3.01/1.59	WORST_CASE(Omega(n^1), O(n^1))
3.01/1.60	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.01/1.60	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.01/1.60	
3.01/1.60	
3.01/1.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.01/1.60	
3.01/1.60	(0) CpxTRS
3.01/1.60	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
3.01/1.60	(2) CpxTRS
3.01/1.60	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 0 ms]
3.01/1.60	    (4) BOUNDS(1, n^1)
3.01/1.60	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.01/1.60	(6) TRS for Loop Detection
3.01/1.60	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.01/1.60	    (8) BEST
3.01/1.60	        (9) proven lower bound
3.01/1.60	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
3.01/1.60	            (11) BOUNDS(n^1, INF)
3.01/1.60	        (12) TRS for Loop Detection
3.01/1.60	
3.01/1.60	
3.01/1.60	----------------------------------------
3.01/1.60	
3.01/1.60	(0)
3.01/1.60	Obligation:
3.01/1.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.01/1.60	
3.01/1.60	
3.01/1.60	The TRS R consists of the following rules:
3.01/1.60	
3.01/1.60	   norm(nil) -> 0
3.01/1.60	   norm(g(x, y)) -> s(norm(x))
3.01/1.60	   f(x, nil) -> g(nil, x)
3.01/1.60	   f(x, g(y, z)) -> g(f(x, y), z)
3.01/1.60	   rem(nil, y) -> nil
3.01/1.60	   rem(g(x, y), 0) -> g(x, y)
3.01/1.60	   rem(g(x, y), s(z)) -> rem(x, z)
3.01/1.60	
3.01/1.60	S is empty.
3.01/1.60	Rewrite Strategy: INNERMOST
3.01/1.60	----------------------------------------
3.01/1.60	
3.01/1.60	(1) RelTrsToTrsProof (UPPER BOUND(ID))
3.01/1.60	transformed relative TRS to TRS
3.01/1.60	----------------------------------------
3.01/1.60	
3.01/1.60	(2)
3.01/1.60	Obligation:
3.01/1.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
3.01/1.60	
3.01/1.60	
3.01/1.60	The TRS R consists of the following rules:
3.01/1.60	
3.01/1.60	   norm(nil) -> 0
3.01/1.60	   norm(g(x, y)) -> s(norm(x))
3.01/1.60	   f(x, nil) -> g(nil, x)
3.01/1.60	   f(x, g(y, z)) -> g(f(x, y), z)
3.01/1.60	   rem(nil, y) -> nil
3.01/1.60	   rem(g(x, y), 0) -> g(x, y)
3.01/1.60	   rem(g(x, y), s(z)) -> rem(x, z)
3.01/1.60	
3.01/1.60	S is empty.
3.01/1.60	Rewrite Strategy: INNERMOST
3.01/1.60	----------------------------------------
3.01/1.60	
3.01/1.60	(3) CpxTrsMatchBoundsTAProof (FINISHED)
3.01/1.60	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. 
3.01/1.60	
3.01/1.60	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
3.01/1.60	final states : [1, 2, 3]
3.01/1.60	transitions: 
3.01/1.60	nil0() -> 0
3.01/1.60	00() -> 0
3.01/1.60	g0(0, 0) -> 0
3.01/1.60	s0(0) -> 0
3.01/1.60	norm0(0) -> 1
3.01/1.60	f0(0, 0) -> 2
3.01/1.60	rem0(0, 0) -> 3
3.01/1.60	01() -> 1
3.01/1.60	norm1(0) -> 4
3.01/1.60	s1(4) -> 1
3.01/1.60	nil1() -> 5
3.01/1.60	g1(5, 0) -> 2
3.01/1.60	f1(0, 0) -> 6
3.01/1.60	g1(6, 0) -> 2
3.01/1.60	nil1() -> 3
3.01/1.60	g1(0, 0) -> 3
3.01/1.60	rem1(0, 0) -> 3
3.01/1.60	01() -> 4
3.01/1.60	s1(4) -> 4
3.01/1.60	g1(5, 0) -> 6
3.01/1.60	g1(6, 0) -> 6
3.01/1.60	
3.01/1.60	----------------------------------------
3.01/1.60	
3.01/1.60	(4)
3.01/1.60	BOUNDS(1, n^1)
3.01/1.60	
3.01/1.60	----------------------------------------
3.01/1.60	
3.01/1.60	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.01/1.60	Transformed a relative TRS into a decreasing-loop problem.
3.01/1.60	----------------------------------------
3.01/1.60	
3.01/1.60	(6)
3.01/1.60	Obligation:
3.01/1.60	Analyzing the following TRS for decreasing loops:
3.01/1.60	
3.01/1.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.01/1.60	
3.01/1.60	
3.01/1.60	The TRS R consists of the following rules:
3.01/1.60	
3.01/1.60	   norm(nil) -> 0
3.01/1.60	   norm(g(x, y)) -> s(norm(x))
3.01/1.60	   f(x, nil) -> g(nil, x)
3.01/1.60	   f(x, g(y, z)) -> g(f(x, y), z)
3.01/1.60	   rem(nil, y) -> nil
3.01/1.60	   rem(g(x, y), 0) -> g(x, y)
3.01/1.60	   rem(g(x, y), s(z)) -> rem(x, z)
3.01/1.60	
3.01/1.60	S is empty.
3.01/1.60	Rewrite Strategy: INNERMOST
3.01/1.60	----------------------------------------
3.01/1.60	
3.01/1.60	(7) DecreasingLoopProof (LOWER BOUND(ID))
3.01/1.60	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.01/1.60	
3.01/1.60	The rewrite sequence
3.01/1.60	
3.01/1.60	rem(g(x, y), s(z)) ->^+ rem(x, z)
3.01/1.60	
3.01/1.60	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
3.01/1.60	
3.01/1.60	The pumping substitution is [x / g(x, y), z / s(z)].
3.01/1.60	
3.01/1.60	The result substitution is [ ].
3.01/1.60	
3.01/1.60	
3.01/1.60	
3.01/1.60	
3.01/1.60	----------------------------------------
3.01/1.60	
3.01/1.60	(8)
3.01/1.60	Complex Obligation (BEST)
3.01/1.60	
3.01/1.60	----------------------------------------
3.01/1.60	
3.01/1.60	(9)
3.01/1.60	Obligation:
3.01/1.60	Proved the lower bound n^1 for the following obligation:
3.01/1.60	
3.01/1.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.01/1.60	
3.01/1.60	
3.01/1.60	The TRS R consists of the following rules:
3.01/1.60	
3.01/1.60	   norm(nil) -> 0
3.01/1.60	   norm(g(x, y)) -> s(norm(x))
3.01/1.60	   f(x, nil) -> g(nil, x)
3.01/1.60	   f(x, g(y, z)) -> g(f(x, y), z)
3.01/1.60	   rem(nil, y) -> nil
3.01/1.60	   rem(g(x, y), 0) -> g(x, y)
3.01/1.60	   rem(g(x, y), s(z)) -> rem(x, z)
3.01/1.60	
3.01/1.60	S is empty.
3.01/1.60	Rewrite Strategy: INNERMOST
3.01/1.60	----------------------------------------
3.01/1.60	
3.01/1.60	(10) LowerBoundPropagationProof (FINISHED)
3.01/1.60	Propagated lower bound.
3.01/1.60	----------------------------------------
3.01/1.60	
3.01/1.60	(11)
3.01/1.60	BOUNDS(n^1, INF)
3.01/1.60	
3.01/1.60	----------------------------------------
3.01/1.60	
3.01/1.60	(12)
3.01/1.60	Obligation:
3.01/1.60	Analyzing the following TRS for decreasing loops:
3.01/1.60	
3.01/1.60	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.01/1.60	
3.01/1.60	
3.01/1.60	The TRS R consists of the following rules:
3.01/1.60	
3.01/1.60	   norm(nil) -> 0
3.01/1.60	   norm(g(x, y)) -> s(norm(x))
3.01/1.60	   f(x, nil) -> g(nil, x)
3.01/1.60	   f(x, g(y, z)) -> g(f(x, y), z)
3.01/1.60	   rem(nil, y) -> nil
3.01/1.60	   rem(g(x, y), 0) -> g(x, y)
3.01/1.60	   rem(g(x, y), s(z)) -> rem(x, z)
3.01/1.60	
3.01/1.60	S is empty.
3.01/1.60	Rewrite Strategy: INNERMOST
3.31/1.63	EOF