3.34/1.62	WORST_CASE(NON_POLY, ?)
3.34/1.63	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.34/1.63	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.34/1.63	
3.34/1.63	
3.34/1.63	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.34/1.63	
3.34/1.63	(0) CpxTRS
3.34/1.63	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.34/1.63	(2) TRS for Loop Detection
3.34/1.63	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.34/1.63	(4) BEST
3.34/1.63	    (5) proven lower bound
3.34/1.63	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.34/1.63	        (7) BOUNDS(n^1, INF)
3.34/1.63	    (8) TRS for Loop Detection
3.34/1.63	        (9) DecreasingLoopProof [FINISHED, 36 ms]
3.34/1.63	        (10) BOUNDS(EXP, INF)
3.34/1.63	
3.34/1.63	
3.34/1.63	----------------------------------------
3.34/1.63	
3.34/1.63	(0)
3.34/1.63	Obligation:
3.34/1.63	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.34/1.63	
3.34/1.63	
3.34/1.63	The TRS R consists of the following rules:
3.34/1.63	
3.34/1.63	   a__eq(0, 0) -> true
3.34/1.63	   a__eq(s(X), s(Y)) -> a__eq(X, Y)
3.34/1.63	   a__eq(X, Y) -> false
3.34/1.63	   a__inf(X) -> cons(X, inf(s(X)))
3.34/1.63	   a__take(0, X) -> nil
3.34/1.63	   a__take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
3.34/1.63	   a__length(nil) -> 0
3.34/1.63	   a__length(cons(X, L)) -> s(length(L))
3.34/1.63	   mark(eq(X1, X2)) -> a__eq(X1, X2)
3.34/1.63	   mark(inf(X)) -> a__inf(mark(X))
3.34/1.63	   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
3.34/1.63	   mark(length(X)) -> a__length(mark(X))
3.34/1.63	   mark(0) -> 0
3.34/1.63	   mark(true) -> true
3.34/1.63	   mark(s(X)) -> s(X)
3.34/1.63	   mark(false) -> false
3.34/1.63	   mark(cons(X1, X2)) -> cons(X1, X2)
3.34/1.63	   mark(nil) -> nil
3.34/1.63	   a__eq(X1, X2) -> eq(X1, X2)
3.34/1.63	   a__inf(X) -> inf(X)
3.34/1.63	   a__take(X1, X2) -> take(X1, X2)
3.34/1.63	   a__length(X) -> length(X)
3.34/1.63	
3.34/1.63	S is empty.
3.34/1.63	Rewrite Strategy: FULL
3.34/1.63	----------------------------------------
3.34/1.63	
3.34/1.63	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.34/1.63	Transformed a relative TRS into a decreasing-loop problem.
3.34/1.63	----------------------------------------
3.34/1.63	
3.34/1.63	(2)
3.34/1.63	Obligation:
3.34/1.63	Analyzing the following TRS for decreasing loops:
3.34/1.63	
3.34/1.63	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.34/1.63	
3.34/1.63	
3.34/1.63	The TRS R consists of the following rules:
3.34/1.63	
3.34/1.63	   a__eq(0, 0) -> true
3.34/1.63	   a__eq(s(X), s(Y)) -> a__eq(X, Y)
3.34/1.63	   a__eq(X, Y) -> false
3.34/1.63	   a__inf(X) -> cons(X, inf(s(X)))
3.34/1.63	   a__take(0, X) -> nil
3.34/1.63	   a__take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
3.34/1.63	   a__length(nil) -> 0
3.34/1.63	   a__length(cons(X, L)) -> s(length(L))
3.34/1.63	   mark(eq(X1, X2)) -> a__eq(X1, X2)
3.34/1.63	   mark(inf(X)) -> a__inf(mark(X))
3.34/1.63	   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
3.34/1.63	   mark(length(X)) -> a__length(mark(X))
3.34/1.63	   mark(0) -> 0
3.34/1.63	   mark(true) -> true
3.34/1.63	   mark(s(X)) -> s(X)
3.34/1.63	   mark(false) -> false
3.34/1.63	   mark(cons(X1, X2)) -> cons(X1, X2)
3.34/1.63	   mark(nil) -> nil
3.34/1.63	   a__eq(X1, X2) -> eq(X1, X2)
3.34/1.63	   a__inf(X) -> inf(X)
3.34/1.63	   a__take(X1, X2) -> take(X1, X2)
3.34/1.63	   a__length(X) -> length(X)
3.34/1.63	
3.34/1.63	S is empty.
3.34/1.63	Rewrite Strategy: FULL
3.34/1.63	----------------------------------------
3.34/1.63	
3.34/1.63	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.34/1.63	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.34/1.63	
3.34/1.63	The rewrite sequence
3.34/1.63	
3.34/1.63	mark(length(X)) ->^+ a__length(mark(X))
3.34/1.63	
3.34/1.63	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.34/1.63	
3.34/1.63	The pumping substitution is [X / length(X)].
3.34/1.63	
3.34/1.63	The result substitution is [ ].
3.34/1.63	
3.34/1.63	
3.34/1.63	
3.34/1.63	
3.34/1.63	----------------------------------------
3.34/1.63	
3.34/1.63	(4)
3.34/1.63	Complex Obligation (BEST)
3.34/1.63	
3.34/1.63	----------------------------------------
3.34/1.63	
3.34/1.63	(5)
3.34/1.63	Obligation:
3.34/1.63	Proved the lower bound n^1 for the following obligation:
3.34/1.63	
3.34/1.63	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.34/1.63	
3.34/1.63	
3.34/1.63	The TRS R consists of the following rules:
3.34/1.63	
3.34/1.63	   a__eq(0, 0) -> true
3.34/1.63	   a__eq(s(X), s(Y)) -> a__eq(X, Y)
3.34/1.63	   a__eq(X, Y) -> false
3.34/1.63	   a__inf(X) -> cons(X, inf(s(X)))
3.34/1.63	   a__take(0, X) -> nil
3.34/1.63	   a__take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
3.34/1.63	   a__length(nil) -> 0
3.34/1.63	   a__length(cons(X, L)) -> s(length(L))
3.34/1.63	   mark(eq(X1, X2)) -> a__eq(X1, X2)
3.34/1.63	   mark(inf(X)) -> a__inf(mark(X))
3.34/1.63	   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
3.34/1.63	   mark(length(X)) -> a__length(mark(X))
3.34/1.63	   mark(0) -> 0
3.34/1.63	   mark(true) -> true
3.34/1.63	   mark(s(X)) -> s(X)
3.34/1.63	   mark(false) -> false
3.34/1.63	   mark(cons(X1, X2)) -> cons(X1, X2)
3.34/1.63	   mark(nil) -> nil
3.34/1.63	   a__eq(X1, X2) -> eq(X1, X2)
3.34/1.63	   a__inf(X) -> inf(X)
3.34/1.63	   a__take(X1, X2) -> take(X1, X2)
3.34/1.63	   a__length(X) -> length(X)
3.34/1.63	
3.34/1.63	S is empty.
3.34/1.63	Rewrite Strategy: FULL
3.34/1.63	----------------------------------------
3.34/1.63	
3.34/1.63	(6) LowerBoundPropagationProof (FINISHED)
3.34/1.63	Propagated lower bound.
3.34/1.63	----------------------------------------
3.34/1.63	
3.34/1.63	(7)
3.34/1.63	BOUNDS(n^1, INF)
3.34/1.63	
3.34/1.63	----------------------------------------
3.34/1.63	
3.34/1.63	(8)
3.34/1.63	Obligation:
3.34/1.63	Analyzing the following TRS for decreasing loops:
3.34/1.63	
3.34/1.63	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.34/1.63	
3.34/1.63	
3.34/1.63	The TRS R consists of the following rules:
3.34/1.63	
3.34/1.63	   a__eq(0, 0) -> true
3.34/1.63	   a__eq(s(X), s(Y)) -> a__eq(X, Y)
3.34/1.63	   a__eq(X, Y) -> false
3.34/1.63	   a__inf(X) -> cons(X, inf(s(X)))
3.34/1.63	   a__take(0, X) -> nil
3.34/1.63	   a__take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
3.34/1.63	   a__length(nil) -> 0
3.34/1.63	   a__length(cons(X, L)) -> s(length(L))
3.34/1.63	   mark(eq(X1, X2)) -> a__eq(X1, X2)
3.34/1.63	   mark(inf(X)) -> a__inf(mark(X))
3.34/1.63	   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
3.34/1.63	   mark(length(X)) -> a__length(mark(X))
3.34/1.63	   mark(0) -> 0
3.34/1.63	   mark(true) -> true
3.34/1.63	   mark(s(X)) -> s(X)
3.34/1.63	   mark(false) -> false
3.34/1.63	   mark(cons(X1, X2)) -> cons(X1, X2)
3.34/1.63	   mark(nil) -> nil
3.34/1.63	   a__eq(X1, X2) -> eq(X1, X2)
3.34/1.63	   a__inf(X) -> inf(X)
3.34/1.63	   a__take(X1, X2) -> take(X1, X2)
3.34/1.63	   a__length(X) -> length(X)
3.34/1.63	
3.34/1.63	S is empty.
3.34/1.63	Rewrite Strategy: FULL
3.34/1.63	----------------------------------------
3.34/1.63	
3.34/1.63	(9) DecreasingLoopProof (FINISHED)
3.34/1.63	The following loop(s) give(s) rise to the lower bound EXP:
3.34/1.63	
3.34/1.63	The rewrite sequence
3.34/1.63	
3.34/1.63	mark(inf(X)) ->^+ cons(mark(X), inf(s(mark(X))))
3.34/1.63	
3.34/1.63	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.34/1.63	
3.34/1.63	The pumping substitution is [X / inf(X)].
3.34/1.63	
3.34/1.63	The result substitution is [ ].
3.34/1.63	
3.34/1.63	
3.34/1.63	
3.34/1.63	The rewrite sequence
3.34/1.63	
3.34/1.63	mark(inf(X)) ->^+ cons(mark(X), inf(s(mark(X))))
3.34/1.63	
3.34/1.63	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
3.34/1.63	
3.34/1.63	The pumping substitution is [X / inf(X)].
3.34/1.63	
3.34/1.63	The result substitution is [ ].
3.34/1.63	
3.34/1.63	
3.34/1.63	
3.34/1.63	
3.34/1.63	----------------------------------------
3.34/1.63	
3.34/1.63	(10)
3.34/1.63	BOUNDS(EXP, INF)
3.59/1.68	EOF