3.46/1.77	WORST_CASE(Omega(n^1), O(n^1))
3.46/1.79	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.46/1.79	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.46/1.79	
3.46/1.79	
3.46/1.79	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.46/1.79	
3.46/1.79	(0) CpxTRS
3.46/1.79	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
3.46/1.79	(2) CpxTRS
3.46/1.79	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 47 ms]
3.46/1.79	    (4) BOUNDS(1, n^1)
3.46/1.79	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.46/1.79	(6) TRS for Loop Detection
3.46/1.79	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.46/1.79	    (8) BEST
3.46/1.79	        (9) proven lower bound
3.46/1.79	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
3.46/1.79	            (11) BOUNDS(n^1, INF)
3.46/1.79	        (12) TRS for Loop Detection
3.46/1.79	
3.46/1.79	
3.46/1.79	----------------------------------------
3.46/1.79	
3.46/1.79	(0)
3.46/1.79	Obligation:
3.46/1.79	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.46/1.79	
3.46/1.79	
3.46/1.79	The TRS R consists of the following rules:
3.46/1.79	
3.46/1.79	   f(a, empty) -> g(a, empty)
3.46/1.79	   f(a, cons(x, k)) -> f(cons(x, a), k)
3.46/1.79	   g(empty, d) -> d
3.46/1.79	   g(cons(x, k), d) -> g(k, cons(x, d))
3.46/1.79	
3.46/1.79	S is empty.
3.46/1.79	Rewrite Strategy: FULL
3.46/1.79	----------------------------------------
3.46/1.79	
3.46/1.79	(1) RelTrsToTrsProof (UPPER BOUND(ID))
3.46/1.79	transformed relative TRS to TRS
3.46/1.79	----------------------------------------
3.46/1.79	
3.46/1.79	(2)
3.46/1.79	Obligation:
3.46/1.79	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
3.46/1.79	
3.46/1.79	
3.46/1.79	The TRS R consists of the following rules:
3.46/1.79	
3.46/1.79	   f(a, empty) -> g(a, empty)
3.46/1.79	   f(a, cons(x, k)) -> f(cons(x, a), k)
3.46/1.79	   g(empty, d) -> d
3.46/1.79	   g(cons(x, k), d) -> g(k, cons(x, d))
3.46/1.79	
3.46/1.79	S is empty.
3.46/1.79	Rewrite Strategy: FULL
3.46/1.79	----------------------------------------
3.46/1.79	
3.46/1.79	(3) CpxTrsMatchBoundsTAProof (FINISHED)
3.46/1.79	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. 
3.46/1.79	
3.46/1.79	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
3.46/1.79	final states : [1, 2]
3.46/1.79	transitions: 
3.46/1.79	empty0() -> 0
3.46/1.79	cons0(0, 0) -> 0
3.46/1.79	f0(0, 0) -> 1
3.46/1.79	g0(0, 0) -> 2
3.46/1.79	empty1() -> 3
3.46/1.79	g1(0, 3) -> 1
3.46/1.79	cons1(0, 0) -> 4
3.46/1.80	f1(4, 0) -> 1
3.46/1.80	cons1(0, 0) -> 5
3.46/1.80	g1(0, 5) -> 2
3.46/1.80	g1(4, 3) -> 1
3.46/1.80	cons1(0, 4) -> 4
3.46/1.80	cons1(0, 3) -> 5
3.46/1.80	g1(0, 5) -> 1
3.46/1.80	cons1(0, 5) -> 5
3.46/1.80	cons2(0, 3) -> 6
3.46/1.80	g2(0, 6) -> 1
3.46/1.80	g2(4, 6) -> 1
3.46/1.80	cons1(0, 6) -> 5
3.46/1.80	cons2(0, 6) -> 6
3.46/1.80	0 -> 2
3.46/1.80	3 -> 1
3.46/1.80	5 -> 2
3.46/1.80	5 -> 1
3.46/1.80	6 -> 1
3.46/1.80	
3.46/1.80	----------------------------------------
3.46/1.80	
3.46/1.80	(4)
3.46/1.80	BOUNDS(1, n^1)
3.46/1.80	
3.46/1.80	----------------------------------------
3.46/1.80	
3.46/1.80	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.46/1.80	Transformed a relative TRS into a decreasing-loop problem.
3.46/1.80	----------------------------------------
3.46/1.80	
3.46/1.80	(6)
3.46/1.80	Obligation:
3.46/1.80	Analyzing the following TRS for decreasing loops:
3.46/1.80	
3.46/1.80	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.46/1.80	
3.46/1.80	
3.46/1.80	The TRS R consists of the following rules:
3.46/1.80	
3.46/1.80	   f(a, empty) -> g(a, empty)
3.46/1.80	   f(a, cons(x, k)) -> f(cons(x, a), k)
3.46/1.80	   g(empty, d) -> d
3.46/1.80	   g(cons(x, k), d) -> g(k, cons(x, d))
3.46/1.80	
3.46/1.80	S is empty.
3.46/1.80	Rewrite Strategy: FULL
3.46/1.80	----------------------------------------
3.46/1.80	
3.46/1.80	(7) DecreasingLoopProof (LOWER BOUND(ID))
3.46/1.80	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.46/1.80	
3.46/1.80	The rewrite sequence
3.46/1.80	
3.46/1.80	f(a, cons(x, k)) ->^+ f(cons(x, a), k)
3.46/1.80	
3.46/1.80	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
3.46/1.80	
3.46/1.80	The pumping substitution is [k / cons(x, k)].
3.46/1.80	
3.46/1.80	The result substitution is [a / cons(x, a)].
3.46/1.80	
3.46/1.80	
3.46/1.80	
3.46/1.80	
3.46/1.80	----------------------------------------
3.46/1.80	
3.46/1.80	(8)
3.46/1.80	Complex Obligation (BEST)
3.46/1.80	
3.46/1.80	----------------------------------------
3.46/1.80	
3.46/1.80	(9)
3.46/1.80	Obligation:
3.46/1.80	Proved the lower bound n^1 for the following obligation:
3.46/1.80	
3.46/1.80	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.46/1.80	
3.46/1.80	
3.46/1.80	The TRS R consists of the following rules:
3.46/1.80	
3.46/1.80	   f(a, empty) -> g(a, empty)
3.46/1.80	   f(a, cons(x, k)) -> f(cons(x, a), k)
3.46/1.80	   g(empty, d) -> d
3.46/1.80	   g(cons(x, k), d) -> g(k, cons(x, d))
3.46/1.80	
3.46/1.80	S is empty.
3.46/1.80	Rewrite Strategy: FULL
3.46/1.80	----------------------------------------
3.46/1.80	
3.46/1.80	(10) LowerBoundPropagationProof (FINISHED)
3.46/1.80	Propagated lower bound.
3.46/1.80	----------------------------------------
3.46/1.80	
3.46/1.80	(11)
3.46/1.80	BOUNDS(n^1, INF)
3.46/1.80	
3.46/1.80	----------------------------------------
3.46/1.80	
3.46/1.80	(12)
3.46/1.80	Obligation:
3.46/1.80	Analyzing the following TRS for decreasing loops:
3.46/1.80	
3.46/1.80	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.46/1.80	
3.46/1.80	
3.46/1.80	The TRS R consists of the following rules:
3.46/1.80	
3.46/1.80	   f(a, empty) -> g(a, empty)
3.46/1.80	   f(a, cons(x, k)) -> f(cons(x, a), k)
3.46/1.80	   g(empty, d) -> d
3.46/1.80	   g(cons(x, k), d) -> g(k, cons(x, d))
3.46/1.80	
3.46/1.80	S is empty.
3.46/1.80	Rewrite Strategy: FULL
3.59/1.83	EOF