3.19/1.57	WORST_CASE(Omega(n^1), O(n^1))
3.19/1.57	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.19/1.57	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.19/1.57	
3.19/1.57	
3.19/1.57	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.19/1.57	
3.19/1.57	(0) CpxTRS
3.19/1.57	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
3.19/1.57	(2) CpxTRS
3.19/1.57	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 0 ms]
3.19/1.57	    (4) BOUNDS(1, n^1)
3.19/1.57	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.19/1.57	(6) TRS for Loop Detection
3.19/1.57	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.19/1.57	    (8) BEST
3.19/1.57	        (9) proven lower bound
3.19/1.57	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
3.19/1.57	            (11) BOUNDS(n^1, INF)
3.19/1.57	        (12) TRS for Loop Detection
3.19/1.57	
3.19/1.57	
3.19/1.57	----------------------------------------
3.19/1.57	
3.19/1.57	(0)
3.19/1.57	Obligation:
3.19/1.57	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.19/1.57	
3.19/1.57	
3.19/1.57	The TRS R consists of the following rules:
3.19/1.57	
3.19/1.57	   p(m, n, s(r)) -> p(m, r, n)
3.19/1.57	   p(m, s(n), 0) -> p(0, n, m)
3.19/1.57	   p(m, 0, 0) -> m
3.19/1.57	
3.19/1.57	S is empty.
3.19/1.57	Rewrite Strategy: FULL
3.19/1.57	----------------------------------------
3.19/1.57	
3.19/1.57	(1) RelTrsToTrsProof (UPPER BOUND(ID))
3.19/1.57	transformed relative TRS to TRS
3.19/1.57	----------------------------------------
3.19/1.57	
3.19/1.57	(2)
3.19/1.57	Obligation:
3.19/1.57	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
3.19/1.57	
3.19/1.57	
3.19/1.57	The TRS R consists of the following rules:
3.19/1.57	
3.19/1.57	   p(m, n, s(r)) -> p(m, r, n)
3.19/1.57	   p(m, s(n), 0) -> p(0, n, m)
3.19/1.57	   p(m, 0, 0) -> m
3.19/1.57	
3.19/1.57	S is empty.
3.19/1.57	Rewrite Strategy: FULL
3.19/1.57	----------------------------------------
3.19/1.57	
3.19/1.57	(3) CpxTrsMatchBoundsTAProof (FINISHED)
3.19/1.57	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. 
3.19/1.57	
3.19/1.57	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
3.19/1.57	final states : [1]
3.19/1.57	transitions: 
3.19/1.57	s0(0) -> 0
3.19/1.57	00() -> 0
3.19/1.57	p0(0, 0, 0) -> 1
3.19/1.57	p1(0, 0, 0) -> 1
3.19/1.57	01() -> 2
3.19/1.57	p1(2, 0, 0) -> 1
3.19/1.57	p1(2, 0, 2) -> 1
3.19/1.57	0 -> 1
3.19/1.57	2 -> 1
3.19/1.57	
3.19/1.57	----------------------------------------
3.19/1.57	
3.19/1.57	(4)
3.19/1.57	BOUNDS(1, n^1)
3.19/1.57	
3.19/1.57	----------------------------------------
3.19/1.57	
3.19/1.57	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.19/1.57	Transformed a relative TRS into a decreasing-loop problem.
3.19/1.57	----------------------------------------
3.19/1.57	
3.19/1.57	(6)
3.19/1.57	Obligation:
3.19/1.57	Analyzing the following TRS for decreasing loops:
3.19/1.57	
3.19/1.57	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.19/1.57	
3.19/1.57	
3.19/1.57	The TRS R consists of the following rules:
3.19/1.57	
3.19/1.57	   p(m, n, s(r)) -> p(m, r, n)
3.19/1.57	   p(m, s(n), 0) -> p(0, n, m)
3.19/1.57	   p(m, 0, 0) -> m
3.19/1.57	
3.19/1.57	S is empty.
3.19/1.57	Rewrite Strategy: FULL
3.19/1.57	----------------------------------------
3.19/1.57	
3.19/1.57	(7) DecreasingLoopProof (LOWER BOUND(ID))
3.19/1.57	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.19/1.57	
3.19/1.57	The rewrite sequence
3.19/1.57	
3.19/1.57	p(m, s(r3_0), s(r)) ->^+ p(m, r3_0, r)
3.19/1.57	
3.19/1.57	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
3.19/1.57	
3.19/1.57	The pumping substitution is [r3_0 / s(r3_0), r / s(r)].
3.19/1.57	
3.19/1.57	The result substitution is [ ].
3.19/1.57	
3.19/1.57	
3.19/1.57	
3.19/1.57	
3.19/1.57	----------------------------------------
3.19/1.57	
3.19/1.57	(8)
3.19/1.57	Complex Obligation (BEST)
3.19/1.57	
3.19/1.57	----------------------------------------
3.19/1.57	
3.19/1.57	(9)
3.19/1.57	Obligation:
3.19/1.57	Proved the lower bound n^1 for the following obligation:
3.19/1.57	
3.19/1.57	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.19/1.57	
3.19/1.57	
3.19/1.57	The TRS R consists of the following rules:
3.19/1.57	
3.19/1.57	   p(m, n, s(r)) -> p(m, r, n)
3.19/1.57	   p(m, s(n), 0) -> p(0, n, m)
3.19/1.57	   p(m, 0, 0) -> m
3.19/1.57	
3.19/1.57	S is empty.
3.19/1.57	Rewrite Strategy: FULL
3.19/1.57	----------------------------------------
3.19/1.57	
3.19/1.57	(10) LowerBoundPropagationProof (FINISHED)
3.19/1.57	Propagated lower bound.
3.19/1.57	----------------------------------------
3.19/1.57	
3.19/1.57	(11)
3.19/1.57	BOUNDS(n^1, INF)
3.19/1.57	
3.19/1.57	----------------------------------------
3.19/1.57	
3.19/1.57	(12)
3.19/1.57	Obligation:
3.19/1.57	Analyzing the following TRS for decreasing loops:
3.19/1.57	
3.19/1.57	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.19/1.57	
3.19/1.57	
3.19/1.57	The TRS R consists of the following rules:
3.19/1.57	
3.19/1.57	   p(m, n, s(r)) -> p(m, r, n)
3.19/1.57	   p(m, s(n), 0) -> p(0, n, m)
3.19/1.57	   p(m, 0, 0) -> m
3.19/1.57	
3.19/1.57	S is empty.
3.19/1.57	Rewrite Strategy: FULL
3.39/1.61	EOF