3.11/1.53	WORST_CASE(Omega(n^1), O(n^1))
3.11/1.54	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.11/1.54	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.11/1.54	
3.11/1.54	
3.11/1.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.11/1.54	
3.11/1.54	(0) CpxTRS
3.11/1.54	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
3.11/1.54	(2) CpxTRS
3.11/1.54	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 0 ms]
3.11/1.54	    (4) BOUNDS(1, n^1)
3.11/1.54	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.11/1.54	(6) TRS for Loop Detection
3.11/1.54	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.11/1.54	    (8) BEST
3.11/1.54	        (9) proven lower bound
3.11/1.54	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
3.11/1.54	            (11) BOUNDS(n^1, INF)
3.11/1.54	        (12) TRS for Loop Detection
3.11/1.54	
3.11/1.54	
3.11/1.54	----------------------------------------
3.11/1.54	
3.11/1.54	(0)
3.11/1.54	Obligation:
3.11/1.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.11/1.54	
3.11/1.54	
3.11/1.54	The TRS R consists of the following rules:
3.11/1.54	
3.11/1.54	   and(tt, X) -> activate(X)
3.11/1.54	   plus(N, 0) -> N
3.11/1.54	   plus(N, s(M)) -> s(plus(N, M))
3.11/1.54	   activate(X) -> X
3.11/1.54	
3.11/1.54	S is empty.
3.11/1.54	Rewrite Strategy: INNERMOST
3.11/1.54	----------------------------------------
3.11/1.54	
3.11/1.54	(1) RelTrsToTrsProof (UPPER BOUND(ID))
3.11/1.54	transformed relative TRS to TRS
3.11/1.54	----------------------------------------
3.11/1.54	
3.11/1.54	(2)
3.11/1.54	Obligation:
3.11/1.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
3.11/1.54	
3.11/1.54	
3.11/1.54	The TRS R consists of the following rules:
3.11/1.54	
3.11/1.54	   and(tt, X) -> activate(X)
3.11/1.54	   plus(N, 0) -> N
3.11/1.54	   plus(N, s(M)) -> s(plus(N, M))
3.11/1.54	   activate(X) -> X
3.11/1.54	
3.11/1.54	S is empty.
3.11/1.54	Rewrite Strategy: INNERMOST
3.11/1.54	----------------------------------------
3.11/1.54	
3.11/1.54	(3) CpxTrsMatchBoundsTAProof (FINISHED)
3.11/1.54	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. 
3.11/1.54	
3.11/1.54	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
3.11/1.54	final states : [1, 2, 3]
3.11/1.54	transitions: 
3.11/1.54	tt0() -> 0
3.11/1.54	00() -> 0
3.11/1.54	s0(0) -> 0
3.11/1.54	and0(0, 0) -> 1
3.11/1.54	plus0(0, 0) -> 2
3.11/1.54	activate0(0) -> 3
3.11/1.54	activate1(0) -> 1
3.11/1.54	plus1(0, 0) -> 4
3.11/1.54	s1(4) -> 2
3.11/1.54	s1(4) -> 4
3.11/1.54	0 -> 2
3.11/1.54	0 -> 3
3.11/1.54	0 -> 4
3.11/1.54	0 -> 1
3.11/1.54	
3.11/1.54	----------------------------------------
3.11/1.54	
3.11/1.54	(4)
3.11/1.54	BOUNDS(1, n^1)
3.11/1.54	
3.11/1.54	----------------------------------------
3.11/1.54	
3.11/1.54	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.11/1.54	Transformed a relative TRS into a decreasing-loop problem.
3.11/1.54	----------------------------------------
3.11/1.54	
3.11/1.54	(6)
3.11/1.54	Obligation:
3.11/1.54	Analyzing the following TRS for decreasing loops:
3.11/1.54	
3.11/1.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.11/1.54	
3.11/1.54	
3.11/1.54	The TRS R consists of the following rules:
3.11/1.54	
3.11/1.54	   and(tt, X) -> activate(X)
3.11/1.54	   plus(N, 0) -> N
3.11/1.54	   plus(N, s(M)) -> s(plus(N, M))
3.11/1.54	   activate(X) -> X
3.11/1.54	
3.11/1.54	S is empty.
3.11/1.54	Rewrite Strategy: INNERMOST
3.11/1.54	----------------------------------------
3.11/1.54	
3.11/1.54	(7) DecreasingLoopProof (LOWER BOUND(ID))
3.11/1.54	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.11/1.54	
3.11/1.54	The rewrite sequence
3.11/1.54	
3.11/1.54	plus(N, s(M)) ->^+ s(plus(N, M))
3.11/1.54	
3.11/1.54	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.11/1.54	
3.11/1.54	The pumping substitution is [M / s(M)].
3.11/1.54	
3.11/1.54	The result substitution is [ ].
3.11/1.55	
3.11/1.55	
3.11/1.55	
3.11/1.55	
3.11/1.55	----------------------------------------
3.11/1.55	
3.11/1.55	(8)
3.11/1.55	Complex Obligation (BEST)
3.11/1.55	
3.11/1.55	----------------------------------------
3.11/1.55	
3.11/1.55	(9)
3.11/1.55	Obligation:
3.11/1.55	Proved the lower bound n^1 for the following obligation:
3.11/1.55	
3.11/1.55	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.11/1.55	
3.11/1.55	
3.11/1.55	The TRS R consists of the following rules:
3.11/1.55	
3.11/1.55	   and(tt, X) -> activate(X)
3.11/1.55	   plus(N, 0) -> N
3.11/1.55	   plus(N, s(M)) -> s(plus(N, M))
3.11/1.55	   activate(X) -> X
3.11/1.55	
3.11/1.55	S is empty.
3.11/1.55	Rewrite Strategy: INNERMOST
3.11/1.55	----------------------------------------
3.11/1.55	
3.11/1.55	(10) LowerBoundPropagationProof (FINISHED)
3.11/1.55	Propagated lower bound.
3.11/1.55	----------------------------------------
3.11/1.55	
3.11/1.55	(11)
3.11/1.55	BOUNDS(n^1, INF)
3.11/1.55	
3.11/1.55	----------------------------------------
3.11/1.55	
3.11/1.55	(12)
3.11/1.55	Obligation:
3.11/1.55	Analyzing the following TRS for decreasing loops:
3.11/1.55	
3.11/1.55	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.11/1.55	
3.11/1.55	
3.11/1.55	The TRS R consists of the following rules:
3.11/1.55	
3.11/1.55	   and(tt, X) -> activate(X)
3.11/1.55	   plus(N, 0) -> N
3.11/1.55	   plus(N, s(M)) -> s(plus(N, M))
3.11/1.55	   activate(X) -> X
3.11/1.55	
3.11/1.55	S is empty.
3.11/1.55	Rewrite Strategy: INNERMOST
3.36/1.57	EOF