3.43/1.68	WORST_CASE(NON_POLY, ?)
3.59/1.70	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.59/1.70	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.59/1.70	
3.59/1.70	
3.59/1.70	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.59/1.70	
3.59/1.70	(0) CpxTRS
3.59/1.70	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.59/1.70	(2) TRS for Loop Detection
3.59/1.70	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.59/1.70	(4) BEST
3.59/1.70	    (5) proven lower bound
3.59/1.70	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.59/1.70	        (7) BOUNDS(n^1, INF)
3.59/1.70	    (8) TRS for Loop Detection
3.59/1.70	        (9) DecreasingLoopProof [FINISHED, 32 ms]
3.59/1.70	        (10) BOUNDS(EXP, INF)
3.59/1.70	
3.59/1.70	
3.59/1.70	----------------------------------------
3.59/1.70	
3.59/1.70	(0)
3.59/1.70	Obligation:
3.59/1.70	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.59/1.70	
3.59/1.70	
3.59/1.70	The TRS R consists of the following rules:
3.59/1.70	
3.59/1.70	   fst(0, Z) -> nil
3.59/1.70	   fst(s(X), cons(Y, Z)) -> cons(Y, n__fst(activate(X), activate(Z)))
3.59/1.70	   from(X) -> cons(X, n__from(n__s(X)))
3.59/1.70	   add(0, X) -> X
3.59/1.70	   add(s(X), Y) -> s(n__add(activate(X), Y))
3.59/1.70	   len(nil) -> 0
3.59/1.70	   len(cons(X, Z)) -> s(n__len(activate(Z)))
3.59/1.70	   fst(X1, X2) -> n__fst(X1, X2)
3.59/1.70	   from(X) -> n__from(X)
3.59/1.70	   s(X) -> n__s(X)
3.59/1.70	   add(X1, X2) -> n__add(X1, X2)
3.59/1.70	   len(X) -> n__len(X)
3.59/1.70	   activate(n__fst(X1, X2)) -> fst(activate(X1), activate(X2))
3.59/1.70	   activate(n__from(X)) -> from(activate(X))
3.59/1.70	   activate(n__s(X)) -> s(X)
3.59/1.70	   activate(n__add(X1, X2)) -> add(activate(X1), activate(X2))
3.59/1.70	   activate(n__len(X)) -> len(activate(X))
3.59/1.70	   activate(X) -> X
3.59/1.70	
3.59/1.70	S is empty.
3.59/1.70	Rewrite Strategy: FULL
3.59/1.70	----------------------------------------
3.59/1.70	
3.59/1.70	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.59/1.70	Transformed a relative TRS into a decreasing-loop problem.
3.59/1.70	----------------------------------------
3.59/1.70	
3.59/1.70	(2)
3.59/1.70	Obligation:
3.59/1.70	Analyzing the following TRS for decreasing loops:
3.59/1.70	
3.59/1.70	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.59/1.70	
3.59/1.70	
3.59/1.70	The TRS R consists of the following rules:
3.59/1.70	
3.59/1.70	   fst(0, Z) -> nil
3.59/1.70	   fst(s(X), cons(Y, Z)) -> cons(Y, n__fst(activate(X), activate(Z)))
3.59/1.70	   from(X) -> cons(X, n__from(n__s(X)))
3.59/1.70	   add(0, X) -> X
3.59/1.70	   add(s(X), Y) -> s(n__add(activate(X), Y))
3.59/1.70	   len(nil) -> 0
3.59/1.70	   len(cons(X, Z)) -> s(n__len(activate(Z)))
3.59/1.70	   fst(X1, X2) -> n__fst(X1, X2)
3.59/1.70	   from(X) -> n__from(X)
3.59/1.70	   s(X) -> n__s(X)
3.59/1.70	   add(X1, X2) -> n__add(X1, X2)
3.59/1.70	   len(X) -> n__len(X)
3.59/1.70	   activate(n__fst(X1, X2)) -> fst(activate(X1), activate(X2))
3.59/1.70	   activate(n__from(X)) -> from(activate(X))
3.59/1.70	   activate(n__s(X)) -> s(X)
3.59/1.70	   activate(n__add(X1, X2)) -> add(activate(X1), activate(X2))
3.59/1.70	   activate(n__len(X)) -> len(activate(X))
3.59/1.70	   activate(X) -> X
3.59/1.70	
3.59/1.70	S is empty.
3.59/1.70	Rewrite Strategy: FULL
3.59/1.70	----------------------------------------
3.59/1.70	
3.59/1.70	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.59/1.70	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.59/1.70	
3.59/1.70	The rewrite sequence
3.59/1.70	
3.59/1.70	activate(n__add(X1, X2)) ->^+ add(activate(X1), activate(X2))
3.59/1.70	
3.59/1.70	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.59/1.70	
3.59/1.70	The pumping substitution is [X1 / n__add(X1, X2)].
3.59/1.70	
3.59/1.70	The result substitution is [ ].
3.59/1.70	
3.59/1.70	
3.59/1.70	
3.59/1.70	
3.59/1.70	----------------------------------------
3.59/1.70	
3.59/1.70	(4)
3.59/1.70	Complex Obligation (BEST)
3.59/1.70	
3.59/1.70	----------------------------------------
3.59/1.70	
3.59/1.70	(5)
3.59/1.70	Obligation:
3.59/1.70	Proved the lower bound n^1 for the following obligation:
3.59/1.70	
3.59/1.70	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.59/1.70	
3.59/1.70	
3.59/1.70	The TRS R consists of the following rules:
3.59/1.70	
3.59/1.70	   fst(0, Z) -> nil
3.59/1.70	   fst(s(X), cons(Y, Z)) -> cons(Y, n__fst(activate(X), activate(Z)))
3.59/1.70	   from(X) -> cons(X, n__from(n__s(X)))
3.59/1.70	   add(0, X) -> X
3.59/1.70	   add(s(X), Y) -> s(n__add(activate(X), Y))
3.59/1.70	   len(nil) -> 0
3.59/1.70	   len(cons(X, Z)) -> s(n__len(activate(Z)))
3.59/1.70	   fst(X1, X2) -> n__fst(X1, X2)
3.59/1.70	   from(X) -> n__from(X)
3.59/1.70	   s(X) -> n__s(X)
3.59/1.70	   add(X1, X2) -> n__add(X1, X2)
3.59/1.70	   len(X) -> n__len(X)
3.59/1.70	   activate(n__fst(X1, X2)) -> fst(activate(X1), activate(X2))
3.59/1.70	   activate(n__from(X)) -> from(activate(X))
3.59/1.70	   activate(n__s(X)) -> s(X)
3.59/1.70	   activate(n__add(X1, X2)) -> add(activate(X1), activate(X2))
3.59/1.70	   activate(n__len(X)) -> len(activate(X))
3.59/1.70	   activate(X) -> X
3.59/1.70	
3.59/1.70	S is empty.
3.59/1.70	Rewrite Strategy: FULL
3.59/1.70	----------------------------------------
3.59/1.70	
3.59/1.70	(6) LowerBoundPropagationProof (FINISHED)
3.59/1.70	Propagated lower bound.
3.59/1.70	----------------------------------------
3.59/1.70	
3.59/1.70	(7)
3.59/1.70	BOUNDS(n^1, INF)
3.59/1.70	
3.59/1.70	----------------------------------------
3.59/1.70	
3.59/1.70	(8)
3.59/1.70	Obligation:
3.59/1.70	Analyzing the following TRS for decreasing loops:
3.59/1.70	
3.59/1.70	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.59/1.70	
3.59/1.70	
3.59/1.70	The TRS R consists of the following rules:
3.59/1.70	
3.59/1.70	   fst(0, Z) -> nil
3.59/1.70	   fst(s(X), cons(Y, Z)) -> cons(Y, n__fst(activate(X), activate(Z)))
3.59/1.70	   from(X) -> cons(X, n__from(n__s(X)))
3.59/1.70	   add(0, X) -> X
3.59/1.70	   add(s(X), Y) -> s(n__add(activate(X), Y))
3.59/1.70	   len(nil) -> 0
3.59/1.70	   len(cons(X, Z)) -> s(n__len(activate(Z)))
3.59/1.70	   fst(X1, X2) -> n__fst(X1, X2)
3.59/1.70	   from(X) -> n__from(X)
3.59/1.70	   s(X) -> n__s(X)
3.59/1.70	   add(X1, X2) -> n__add(X1, X2)
3.59/1.70	   len(X) -> n__len(X)
3.59/1.70	   activate(n__fst(X1, X2)) -> fst(activate(X1), activate(X2))
3.59/1.70	   activate(n__from(X)) -> from(activate(X))
3.59/1.70	   activate(n__s(X)) -> s(X)
3.59/1.70	   activate(n__add(X1, X2)) -> add(activate(X1), activate(X2))
3.59/1.70	   activate(n__len(X)) -> len(activate(X))
3.59/1.70	   activate(X) -> X
3.59/1.70	
3.59/1.70	S is empty.
3.59/1.70	Rewrite Strategy: FULL
3.59/1.70	----------------------------------------
3.59/1.70	
3.59/1.70	(9) DecreasingLoopProof (FINISHED)
3.59/1.70	The following loop(s) give(s) rise to the lower bound EXP:
3.59/1.70	
3.59/1.70	The rewrite sequence
3.59/1.70	
3.59/1.70	activate(n__from(X)) ->^+ cons(activate(X), n__from(n__s(activate(X))))
3.59/1.70	
3.59/1.70	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.59/1.70	
3.59/1.70	The pumping substitution is [X / n__from(X)].
3.59/1.70	
3.59/1.70	The result substitution is [ ].
3.59/1.70	
3.59/1.70	
3.59/1.70	
3.59/1.70	The rewrite sequence
3.59/1.70	
3.59/1.70	activate(n__from(X)) ->^+ cons(activate(X), n__from(n__s(activate(X))))
3.59/1.70	
3.59/1.70	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
3.59/1.70	
3.59/1.70	The pumping substitution is [X / n__from(X)].
3.59/1.70	
3.59/1.70	The result substitution is [ ].
3.59/1.70	
3.59/1.70	
3.59/1.70	
3.59/1.70	
3.59/1.70	----------------------------------------
3.59/1.70	
3.59/1.70	(10)
3.59/1.70	BOUNDS(EXP, INF)
3.59/1.74	EOF