4.03/1.80	WORST_CASE(Omega(n^1), O(n^1))
4.03/1.81	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
4.03/1.81	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.03/1.81	
4.03/1.81	
4.03/1.81	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1).
4.03/1.81	
4.03/1.81	(0) CpxRelTRS
4.03/1.81	(1) STerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 152 ms]
4.03/1.81	(2) CpxRelTRS
4.03/1.81	(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
4.03/1.81	(4) CpxTRS
4.03/1.81	    (5) CpxTrsMatchBoundsTAProof [FINISHED, 38 ms]
4.03/1.81	    (6) BOUNDS(1, n^1)
4.03/1.81	(7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
4.03/1.81	(8) TRS for Loop Detection
4.03/1.81	    (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
4.03/1.81	    (10) BEST
4.03/1.81	        (11) proven lower bound
4.03/1.81	            (12) LowerBoundPropagationProof [FINISHED, 0 ms]
4.03/1.81	            (13) BOUNDS(n^1, INF)
4.03/1.81	        (14) TRS for Loop Detection
4.03/1.81	
4.03/1.81	
4.03/1.81	----------------------------------------
4.03/1.81	
4.03/1.81	(0)
4.03/1.81	Obligation:
4.03/1.81	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1).
4.03/1.81	
4.03/1.81	
4.03/1.81	The TRS R consists of the following rules:
4.03/1.81	
4.03/1.81	   div2(S(S(x))) -> +(S(0), div2(x))
4.03/1.81	   div2(S(0)) -> 0
4.03/1.81	   div2(0) -> 0
4.03/1.81	
4.03/1.81	The (relative) TRS S consists of the following rules:
4.03/1.81	
4.03/1.81	   +(x, S(0)) -> S(x)
4.03/1.81	   +(S(0), y) -> S(y)
4.03/1.81	
4.03/1.81	Rewrite Strategy: INNERMOST
4.03/1.81	----------------------------------------
4.03/1.81	
4.03/1.81	(1) STerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
4.03/1.81	proved termination of relative rules
4.03/1.81	----------------------------------------
4.03/1.81	
4.03/1.81	(2)
4.03/1.81	Obligation:
4.03/1.81	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1).
4.03/1.81	
4.03/1.81	
4.03/1.81	The TRS R consists of the following rules:
4.03/1.81	
4.03/1.81	   div2(S(S(x))) -> +(S(0), div2(x))
4.03/1.81	   div2(S(0)) -> 0
4.03/1.81	   div2(0) -> 0
4.03/1.81	
4.03/1.81	The (relative) TRS S consists of the following rules:
4.03/1.81	
4.03/1.81	   +(x, S(0)) -> S(x)
4.03/1.81	   +(S(0), y) -> S(y)
4.03/1.81	
4.03/1.81	Rewrite Strategy: INNERMOST
4.03/1.81	----------------------------------------
4.03/1.81	
4.03/1.81	(3) RelTrsToTrsProof (UPPER BOUND(ID))
4.03/1.81	transformed relative TRS to TRS
4.03/1.81	----------------------------------------
4.03/1.81	
4.03/1.81	(4)
4.03/1.81	Obligation:
4.03/1.81	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
4.03/1.81	
4.03/1.81	
4.03/1.81	The TRS R consists of the following rules:
4.03/1.81	
4.03/1.81	   div2(S(S(x))) -> +(S(0), div2(x))
4.03/1.81	   div2(S(0)) -> 0
4.03/1.81	   div2(0) -> 0
4.03/1.81	   +(x, S(0)) -> S(x)
4.03/1.81	   +(S(0), y) -> S(y)
4.03/1.81	
4.03/1.81	S is empty.
4.03/1.81	Rewrite Strategy: INNERMOST
4.03/1.81	----------------------------------------
4.03/1.81	
4.03/1.81	(5) CpxTrsMatchBoundsTAProof (FINISHED)
4.03/1.81	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 2. 
4.03/1.81	
4.03/1.81	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
4.03/1.81	final states : [1, 2]
4.03/1.81	transitions: 
4.03/1.81	S0(0) -> 0
4.03/1.81	00() -> 0
4.03/1.81	div20(0) -> 1
4.03/1.81	+0(0, 0) -> 2
4.03/1.81	01() -> 4
4.03/1.81	S1(4) -> 3
4.03/1.81	div21(0) -> 5
4.03/1.81	+1(3, 5) -> 1
4.03/1.81	01() -> 1
4.03/1.81	S1(0) -> 2
4.03/1.81	+1(3, 5) -> 5
4.03/1.81	01() -> 5
4.03/1.81	S2(5) -> 1
4.03/1.81	S2(5) -> 5
4.03/1.81	S2(3) -> 1
4.03/1.81	S2(3) -> 5
4.03/1.81	
4.03/1.81	----------------------------------------
4.03/1.81	
4.03/1.81	(6)
4.03/1.81	BOUNDS(1, n^1)
4.03/1.81	
4.03/1.81	----------------------------------------
4.03/1.81	
4.03/1.81	(7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
4.03/1.81	Transformed a relative TRS into a decreasing-loop problem.
4.03/1.81	----------------------------------------
4.03/1.81	
4.03/1.81	(8)
4.03/1.81	Obligation:
4.03/1.81	Analyzing the following TRS for decreasing loops:
4.03/1.81	
4.03/1.81	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1).
4.03/1.81	
4.03/1.81	
4.03/1.81	The TRS R consists of the following rules:
4.03/1.81	
4.03/1.81	   div2(S(S(x))) -> +(S(0), div2(x))
4.03/1.81	   div2(S(0)) -> 0
4.03/1.81	   div2(0) -> 0
4.03/1.81	
4.03/1.81	The (relative) TRS S consists of the following rules:
4.03/1.81	
4.03/1.81	   +(x, S(0)) -> S(x)
4.03/1.81	   +(S(0), y) -> S(y)
4.03/1.81	
4.03/1.81	Rewrite Strategy: INNERMOST
4.03/1.81	----------------------------------------
4.03/1.81	
4.03/1.81	(9) DecreasingLoopProof (LOWER BOUND(ID))
4.03/1.81	The following loop(s) give(s) rise to the lower bound Omega(n^1):
4.03/1.81	
4.03/1.81	The rewrite sequence
4.03/1.81	
4.03/1.81	div2(S(S(x))) ->^+ +(S(0), div2(x))
4.03/1.81	
4.03/1.81	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
4.03/1.81	
4.03/1.81	The pumping substitution is [x / S(S(x))].
4.03/1.81	
4.03/1.81	The result substitution is [ ].
4.03/1.81	
4.03/1.81	
4.03/1.81	
4.03/1.81	
4.03/1.81	----------------------------------------
4.03/1.81	
4.03/1.81	(10)
4.03/1.81	Complex Obligation (BEST)
4.03/1.81	
4.03/1.81	----------------------------------------
4.03/1.81	
4.03/1.81	(11)
4.03/1.81	Obligation:
4.03/1.81	Proved the lower bound n^1 for the following obligation:
4.03/1.81	
4.03/1.81	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1).
4.03/1.81	
4.03/1.81	
4.03/1.81	The TRS R consists of the following rules:
4.03/1.81	
4.03/1.81	   div2(S(S(x))) -> +(S(0), div2(x))
4.03/1.81	   div2(S(0)) -> 0
4.03/1.81	   div2(0) -> 0
4.03/1.81	
4.03/1.81	The (relative) TRS S consists of the following rules:
4.03/1.81	
4.03/1.81	   +(x, S(0)) -> S(x)
4.03/1.81	   +(S(0), y) -> S(y)
4.03/1.81	
4.03/1.81	Rewrite Strategy: INNERMOST
4.03/1.81	----------------------------------------
4.03/1.81	
4.03/1.81	(12) LowerBoundPropagationProof (FINISHED)
4.03/1.81	Propagated lower bound.
4.03/1.81	----------------------------------------
4.03/1.81	
4.03/1.81	(13)
4.03/1.81	BOUNDS(n^1, INF)
4.03/1.81	
4.03/1.81	----------------------------------------
4.03/1.81	
4.03/1.81	(14)
4.03/1.81	Obligation:
4.03/1.81	Analyzing the following TRS for decreasing loops:
4.03/1.81	
4.03/1.81	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1).
4.03/1.81	
4.03/1.81	
4.03/1.81	The TRS R consists of the following rules:
4.03/1.81	
4.03/1.81	   div2(S(S(x))) -> +(S(0), div2(x))
4.03/1.81	   div2(S(0)) -> 0
4.03/1.81	   div2(0) -> 0
4.03/1.81	
4.03/1.81	The (relative) TRS S consists of the following rules:
4.03/1.81	
4.03/1.81	   +(x, S(0)) -> S(x)
4.03/1.81	   +(S(0), y) -> S(y)
4.03/1.81	
4.03/1.81	Rewrite Strategy: INNERMOST
4.03/1.83	EOF