3.14/1.57	WORST_CASE(NON_POLY, ?)
3.14/1.58	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.14/1.58	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.14/1.58	
3.14/1.58	
3.14/1.58	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.14/1.58	
3.14/1.58	(0) CpxTRS
3.14/1.58	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.14/1.58	(2) TRS for Loop Detection
3.14/1.58	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.14/1.58	(4) BEST
3.14/1.58	    (5) proven lower bound
3.14/1.58	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.14/1.58	        (7) BOUNDS(n^1, INF)
3.14/1.58	    (8) TRS for Loop Detection
3.14/1.58	        (9) DecreasingLoopProof [FINISHED, 39 ms]
3.14/1.58	        (10) BOUNDS(EXP, INF)
3.14/1.58	
3.14/1.58	
3.14/1.58	----------------------------------------
3.14/1.58	
3.14/1.58	(0)
3.14/1.58	Obligation:
3.14/1.58	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.14/1.58	
3.14/1.58	
3.14/1.58	The TRS R consists of the following rules:
3.14/1.58	
3.14/1.58	   a__f(X) -> cons(mark(X), f(g(X)))
3.14/1.58	   a__g(0) -> s(0)
3.14/1.58	   a__g(s(X)) -> s(s(a__g(mark(X))))
3.14/1.58	   a__sel(0, cons(X, Y)) -> mark(X)
3.14/1.58	   a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z))
3.14/1.58	   mark(f(X)) -> a__f(mark(X))
3.14/1.58	   mark(g(X)) -> a__g(mark(X))
3.14/1.58	   mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
3.14/1.58	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.14/1.58	   mark(0) -> 0
3.14/1.58	   mark(s(X)) -> s(mark(X))
3.14/1.58	   a__f(X) -> f(X)
3.14/1.58	   a__g(X) -> g(X)
3.14/1.58	   a__sel(X1, X2) -> sel(X1, X2)
3.14/1.58	
3.14/1.58	S is empty.
3.14/1.58	Rewrite Strategy: FULL
3.14/1.58	----------------------------------------
3.14/1.58	
3.14/1.58	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.14/1.58	Transformed a relative TRS into a decreasing-loop problem.
3.14/1.58	----------------------------------------
3.14/1.58	
3.14/1.58	(2)
3.14/1.58	Obligation:
3.14/1.58	Analyzing the following TRS for decreasing loops:
3.14/1.58	
3.14/1.58	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.14/1.58	
3.14/1.58	
3.14/1.58	The TRS R consists of the following rules:
3.14/1.58	
3.14/1.58	   a__f(X) -> cons(mark(X), f(g(X)))
3.14/1.58	   a__g(0) -> s(0)
3.14/1.58	   a__g(s(X)) -> s(s(a__g(mark(X))))
3.14/1.58	   a__sel(0, cons(X, Y)) -> mark(X)
3.14/1.58	   a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z))
3.14/1.58	   mark(f(X)) -> a__f(mark(X))
3.14/1.58	   mark(g(X)) -> a__g(mark(X))
3.14/1.58	   mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
3.14/1.58	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.14/1.58	   mark(0) -> 0
3.14/1.58	   mark(s(X)) -> s(mark(X))
3.14/1.58	   a__f(X) -> f(X)
3.14/1.58	   a__g(X) -> g(X)
3.14/1.58	   a__sel(X1, X2) -> sel(X1, X2)
3.14/1.58	
3.14/1.58	S is empty.
3.14/1.58	Rewrite Strategy: FULL
3.14/1.58	----------------------------------------
3.14/1.58	
3.14/1.58	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.14/1.58	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.14/1.58	
3.14/1.58	The rewrite sequence
3.14/1.58	
3.14/1.58	mark(f(X)) ->^+ a__f(mark(X))
3.14/1.58	
3.14/1.58	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.14/1.58	
3.14/1.58	The pumping substitution is [X / f(X)].
3.14/1.58	
3.14/1.58	The result substitution is [ ].
3.14/1.58	
3.14/1.58	
3.14/1.58	
3.14/1.58	
3.14/1.58	----------------------------------------
3.14/1.58	
3.14/1.58	(4)
3.14/1.58	Complex Obligation (BEST)
3.14/1.58	
3.14/1.58	----------------------------------------
3.14/1.58	
3.14/1.58	(5)
3.14/1.58	Obligation:
3.14/1.58	Proved the lower bound n^1 for the following obligation:
3.14/1.58	
3.14/1.58	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.14/1.58	
3.14/1.58	
3.14/1.58	The TRS R consists of the following rules:
3.14/1.58	
3.14/1.58	   a__f(X) -> cons(mark(X), f(g(X)))
3.14/1.58	   a__g(0) -> s(0)
3.14/1.58	   a__g(s(X)) -> s(s(a__g(mark(X))))
3.14/1.58	   a__sel(0, cons(X, Y)) -> mark(X)
3.14/1.58	   a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z))
3.14/1.58	   mark(f(X)) -> a__f(mark(X))
3.14/1.58	   mark(g(X)) -> a__g(mark(X))
3.14/1.58	   mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
3.14/1.58	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.14/1.58	   mark(0) -> 0
3.14/1.58	   mark(s(X)) -> s(mark(X))
3.14/1.58	   a__f(X) -> f(X)
3.14/1.58	   a__g(X) -> g(X)
3.14/1.58	   a__sel(X1, X2) -> sel(X1, X2)
3.14/1.58	
3.14/1.58	S is empty.
3.14/1.58	Rewrite Strategy: FULL
3.14/1.58	----------------------------------------
3.14/1.58	
3.14/1.58	(6) LowerBoundPropagationProof (FINISHED)
3.14/1.58	Propagated lower bound.
3.14/1.58	----------------------------------------
3.14/1.58	
3.14/1.58	(7)
3.14/1.58	BOUNDS(n^1, INF)
3.14/1.58	
3.14/1.58	----------------------------------------
3.14/1.58	
3.14/1.58	(8)
3.14/1.58	Obligation:
3.14/1.58	Analyzing the following TRS for decreasing loops:
3.14/1.58	
3.14/1.58	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.14/1.58	
3.14/1.58	
3.14/1.58	The TRS R consists of the following rules:
3.14/1.58	
3.14/1.58	   a__f(X) -> cons(mark(X), f(g(X)))
3.14/1.58	   a__g(0) -> s(0)
3.14/1.58	   a__g(s(X)) -> s(s(a__g(mark(X))))
3.14/1.58	   a__sel(0, cons(X, Y)) -> mark(X)
3.14/1.58	   a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z))
3.14/1.58	   mark(f(X)) -> a__f(mark(X))
3.14/1.58	   mark(g(X)) -> a__g(mark(X))
3.14/1.58	   mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
3.14/1.58	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.14/1.58	   mark(0) -> 0
3.14/1.58	   mark(s(X)) -> s(mark(X))
3.14/1.58	   a__f(X) -> f(X)
3.14/1.58	   a__g(X) -> g(X)
3.14/1.58	   a__sel(X1, X2) -> sel(X1, X2)
3.14/1.58	
3.14/1.58	S is empty.
3.14/1.58	Rewrite Strategy: FULL
3.14/1.58	----------------------------------------
3.14/1.58	
3.14/1.58	(9) DecreasingLoopProof (FINISHED)
3.14/1.58	The following loop(s) give(s) rise to the lower bound EXP:
3.14/1.58	
3.14/1.58	The rewrite sequence
3.14/1.58	
3.14/1.58	mark(f(X)) ->^+ cons(mark(mark(X)), f(g(mark(X))))
3.14/1.58	
3.14/1.58	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
3.14/1.58	
3.14/1.58	The pumping substitution is [X / f(X)].
3.14/1.58	
3.14/1.58	The result substitution is [ ].
3.14/1.58	
3.14/1.58	
3.14/1.58	
3.14/1.58	The rewrite sequence
3.14/1.58	
3.14/1.58	mark(f(X)) ->^+ cons(mark(mark(X)), f(g(mark(X))))
3.14/1.58	
3.14/1.58	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
3.14/1.58	
3.14/1.58	The pumping substitution is [X / f(X)].
3.14/1.58	
3.14/1.58	The result substitution is [ ].
3.14/1.58	
3.14/1.58	
3.14/1.58	
3.14/1.58	
3.14/1.58	----------------------------------------
3.14/1.58	
3.14/1.58	(10)
3.14/1.58	BOUNDS(EXP, INF)
3.14/1.61	EOF