].
3.14/1.55	
3.14/1.55	The result substitution is [ ].
3.14/1.55	
3.14/1.55	
3.14/1.55	
3.14/1.55	
3.14/1.55	----------------------------------------
3.14/1.55	
3.14/1.55	(4)
3.14/1.55	Complex Obligation (BEST)
3.14/1.55	
3.14/1.55	----------------------------------------
3.14/1.55	
3.14/1.55	(5)
3.14/1.55	Obligation:
3.14/1.55	Proved the lower bound n^1 for the following obligation:
3.14/1.55	
3.14/1.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.14/1.55	
3.14/1.55	
3.14/1.55	The TRS R consists of the following rules:
3.14/1.55	
3.14/1.55	   and(true, X) -> X
3.14/1.55	   and(false, Y) -> false
3.14/1.55	   if(true, X, Y) -> X
3.14/1.55	   if(false, X, Y) -> Y
3.14/1.55	   add(0, X) -> X
3.14/1.55	   add(s(X), Y) -> s(add(X, Y))
3.14/1.55	   first(0, X) -> nil
3.14/1.55	   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
3.14/1.55	   from(X) -> cons(X, from(s(X)))
3.14/1.55	
3.14/1.55	S is empty.
3.14/1.55	Rewrite Strategy: FULL
3.14/1.55	----------------------------------------
3.14/1.55	
3.14/1.55	(6) LowerBoundPropagationProof (FINISHED)
3.14/1.55	Propagated lower bound.
3.14/1.55	----------------------------------------
3.14/1.55	
3.14/1.55	(7)
3.14/1.55	BOUNDS(n^1, INF)
3.14/1.55	
3.14/1.55	----------------------------------------
3.14/1.55	
3.14/1.55	(8)
3.14/1.55	Obligation:
3.14/1.55	Analyzing the following TRS for decreasing loops:
3.14/1.55	
3.14/1.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.14/1.55	
3.14/1.55	
3.14/1.55	The TRS R consists of the following rules:
3.14/1.55	
3.14/1.55	   and(true, X) -> X
3.14/1.55	   and(false, Y) -> false
3.14/1.55	   if(true, X, Y) -> X
3.14/1.55	   if(false, X, Y) -> Y
3.14/1.55	   add(0, X) -> X
3.14/1.55	   add(s(X), Y) -> s(add(X, Y))
3.14/1.55	   first(0, X) -> nil
3.14/1.55	   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
3.14/1.55	   from(X) -> cons(X, from(s(X)))
3.14/1.55	
3.14/1.55	S is empty.
3.14/1.55	Rewrite Strategy: FULL
3.14/1.55	----------------------------------------
3.14/1.55	
3.14/1.55	(9) InfiniteLowerBoundProof (FINISHED)
3.14/1.55	The following loop proves infinite runtime complexity:
3.14/1.55	
3.14/1.55	The rewrite sequence
3.14/1.55	
3.14/1.55	from(X) ->^+ cons(X, from(s(X)))
3.14/1.55	
3.14/1.55	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
3.14/1.55	
3.14/1.55	The pumping substitution is [ ].
3.14/1.55	
3.14/1.55	The result substitution is [X / s(X)].
3.14/1.55	
3.14/1.55	
3.14/1.55	
3.14/1.55	
3.14/1.55	----------------------------------------
3.14/1.55	
3.14/1.55	(10)
3.14/1.55	BOUNDS(INF, INF)
3.31/1.58	EOF