3.61/1.72	WORST_CASE(NON_POLY, ?)
3.61/1.72	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.61/1.72	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.61/1.72	
3.61/1.72	
3.61/1.72	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.61/1.72	
3.61/1.72	(0) CpxTRS
3.61/1.72	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.61/1.72	(2) TRS for Loop Detection
3.61/1.72	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.61/1.72	(4) BEST
3.61/1.72	    (5) proven lower bound
3.61/1.72	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.61/1.72	        (7) BOUNDS(n^1, INF)
3.61/1.72	    (8) TRS for Loop Detection
3.61/1.72	        (9) DecreasingLoopProof [FINISHED, 18 ms]
3.61/1.72	        (10) BOUNDS(EXP, INF)
3.61/1.72	
3.61/1.72	
3.61/1.72	----------------------------------------
3.61/1.72	
3.61/1.72	(0)
3.61/1.72	Obligation:
3.61/1.72	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.61/1.72	
3.61/1.72	
3.61/1.72	The TRS R consists of the following rules:
3.61/1.72	
3.61/1.72	   a__fst(0, Z) -> nil
3.61/1.72	   a__fst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z))
3.61/1.72	   a__from(X) -> cons(mark(X), from(s(X)))
3.61/1.72	   a__add(0, X) -> mark(X)
3.61/1.72	   a__add(s(X), Y) -> s(add(X, Y))
3.61/1.72	   a__len(nil) -> 0
3.61/1.72	   a__len(cons(X, Z)) -> s(len(Z))
3.61/1.72	   mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2))
3.61/1.72	   mark(from(X)) -> a__from(mark(X))
3.61/1.72	   mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
3.61/1.72	   mark(len(X)) -> a__len(mark(X))
3.61/1.72	   mark(0) -> 0
3.61/1.72	   mark(s(X)) -> s(X)
3.61/1.72	   mark(nil) -> nil
3.61/1.72	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.61/1.72	   a__fst(X1, X2) -> fst(X1, X2)
3.61/1.72	   a__from(X) -> from(X)
3.61/1.72	   a__add(X1, X2) -> add(X1, X2)
3.61/1.72	   a__len(X) -> len(X)
3.61/1.72	
3.61/1.72	S is empty.
3.61/1.72	Rewrite Strategy: FULL
3.61/1.72	----------------------------------------
3.61/1.72	
3.61/1.72	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.61/1.72	Transformed a relative TRS into a decreasing-loop problem.
3.61/1.72	----------------------------------------
3.61/1.72	
3.61/1.72	(2)
3.61/1.72	Obligation:
3.61/1.72	Analyzing the following TRS for decreasing loops:
3.61/1.72	
3.61/1.72	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.61/1.72	
3.61/1.72	
3.61/1.72	The TRS R consists of the following rules:
3.61/1.72	
3.61/1.72	   a__fst(0, Z) -> nil
3.61/1.72	   a__fst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z))
3.61/1.72	   a__from(X) -> cons(mark(X), from(s(X)))
3.61/1.72	   a__add(0, X) -> mark(X)
3.61/1.72	   a__add(s(X), Y) -> s(add(X, Y))
3.61/1.72	   a__len(nil) -> 0
3.61/1.72	   a__len(cons(X, Z)) -> s(len(Z))
3.61/1.72	   mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2))
3.61/1.72	   mark(from(X)) -> a__from(mark(X))
3.61/1.72	   mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
3.61/1.72	   mark(len(X)) -> a__len(mark(X))
3.61/1.72	   mark(0) -> 0
3.61/1.72	   mark(s(X)) -> s(X)
3.61/1.72	   mark(nil) -> nil
3.61/1.72	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.61/1.72	   a__fst(X1, X2) -> fst(X1, X2)
3.61/1.72	   a__from(X) -> from(X)
3.61/1.72	   a__add(X1, X2) -> add(X1, X2)
3.61/1.72	   a__len(X) -> len(X)
3.61/1.72	
3.61/1.72	S is empty.
3.61/1.72	Rewrite Strategy: FULL
3.61/1.72	----------------------------------------
3.61/1.72	
3.61/1.72	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.61/1.72	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.61/1.72	
3.61/1.72	The rewrite sequence
3.61/1.72	
3.61/1.72	mark(len(X)) ->^+ a__len(mark(X))
3.61/1.72	
3.61/1.72	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.61/1.72	
3.61/1.72	The pumping substitution is [X / len(X)].
3.61/1.72	
3.61/1.72	The result substitution is [ ].
3.61/1.72	
3.61/1.72	
3.61/1.72	
3.61/1.72	
3.61/1.72	----------------------------------------
3.61/1.72	
3.61/1.72	(4)
3.61/1.72	Complex Obligation (BEST)
3.61/1.72	
3.61/1.72	----------------------------------------
3.61/1.72	
3.61/1.72	(5)
3.61/1.72	Obligation:
3.61/1.72	Proved the lower bound n^1 for the following obligation:
3.61/1.72	
3.61/1.72	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.61/1.72	
3.61/1.72	
3.61/1.72	The TRS R consists of the following rules:
3.61/1.72	
3.61/1.72	   a__fst(0, Z) -> nil
3.61/1.72	   a__fst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z))
3.61/1.72	   a__from(X) -> cons(mark(X), from(s(X)))
3.61/1.72	   a__add(0, X) -> mark(X)
3.61/1.72	   a__add(s(X), Y) -> s(add(X, Y))
3.61/1.72	   a__len(nil) -> 0
3.61/1.72	   a__len(cons(X, Z)) -> s(len(Z))
3.61/1.72	   mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2))
3.61/1.72	   mark(from(X)) -> a__from(mark(X))
3.61/1.72	   mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
3.61/1.72	   mark(len(X)) -> a__len(mark(X))
3.61/1.72	   mark(0) -> 0
3.61/1.72	   mark(s(X)) -> s(X)
3.61/1.72	   mark(nil) -> nil
3.61/1.72	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.61/1.72	   a__fst(X1, X2) -> fst(X1, X2)
3.61/1.72	   a__from(X) -> from(X)
3.61/1.72	   a__add(X1, X2) -> add(X1, X2)
3.61/1.72	   a__len(X) -> len(X)
3.61/1.72	
3.61/1.72	S is empty.
3.61/1.72	Rewrite Strategy: FULL
3.61/1.72	----------------------------------------
3.61/1.72	
3.61/1.72	(6) LowerBoundPropagationProof (FINISHED)
3.61/1.72	Propagated lower bound.
3.61/1.72	----------------------------------------
3.61/1.72	
3.61/1.72	(7)
3.61/1.72	BOUNDS(n^1, INF)
3.61/1.72	
3.61/1.72	----------------------------------------
3.61/1.72	
3.61/1.72	(8)
3.61/1.72	Obligation:
3.61/1.72	Analyzing the following TRS for decreasing loops:
3.61/1.72	
3.61/1.72	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.61/1.72	
3.61/1.72	
3.61/1.72	The TRS R consists of the following rules:
3.61/1.72	
3.61/1.72	   a__fst(0, Z) -> nil
3.61/1.72	   a__fst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z))
3.61/1.72	   a__from(X) -> cons(mark(X), from(s(X)))
3.61/1.72	   a__add(0, X) -> mark(X)
3.61/1.72	   a__add(s(X), Y) -> s(add(X, Y))
3.61/1.72	   a__len(nil) -> 0
3.61/1.72	   a__len(cons(X, Z)) -> s(len(Z))
3.61/1.72	   mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2))
3.61/1.72	   mark(from(X)) -> a__from(mark(X))
3.61/1.72	   mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
3.61/1.72	   mark(len(X)) -> a__len(mark(X))
3.61/1.72	   mark(0) -> 0
3.61/1.72	   mark(s(X)) -> s(X)
3.61/1.72	   mark(nil) -> nil
3.61/1.72	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.61/1.72	   a__fst(X1, X2) -> fst(X1, X2)
3.61/1.72	   a__from(X) -> from(X)
3.61/1.72	   a__add(X1, X2) -> add(X1, X2)
3.61/1.72	   a__len(X) -> len(X)
3.61/1.72	
3.61/1.72	S is empty.
3.61/1.72	Rewrite Strategy: FULL
3.61/1.72	----------------------------------------
3.61/1.72	
3.61/1.72	(9) DecreasingLoopProof (FINISHED)
3.61/1.72	The following loop(s) give(s) rise to the lower bound EXP:
3.61/1.72	
3.61/1.72	The rewrite sequence
3.61/1.72	
3.61/1.72	mark(from(X)) ->^+ cons(mark(mark(X)), from(s(mark(X))))
3.61/1.72	
3.61/1.72	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
3.61/1.72	
3.61/1.72	The pumping substitution is [X / from(X)].
3.61/1.72	
3.61/1.72	The result substitution is [ ].
3.61/1.72	
3.61/1.72	
3.61/1.72	
3.61/1.72	The rewrite sequence
3.61/1.72	
3.61/1.72	mark(from(X)) ->^+ cons(mark(mark(X)), from(s(mark(X))))
3.61/1.72	
3.61/1.72	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
3.61/1.72	
3.61/1.72	The pumping substitution is [X / from(X)].
3.61/1.72	
3.61/1.72	The result substitution is [ ].
3.61/1.72	
3.61/1.72	
3.61/1.72	
3.61/1.72	
3.61/1.72	----------------------------------------
3.61/1.72	
3.61/1.72	(10)
3.61/1.72	BOUNDS(EXP, INF)
3.61/1.76	EOF