3.37/1.59	WORST_CASE(Omega(n^1), O(n^1))
3.37/1.60	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.37/1.60	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.37/1.60	
3.37/1.60	
3.37/1.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.37/1.60	
3.37/1.60	(0) CpxTRS
3.37/1.60	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
3.37/1.60	(2) CpxTRS
3.37/1.60	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 36 ms]
3.37/1.60	    (4) BOUNDS(1, n^1)
3.37/1.60	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.37/1.60	(6) TRS for Loop Detection
3.37/1.60	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.37/1.60	    (8) BEST
3.37/1.60	        (9) proven lower bound
3.37/1.60	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
3.37/1.60	            (11) BOUNDS(n^1, INF)
3.37/1.60	        (12) TRS for Loop Detection
3.37/1.60	
3.37/1.60	
3.37/1.60	----------------------------------------
3.37/1.60	
3.37/1.60	(0)
3.37/1.60	Obligation:
3.37/1.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.37/1.60	
3.37/1.60	
3.37/1.60	The TRS R consists of the following rules:
3.37/1.60	
3.37/1.60	   dbl(S(0), S(0)) -> S(S(S(S(0))))
3.37/1.60	   unsafe(S(x)) -> dbl(unsafe(x), 0)
3.37/1.60	   unsafe(0) -> 0
3.37/1.60	   dbl(0, y) -> y
3.37/1.60	
3.37/1.60	S is empty.
3.37/1.60	Rewrite Strategy: INNERMOST
3.37/1.60	----------------------------------------
3.37/1.60	
3.37/1.60	(1) RelTrsToTrsProof (UPPER BOUND(ID))
3.37/1.60	transformed relative TRS to TRS
3.37/1.60	----------------------------------------
3.37/1.60	
3.37/1.60	(2)
3.37/1.60	Obligation:
3.37/1.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
3.37/1.60	
3.37/1.60	
3.37/1.60	The TRS R consists of the following rules:
3.37/1.60	
3.37/1.60	   dbl(S(0), S(0)) -> S(S(S(S(0))))
3.37/1.60	   unsafe(S(x)) -> dbl(unsafe(x), 0)
3.37/1.60	   unsafe(0) -> 0
3.37/1.60	   dbl(0, y) -> y
3.37/1.60	
3.37/1.60	S is empty.
3.37/1.60	Rewrite Strategy: INNERMOST
3.37/1.60	----------------------------------------
3.37/1.60	
3.37/1.60	(3) CpxTrsMatchBoundsTAProof (FINISHED)
3.37/1.60	A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 1. 
3.37/1.60	
3.37/1.60	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
3.37/1.60	final states : [1, 2]
3.37/1.60	transitions: 
3.37/1.60	S0(0) -> 0
3.37/1.60	00() -> 0
3.37/1.60	dbl0(0, 0) -> 1
3.37/1.60	unsafe0(0) -> 2
3.37/1.60	01() -> 5
3.37/1.60	S1(5) -> 4
3.37/1.60	S1(4) -> 3
3.37/1.60	S1(3) -> 3
3.37/1.60	S1(3) -> 1
3.37/1.60	unsafe1(0) -> 6
3.37/1.60	01() -> 7
3.37/1.60	dbl1(6, 7) -> 2
3.37/1.60	01() -> 2
3.37/1.60	dbl1(6, 7) -> 6
3.37/1.60	01() -> 6
3.37/1.60	0 -> 1
3.37/1.60	7 -> 2
3.37/1.60	7 -> 6
3.37/1.60	
3.37/1.60	----------------------------------------
3.37/1.60	
3.37/1.60	(4)
3.37/1.60	BOUNDS(1, n^1)
3.37/1.60	
3.37/1.60	----------------------------------------
3.37/1.60	
3.37/1.60	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.37/1.60	Transformed a relative TRS into a decreasing-loop problem.
3.37/1.60	----------------------------------------
3.37/1.60	
3.37/1.60	(6)
3.37/1.60	Obligation:
3.37/1.60	Analyzing the following TRS for decreasing loops:
3.37/1.60	
3.37/1.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.37/1.60	
3.37/1.60	
3.37/1.60	The TRS R consists of the following rules:
3.37/1.60	
3.37/1.60	   dbl(S(0), S(0)) -> S(S(S(S(0))))
3.37/1.60	   unsafe(S(x)) -> dbl(unsafe(x), 0)
3.37/1.60	   unsafe(0) -> 0
3.37/1.60	   dbl(0, y) -> y
3.37/1.60	
3.37/1.60	S is empty.
3.37/1.60	Rewrite Strategy: INNERMOST
3.37/1.60	----------------------------------------
3.37/1.60	
3.37/1.60	(7) DecreasingLoopProof (LOWER BOUND(ID))
3.37/1.60	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.37/1.60	
3.37/1.60	The rewrite sequence
3.37/1.60	
3.37/1.60	unsafe(S(x)) ->^+ dbl(unsafe(x), 0)
3.37/1.60	
3.37/1.60	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.37/1.60	
3.37/1.60	The pumping substitution is [x / S(x)].
3.37/1.61	
3.37/1.61	The result substitution is [ ].
3.37/1.61	
3.37/1.61	
3.37/1.61	
3.37/1.61	
3.37/1.61	----------------------------------------
3.37/1.61	
3.37/1.61	(8)
3.37/1.61	Complex Obligation (BEST)
3.37/1.61	
3.37/1.61	----------------------------------------
3.37/1.61	
3.37/1.61	(9)
3.37/1.61	Obligation:
3.37/1.61	Proved the lower bound n^1 for the following obligation:
3.37/1.61	
3.37/1.61	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.37/1.61	
3.37/1.61	
3.37/1.61	The TRS R consists of the following rules:
3.37/1.61	
3.37/1.61	   dbl(S(0), S(0)) -> S(S(S(S(0))))
3.37/1.61	   unsafe(S(x)) -> dbl(unsafe(x), 0)
3.37/1.61	   unsafe(0) -> 0
3.37/1.61	   dbl(0, y) -> y
3.37/1.61	
3.37/1.61	S is empty.
3.37/1.61	Rewrite Strategy: INNERMOST
3.37/1.61	----------------------------------------
3.37/1.61	
3.37/1.61	(10) LowerBoundPropagationProof (FINISHED)
3.37/1.61	Propagated lower bound.
3.37/1.61	----------------------------------------
3.37/1.61	
3.37/1.61	(11)
3.37/1.61	BOUNDS(n^1, INF)
3.37/1.61	
3.37/1.61	----------------------------------------
3.37/1.61	
3.37/1.61	(12)
3.37/1.61	Obligation:
3.37/1.61	Analyzing the following TRS for decreasing loops:
3.37/1.61	
3.37/1.61	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.37/1.61	
3.37/1.61	
3.37/1.61	The TRS R consists of the following rules:
3.37/1.61	
3.37/1.61	   dbl(S(0), S(0)) -> S(S(S(S(0))))
3.37/1.61	   unsafe(S(x)) -> dbl(unsafe(x), 0)
3.37/1.61	   unsafe(0) -> 0
3.37/1.61	   dbl(0, y) -> y
3.37/1.61	
3.37/1.61	S is empty.
3.37/1.61	Rewrite Strategy: INNERMOST
3.69/1.64	EOF