3.60/1.77	WORST_CASE(Omega(n^1), O(n^1))
3.60/1.77	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.60/1.77	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.60/1.77	
3.60/1.77	
3.60/1.77	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.60/1.77	
3.60/1.77	(0) CpxTRS
3.60/1.77	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
3.60/1.77	(2) CpxTRS
3.60/1.77	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 99 ms]
3.60/1.77	    (4) BOUNDS(1, n^1)
3.60/1.77	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.60/1.77	(6) TRS for Loop Detection
3.60/1.77	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.60/1.77	    (8) BEST
3.60/1.77	        (9) proven lower bound
3.60/1.77	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
3.60/1.77	            (11) BOUNDS(n^1, INF)
3.60/1.77	        (12) TRS for Loop Detection
3.60/1.77	
3.60/1.77	
3.60/1.77	----------------------------------------
3.60/1.77	
3.60/1.77	(0)
3.60/1.77	Obligation:
3.60/1.77	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.60/1.77	
3.60/1.77	
3.60/1.77	The TRS R consists of the following rules:
3.60/1.77	
3.60/1.77	   max(L(x)) -> x
3.60/1.77	   max(N(L(0), L(y))) -> y
3.60/1.77	   max(N(L(s(x)), L(s(y)))) -> s(max(N(L(x), L(y))))
3.60/1.77	   max(N(L(x), N(y, z))) -> max(N(L(x), L(max(N(y, z)))))
3.60/1.77	
3.60/1.77	S is empty.
3.60/1.77	Rewrite Strategy: FULL
3.60/1.77	----------------------------------------
3.60/1.77	
3.60/1.77	(1) RelTrsToTrsProof (UPPER BOUND(ID))
3.60/1.77	transformed relative TRS to TRS
3.60/1.77	----------------------------------------
3.60/1.77	
3.60/1.77	(2)
3.60/1.77	Obligation:
3.60/1.77	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
3.60/1.77	
3.60/1.77	
3.60/1.77	The TRS R consists of the following rules:
3.60/1.77	
3.60/1.77	   max(L(x)) -> x
3.60/1.77	   max(N(L(0), L(y))) -> y
3.60/1.77	   max(N(L(s(x)), L(s(y)))) -> s(max(N(L(x), L(y))))
3.60/1.77	   max(N(L(x), N(y, z))) -> max(N(L(x), L(max(N(y, z)))))
3.60/1.77	
3.60/1.77	S is empty.
3.60/1.77	Rewrite Strategy: FULL
3.60/1.77	----------------------------------------
3.60/1.77	
3.60/1.77	(3) CpxTrsMatchBoundsTAProof (FINISHED)
3.60/1.77	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. 
3.60/1.77	
3.60/1.77	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
3.60/1.77	final states : [1]
3.60/1.77	transitions: 
3.60/1.77	L0(0) -> 0
3.60/1.77	N0(0, 0) -> 0
3.60/1.77	00() -> 0
3.60/1.77	s0(0) -> 0
3.60/1.77	max0(0) -> 1
3.60/1.77	L1(0) -> 4
3.60/1.77	L1(0) -> 5
3.60/1.77	N1(4, 5) -> 3
3.60/1.77	max1(3) -> 2
3.60/1.77	s1(2) -> 1
3.60/1.77	N1(0, 0) -> 8
3.60/1.77	max1(8) -> 7
3.60/1.77	L1(7) -> 6
3.60/1.77	N1(4, 6) -> 3
3.60/1.77	max1(3) -> 1
3.60/1.77	s1(2) -> 7
3.60/1.77	max1(3) -> 7
3.60/1.77	L1(2) -> 5
3.60/1.77	0 -> 1
3.60/1.77	0 -> 7
3.60/1.77	0 -> 2
3.60/1.77	7 -> 1
3.60/1.77	7 -> 2
3.60/1.77	2 -> 1
3.60/1.77	2 -> 7
3.60/1.77	
3.60/1.77	----------------------------------------
3.60/1.77	
3.60/1.77	(4)
3.60/1.77	BOUNDS(1, n^1)
3.60/1.77	
3.60/1.77	----------------------------------------
3.60/1.77	
3.60/1.77	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.60/1.77	Transformed a relative TRS into a decreasing-loop problem.
3.60/1.77	----------------------------------------
3.60/1.77	
3.60/1.77	(6)
3.60/1.77	Obligation:
3.60/1.77	Analyzing the following TRS for decreasing loops:
3.60/1.77	
3.60/1.77	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.60/1.77	
3.60/1.77	
3.60/1.77	The TRS R consists of the following rules:
3.60/1.77	
3.60/1.77	   max(L(x)) -> x
3.60/1.77	   max(N(L(0), L(y))) -> y
3.60/1.77	   max(N(L(s(x)), L(s(y)))) -> s(max(N(L(x), L(y))))
3.60/1.77	   max(N(L(x), N(y, z))) -> max(N(L(x), L(max(N(y, z)))))
3.60/1.77	
3.60/1.77	S is empty.
3.60/1.77	Rewrite Strategy: FULL
3.60/1.77	----------------------------------------
3.60/1.77	
3.60/1.77	(7) DecreasingLoopProof (LOWER BOUND(ID))
3.60/1.77	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.60/1.77	
3.60/1.77	The rewrite sequence
3.60/1.77	
3.60/1.77	max(N(L(s(x)), L(s(y)))) ->^+ s(max(N(L(x), L(y))))
3.60/1.77	
3.60/1.77	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.60/1.77	
3.60/1.77	The pumping substitution is [x / s(x), y / s(y)].
3.60/1.77	
3.60/1.77	The result substitution is [ ].
3.60/1.77	
3.60/1.77	
3.60/1.77	
3.60/1.77	
3.60/1.77	----------------------------------------
3.60/1.77	
3.60/1.77	(8)
3.60/1.77	Complex Obligation (BEST)
3.60/1.77	
3.60/1.77	----------------------------------------
3.60/1.77	
3.60/1.77	(9)
3.60/1.77	Obligation:
3.60/1.77	Proved the lower bound n^1 for the following obligation:
3.60/1.77	
3.60/1.77	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.60/1.77	
3.60/1.77	
3.60/1.77	The TRS R consists of the following rules:
3.60/1.77	
3.60/1.77	   max(L(x)) -> x
3.60/1.77	   max(N(L(0), L(y))) -> y
3.60/1.77	   max(N(L(s(x)), L(s(y)))) -> s(max(N(L(x), L(y))))
3.60/1.77	   max(N(L(x), N(y, z))) -> max(N(L(x), L(max(N(y, z)))))
3.60/1.77	
3.60/1.77	S is empty.
3.60/1.77	Rewrite Strategy: FULL
3.60/1.77	----------------------------------------
3.60/1.77	
3.60/1.77	(10) LowerBoundPropagationProof (FINISHED)
3.60/1.77	Propagated lower bound.
3.60/1.77	----------------------------------------
3.60/1.77	
3.60/1.77	(11)
3.60/1.77	BOUNDS(n^1, INF)
3.60/1.77	
3.60/1.77	----------------------------------------
3.60/1.77	
3.60/1.77	(12)
3.60/1.77	Obligation:
3.60/1.77	Analyzing the following TRS for decreasing loops:
3.60/1.77	
3.60/1.77	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.60/1.77	
3.60/1.77	
3.60/1.77	The TRS R consists of the following rules:
3.60/1.77	
3.60/1.77	   max(L(x)) -> x
3.60/1.77	   max(N(L(0), L(y))) -> y
3.60/1.77	   max(N(L(s(x)), L(s(y)))) -> s(max(N(L(x), L(y))))
3.60/1.77	   max(N(L(x), N(y, z))) -> max(N(L(x), L(max(N(y, z)))))
3.60/1.77	
3.60/1.77	S is empty.
3.60/1.77	Rewrite Strategy: FULL
3.92/1.81	EOF